Introduction
Let Δ ⊂ C \Delta\subset\mathbb{C} Δ ⊂ C 1 1 1 1 1 1 [19 §§1–3] . Their principal theorem identifies each connected component of the resulting coarse moduli space with an affine space when the prescribed diagonal monodromy eigenvalues are pairwise distinct and different from 1 1 1 [19 Theorem 1.1 and Theorem 4.1] .
The underlying regular-singular connection belongs to the classical Levelt–Deligne theory [20 , 9 , 8 , 26 ] ; the prescribed lattice, however, is additional structure and need not be a Deligne or Levelt lattice. Closely related enriched Riemann–Hilbert problems retain parabolic or parahoric data [5 , 13 , 14 ] , while quotient-stack descriptions of logarithmic connections arise in the Lie-groupoid formalism of Bischoff [2 , 3 ] . The moduli problem considered here is more rigid: the full ordered frame is retained, and morphisms are required to act on the connection and on that frame simultaneously. This rigidification is the source of the spectrum-independent triangularity proved below. As usual, we use the horizontal convention
d Y + A ( z ) Y d z = 0 , dY + A(z)Y\,dz = 0, d Y + A ( z ) Y d z = 0 , so that a scalar residue μ \mu μ e − 2 π i μ e^{-2\pi i\mu} e − 2 π i μ
Log − ( c ) : = { μ ∈ C ∣ e − 2 π i μ = c } . \operatorname{Log}_{-}(c) := \{\mu\in\mathbb{C} \mid e^{-2\pi i\mu}=c\}. Log − ( c ) := { μ ∈ C ∣ e − 2 π i μ = c } . For a groupoid G \mathcal{G} G ∣ G ∣ |\mathcal{G}| ∣ G ∣ X X X X d i s c X_{\mathrm{disc}} X disc X X X
Our first theorem removes both spectral restrictions and upgrades the orbit classification to an equivalence of groupoids.
Theorem 1.1. Let r ≥ 2 r \ge2 r ≥ 2 c = ( c 1 , … , c r ) ∈ ( C × ) r \mathbf{c}=(c_1,\ldots,c_r)\in(\mathbb{C}^{\times})^r c = ( c 1 , … , c r ) ∈ ( C × ) r
res Δ : N S Q C f r ⟶ ( ∏ i = 1 r Log − ( c i ) ) d i s c , \operatorname{res}_{\Delta}\colon\mathrm{NSQ}_{\mathbb{C}}^{\mathrm{fr}}\longrightarrow\left(\prod_{i=1}^{r}\operatorname{Log}_{-}(c_i)\right)_{\mathrm{disc}}, res Δ : NSQ C fr ⟶ ( i = 1 ∏ r Log − ( c i ) ) disc , which is essentially surjective. For every ordered residue vector μ = ( μ 1 , … , μ r ) ∈ ∏ i Log − ( c i ) \boldsymbol{\mu}=(\mu_1,\ldots,\mu_r)\in\prod_i\operatorname{Log}_{-}(c_i) μ = ( μ 1 , … , μ r ) ∈ ∏ i Log − ( c i ) res Δ − 1 ( μ ) \operatorname{res}_{\Delta}^{-1}(\boldsymbol{\mu}) res Δ − 1 ( μ )
n ⊕ n ≅ A C r ( r − 1 ) . \mathfrak{n}\oplus\mathfrak{n}\cong\mathbb{A}_{\mathbb{C}}^{r(r-1)}. n ⊕ n ≅ A C r ( r − 1 ) . Every object of the fibre is isomorphic to a unique connection
∇ α , β = d + [ M μ z + ∑ 1 ≤ j < i ≤ r ( α j i z − 2 + β j i ρ λ j i ( z ) ) E j i ] d z , (1) \nabla_{\alpha,\beta}=d+\left[\frac{M_\mu}{z}+\sum_{1\le j<i\le r}\left(\alpha_{ji}z^{-2}+\beta_{ji}\rho_{\lambda_{ji}}(z)\right)E_{ji}\right]dz, \tag*{(1)} ∇ α , β = d + [ z M μ + 1 ≤ j < i ≤ r ∑ ( α j i z − 2 + β j i ρ λ j i ( z ) ) E j i ] d z , (1) where M μ = diag ( μ 1 , … , μ r ) M_\mu=\operatorname{diag}(\mu_1,\ldots,\mu_r) M μ = diag ( μ 1 , … , μ r ) λ j i = μ i − μ j \lambda_{ji}=\mu_i-\mu_j λ j i = μ i − μ j
ρ λ ( z ) = { z λ − 1 , λ ∈ Z ≥ 0 , z − 1 , λ ∉ Z ≥ 0 . \rho_\lambda(z)=
\begin{cases}
z^{\lambda-1}, & \lambda\in\mathbb{Z}_{\ge0},\\
z^{-1}, & \lambda\notin\mathbb{Z}_{\ge0}.
\end{cases} ρ λ ( z ) = { z λ − 1 , z − 1 , λ ∈ Z ≥ 0 , λ ∈ / Z ≥ 0 . The coordinates ( α , β ) (\alpha,\beta) ( α , β )
N S Q C c f r ≅ ∐ μ ∈ ∏ i L o g − ( c i ) ( A C r ( r − 1 ) ) d i s c . \mathrm{NSQC}^{\mathrm{fr}}_c\cong\coprod_{\mu\in\prod_i\mathrm{Log}_-(c_i)}\left(\mathbb{A}_{\mathbb{C}}^{r(r-1)}\right)_{\mathrm{disc}}. NSQC c fr ≅ μ ∈ ∏ i Log − ( c i ) ∐ ( A C r ( r − 1 ) ) disc . In particular, repeated monodromy eigenvalues and the eigenvalue 1 1 1
The proof is given in § 4 and § 5 . The residue functor is constructed in Proposition 4.4 ; framed Borel rigidity and diagonal reduction identify each fibre with a unipotent action groupoid; and Theorem 5.9 provides the unique affine slice and proves that the stabilisers are trivial.
There are two independent steps. First, the monodromy-invariant flag is extended to meromorphic flat subbundles; these are then intersected with the prescribed lattice and saturated. This produces an upper triangular holomorphic frame without any hypothesis on the eigenvalues. Second, if two such representatives are related by a holomorphic gauge g g g g ( 1 ) = I r g(1)=I_r g ( 1 ) = I r g g g g l r / b \mathfrak{gl}_r/\mathfrak{b} gl r / b g g g [19 §4] , then apply to every residue spectrum.
The theorem classifies the entire complex fibre groupoid, not merely its set of orbits. It does not, however, assert representability of the unsliced relative functor over arbitrary bases. A relative affine quotient is proved in § 9 after a triangular reduction and a locally constant residue vector have been supplied. This distinction accords with the general theory of regular-singular connections with parameters, in which the existence of logarithmic models and their behaviour under base change require separate hypotheses [11 , 10 , 28 ] .
We take the first step beyond the fixed-stratum theorem by constructing the elementary wall charts.
Theorem 1.2. Let λ \lambda λ H t H_t H t (28) . If λ ∈ Z ≥ 1 \lambda\in\mathbb{Z}_{\ge1} λ ∈ Z ≥ 1 λ ∉ Z ≥ 1 \lambda\notin\mathbb{Z}_{\ge1} λ ∈ / Z ≥ 1
α = λ + 1 t + a , β = − λ t + b , a , b ∈ O S , \alpha=\frac{\lambda+1}{t}+a,\qquad\beta=-\frac{\lambda}{t}+b,\qquad a,b\in\mathcal{O}_S, α = t λ + 1 + a , β = − t λ + b , a , b ∈ O S , and its special-fibre normal coordinates are given by the scalar projection ( 0 , σ λ + 2 ( a ( 0 ) , b ( 0 ) ) ) (0,\sigma_{\lambda+2}(a(0),b(0))) ( 0 , σ λ + 2 ( a ( 0 ) , b ( 0 )))
I 3 + s t z − 2 E 31 . I_3+stz^{-2}E_{31}. I 3 + s t z − 2 E 31 . Over C [ ε ] / ( ε 2 ) \mathbb{C}[\varepsilon]/(\varepsilon^2) C [ ε ] / ( ε 2 )
The theorem is proved in § 10 . A positive integral resonance barrier appears already in rank two, and the rank-three mixing can be absorbed into polynomial triangular coordinates along a single wall. The first order-dependent compatibility datum occurs when two adjacent walls meet.
The low-rank calculations extend to the full connection problem in arbitrary rank.
Theorem 1.3. Fix a simple wall supported in positions i , i + 1 i,i+1 i , i + 1 μ \mu μ N = r ( r − 1 ) N=r(r-1) N = r ( r − 1 ) λ i = μ i + 1 − μ i \lambda_i=\mu_{i+1}-\mu_i λ i = μ i + 1 − μ i λ i ∈ Z ≥ 1 \lambda_i\in\mathbb{Z}_{\ge1} λ i ∈ Z ≥ 1 S = ( Δ , t ) S=(\Delta,t) S = ( Δ , t ) O S \mathcal{O}_S O S L i , μ \mathcal{L}_{i,\mu} L i , μ N N N L i , μ \mathcal{L}_{i,\mu} L i , μ S × A C N S\times\mathbb{A}_{\mathbb{C}}^N S × A C N
The module L i , μ \mathcal{L}_{i,\mu} L i , μ
sp i , μ : V ( L i , μ ) 0 ⟶ S μ − e i + e i + 1 , \operatorname{sp}_{i,\mu}: \mathop{\mathrm{V}}(\mathcal{L}_{i,\mu})_{0} \longrightarrow S_{\mu-e_i+e_{i+1}}, sp i , μ : V ( L i , μ ) 0 ⟶ S μ − e i + e i + 1 , which is triangular with respect to root height. Each root coordinate of height h h h h h h
This theorem is proved in Theorem 10.8. It extends the rank-two and rank-three calculations to a relative representability statement for every simple wall and every rank. The lattice-theoretic structure in arbitrary rank is independent of the normal-form calculation.
Theorem 1.4. Put
D r = diag ( 1 , z , … , z r − 1 ) D_r=\operatorname{diag}(1,z,\ldots,z^{r-1}) D r = diag ( 1 , z , … , z r − 1 ) and let U r − U_r^{-} U r − D r D_r D r U r − U_r^{-} U r −
U r w a l l = D r − 1 U r − D r = { I r + ∑ j > i u j i z − ( j − i ) E j i } ⊂ G L r ( C ( ( z ) ) ) . U_r^{\mathrm{wall}}=D_r^{-1}U_r^{-}D_r=\left\{I_r+\sum_{j>i}u_{ji}z^{-(j-i)}E_{ji}\right\}\subset GL_r(\mathbb{C}((z))). U r wall = D r − 1 U r − D r = { I r + j > i ∑ u j i z − ( j − i ) E j i } ⊂ G L r ( C (( z ))) . For X j i ( a ) = I r + a z − ( j − i ) E j i X_{ji}(a)=I_r+az^{-(j-i)}E_{ji} X j i ( a ) = I r + a z − ( j − i ) E j i j > i j>i j > i
X k j ( b ) X j i ( a ) = X k i ( a b ) X j i ( a ) X k j ( b ) ( k > j > i ) , X_{kj}(b)X_{ji}(a)=X_{ki}(ab)X_{ji}(a)X_{kj}(b)\qquad(k>j>i), X k j ( b ) X j i ( a ) = X k i ( ab ) X j i ( a ) X k j ( b ) ( k > j > i ) , while two root subgroups commute whenever neither ordered product of their matrix units is non-zero. In particular, the pairwise compatibility relations between elementary wall matrices in arbitrary rank are determined within rank-two and rank-three Levi blocks. In rank two, the meromorphic matrix
G ( t ) = I 2 + t z − 1 E 21 G(t)=I_2+tz^{-1}E_{21} G ( t ) = I 2 + t z − 1 E 21 defines the opposite root-subgroup chart of the minimal affine Schubert curve in the affine flag variety of the rank-two Levi. For t ≠ 0 t\ne0 t = 0 z − 1 z^{-1} z − 1 z z z G ( 0 ) G(0) G ( 0 ) H t H_t H t G L 2 ( C [ [ z ] ] ) GL_2(\mathbb{C[[z]]}) G L 2 ( C [[ z ]] ) t ≠ 0 t\ne0 t = 0
The theorem is proved in Theorem 10.11 and Proposition 10.13. It relates the rank-three cocycle to the standard type-A commutator relation for root subgroups and identifies the matrix that records the lattice position.
Loop-group and affine flag geometry behind the wall charts
Affine flag varieties and their Schubert varieties provide the standard local models for relative positions of lattices and lattice chains [24 ] . Put
K = C ( ( z ) ) , O = C [ [ z ] ] , K=\mathbb{C}((z)),\qquad O=\mathbb{C[[z]]}, K = C (( z )) , O = C [[ z ]] , and write
Gr r = G L r ( K ) / G L r ( O ) , Fl r = G L r ( K ) / I , \operatorname{Gr}_r=GL_r(K)/GL_r(O),\qquad\operatorname{Fl}_r=GL_r(K)/I, Gr r = G L r ( K ) / G L r ( O ) , Fl r = G L r ( K ) / I , where I ⊂ G L r ( O ) I\subset GL_r(O) I ⊂ G L r ( O ) z z z O O O K r K^r K r
q : Fl r ⟶ Gr r q:\operatorname{Fl}_r\longrightarrow\operatorname{Gr}_r q : Fl r ⟶ Gr r forgets the flag.
Three pieces of data must be distinguished in the wall calculation. The prescribed lattice E \mathcal{E} E G ( t ) = I 2 + t z − 1 E 21 G(t)=I_2+tz^{-1}E_{21} G ( t ) = I 2 + t z − 1 E 21 H t H_t H t H t ∈ G L 2 ( O ) H_t\in GL_2(O) H t ∈ G L 2 ( O )
The result is Theorem 1.4 . In the rank-two Levi, G ( t ) G(t) G ( t ) U r w a l l U_r^{\mathrm{wall}} U r wall A 1 A_1 A 1 A 1 × A 1 A_1\times A_1 A 1 × A 1 A 2 A_2 A 2
The connection carries information not encoded by the lattice position alone. The positive integral resonance barrier of Theorem 10.3 determines which root-subgroup arcs lift to the distinguished normal slice; the scalar projections determine the limiting connection coordinate; and the square-zero calculation detects infinitesimal saturation data invisible on the reduced affine flag variety. A global position morphism would, moreover, require a compatible logarithmic or comparison lattice, which is not part of the original unsliced functor. Constructing the appropriate enhanced local-model stack and proving that its charts descend when the auxiliary comparison data are changed remain open.
Although the affine orbit type is independent of the spectrum, the monodromy map still depends on the residue stratum. In rank two we compute the monodromy and the re-framing action in all integral resonance regimes. In arbitrary rank we construct flat isomonodromic families whose punctured meromorphic connections are all meromorphically isomorphic to a fixed connection while the prescribed lattice varies. If F : S → a F:S\to\mathfrak{a} F : S → a a \mathfrak{a} a ad R \operatorname{ad}_{R} ad R
∇ R , F = d + d S F u + R u d u + [ R , F ] − F u 2 d u (2) \nabla_{R,F}=d+\frac{d_{S}F}{u}+\frac{R}{u}du+\frac{[R,F]-F}{u^{2}}du \tag*{(2)} ∇ R , F = d + u d S F + u R d u + u 2 [ R , F ] − F d u (2) is flat and satisfies the pole conditions of an F F F ( ad R − I ) F (\operatorname{ad}_{R}-I)F ( ad R − I ) F [22 , 26 ] , the datum retained here is different: the local system is fixed, whereas its extension across the pole varies.
The global theory is most naturally expressed in the notation of Katzarkov–Kontsevich–Pantev–Yu [17 §§3–5] . If ( H , ∇ ) / B (\mathcal{H},\nabla)/B ( H , ∇ ) / B F F F
κ ∇ : = ∇ u 2 ∂ u ∣ H / u H . \kappa_{\nabla}:=\left.\nabla_{u^{2}\partial_{u}}\right|_{\mathcal{H}/u\mathcal{H}}. κ ∇ := ∇ u 2 ∂ u ∣ H / u H . In the formal and non-Archimedean categories, the spectral decomposition theorem of Hinault–Yu–Zhang–Zhang decomposes a maximal F F F κ ∇ \kappa_{\nabla} κ ∇ [12 Theorems 3.31 and 3.41] . The connected components of the unramified reduced spectral cover give the local atoms of [17 §5] .
For a rank-n n n ( T + a ) n (T+a)^{n} ( T + a ) n a = − n − 1 tr ( κ ∇ ) a=-n^{-1}\operatorname{tr}(\kappa_{\nabla}) a = − n − 1 tr ( κ ∇ ) d ( a / u ) d(a/u) d ( a / u ) F F F F F F [17 §6.4] . Thus every geometric atomic F F F
( A , ∇ ) ⟷ ( a , A , ∇ − d ( a / u ) I ) . (\mathcal{A},\nabla)\longleftrightarrow(a,\mathcal{A},\nabla-d(a/u)I). ( A , ∇ ) ⟷ ( a , A , ∇ − d ( a / u ) I ) . The branch is retained as part of the residual atom; the zero-branch factor determines a classifying morphism to the unframed residual stack.
Let RAtoms G K \operatorname{RAtoms}_{G}^{K} RAtoms G K G G G K K K G G G F F F X X X
CF G r e s ( X ) ∈ N ( RAtoms G K ) \operatorname{CF}_{G}^{\mathrm{res}}(X)\in\mathbb{N}^{(\operatorname{RAtoms}_{G}^{K})} CF G res ( X ) ∈ N ( RAtoms G K ) is the finite multiset of its residual atoms with their spectral multiplicities. Its group completion lies in
Z ( RAtoms G K ) . \mathbb{Z}^{(\operatorname{RAtoms}_{G}^{K})}. Z ( RAtoms G K ) . Theorem 1.5. The branch is recovered intrinsically from the residual endomorphism, and the branch-residual splitting is an equivalence of stacks, functorial under arbitrary base change and symmetric monoidal with respect to addition of scalar branches. It induces a bijection between geometric G G G F F F G G G F F F CF G r e s ( X ) \operatorname{CF}_{G}^{\mathrm{res}}(X) CF G res ( X )
CF G r e s ( X 1 ⊔ X 2 ) = CF G r e s ( X 1 ) + CF G r e s ( X 2 ) , \operatorname{CF}_{G}^{\mathrm{res}}(X_{1}\sqcup X_{2})=\operatorname{CF}_{G}^{\mathrm{res}}(X_{1})+\operatorname{CF}_{G}^{\mathrm{res}}(X_{2}), CF G res ( X 1 ⊔ X 2 ) = CF G res ( X 1 ) + CF G res ( X 2 ) , CF G r e s ( Bl Z X ) = CF G r e s ( X ) + ( c − 1 ) CF G r e s ( Z ) , \operatorname{CF}_{G}^{\mathrm{res}}(\operatorname{Bl}_{Z}X)=\operatorname{CF}_{G}^{\mathrm{res}}(X)+(c-1)\operatorname{CF}_{G}^{\mathrm{res}}(Z), CF G res ( Bl Z X ) = CF G res ( X ) + ( c − 1 ) CF G res ( Z ) , CF G r e s ( P X ( V ) ) = m CF G r e s ( X ) , \operatorname{CF}_{G}^{\mathrm{res}}(\mathbb{P}_{X}(V))=m\operatorname{CF}_{G}^{\mathrm{res}}(X), CF G res ( P X ( V )) = m CF G res ( X ) , where Z ⊂ X Z\subset X Z ⊂ X c c c V V V m m m F F F Z ( RAtoms G K ) \mathbb{Z}^{(\operatorname{RAtoms}_{G}^{K})} Z ( RAtoms G K )
cf G r e s : K 0 ( Var K ) ⟶ Z ( RAtoms G K ) . \operatorname{cf}_{G}^{\mathrm{res}}:K_{0}(\operatorname{Var}_{K})\longrightarrow\mathbb{Z}^{(\operatorname{RAtoms}_{G}^{K})}. cf G res : K 0 ( Var K ) ⟶ Z ( RAtoms G K ) . Proof. Intrinsic branch extraction and the branch-residual equivalence are proved in Lemma 11.4 and Theorem 11.5 ; compatibility with enhanced structures is established in Proposition 11.6 , and compatibility with atomic correspondences in Proposition 11.8 . The independence and the three displayed identities are established in Theorem 11.10 ; the ring statement follows in Corollary 11.11 . □
The branch-residual presentation determines several further structures on a residual family. On the zero-branch factor, the nilpotent endomorphism κ r e s \kappa^{\mathrm{res}} κ res
A second application concerns the universal unit direction of the A-model. If t 0 t^0 t 0 t 0 = 0 t^0=0 t 0 = 0 d ( t 0 / u ) \mathrm{d}(t^0/u) d ( t 0 / u ) t 0 t^0 t 0 t 0 t^0 t 0 Q ≥ 0 / ( Q > 0 ) 3 \mathbb{Q}_{\geq0}/(\mathbb{Q}_{>0})^3 Q ≥ 0 / ( Q > 0 ) 3 Z / 2 \mathbb{Z}/2 Z /2 (2) shows that the residual lattice is not determined by monodromy: the local system on the punctured disc may remain constant while the lattice Jordan profile varies.
Structure of the paper
Sections 2 to 5 treat the local prescribed-lattice problem and its normal form for arbitrary spectra. Sections 6 to 9 discuss resonant examples, framed descent, isomonodromic variation of the lattice, and the fixed-stratum relative slice. Section 10 classifies the oriented rank-two wall for every residue difference and constructs two polynomial rank-three wall charts. It then proves that every permitted simple wall in arbitrary rank is represented by a relative affine torsor, with a polynomial specialisation map triangular by root height. The same section computes the adjacent two-wall compatibility factor, the saturation over the dual numbers, the root-subgroup compatibility law in the loop group, and the rank-two affine position of the meromorphic comparison lattice. Section 11 develops the branch-residual splitting and its applications to projective varieties. The generic-spectrum case is due to Kontsevich and Szabó [19 ] ; all other dependencies are cited where they enter.
Notation and conventions
We follow the notation of [17 §§3–5] whenever the global theory of atoms is involved. The local complex coordinate is denoted by z z z u u u F F F
( H , ∇ ) / S (\mathcal{H},\nabla)/S ( H , ∇ ) / S F F F S S S
H 0 \mathcal{H}_0 H 0 H / u H \mathcal{H}/u\mathcal{H} H / u H
κ r e s \kappa^{\mathrm{res}} κ res ∇ u 2 ∂ u H 0 \nabla_{u^2\partial_u}\mathcal{H}_0 ∇ u 2 ∂ u H 0
U ~ X → U X \widetilde{U}_X\to U_X U X → U X F F F X X X
Atoms G K , F \operatorname{Atoms}^{K,F}_G Atoms G K , F G G G F F F K K K
CF G F ( X ) \operatorname{CF}^{F}_G(X) CF G F ( X ) F F F
RAtoms G K \operatorname{RAtoms}^{K}_G RAtoms G K G G G § 11 ;
CF G r e s ( X ) \operatorname{CF}^{\mathrm{res}}_G(X) CF G res ( X ) X X X
For a set S S S N ( S ) \mathbb{N}^{(S)} N ( S ) Z ( S ) \mathbb{Z}^{(S)} Z ( S ) S S S ⨆ \bigsqcup ⨆ ∐ \coprod ∐
The prescribed-lattice groupoid and compatible logarithms
Let B ⊂ G L r ( C ) B\subset\mathrm{GL}_r(\mathbb{C}) B ⊂ GL r ( C ) U U U b \mathfrak{b} b n \mathfrak{n} n
( d + A d z ) ⋅ g = d + ( g − 1 A g + g − 1 g ′ ) d z . (3) (d+A\,dz)\cdot g=d+\left(g^{-1}Ag+g^{-1}g'\right)dz. \tag*{(3)} ( d + A d z ) ⋅ g = d + ( g − 1 A g + g − 1 g ′ ) d z . (3) Definition 3.1. Kontsevich and Szabó define this groupoid under the assumption c i ≠ 1 c_i \ne1 c i = 1 i i i
c = ( c 1 , … , c r ) ∈ ( C × ) r . \mathbf{c}=(c_1,\ldots,c_r)\in(\mathbb{C}^{\times})^r. c = ( c 1 , … , c r ) ∈ ( C × ) r . The enlarged groupoid
N S Q C c f r \mathrm{NSQC}_{\mathbf{c}}^{\mathrm{fr}} NSQC c fr has as objects triples ( E , ∇ , v ) (E,\nabla,\mathbf{v}) ( E , ∇ , v )
(i) E = O ⊕ r E=\mathcal{O}^{\oplus r} E = O ⊕ r
(ii) ∇ \nabla ∇ E E E 0 0 0
(iii) In the prescribed lattice E E E 0 0 0
(iv) v = ( v 1 ( 1 ) , … , v r ( 1 ) ) \mathbf{v}=(v_1(1),\ldots,v_r(1)) v = ( v 1 ( 1 ) , … , v r ( 1 )) E ∣ z = 1 E|_{z=1} E ∣ z = 1
(v) The monodromy preserves the full flag defined by v \mathbf{v} v c 1 , … , c r c_1,\ldots,c_r c 1 , … , c r
We regard an ordered basis as a row v = ( v 1 ( 1 ) , … , v r ( 1 ) ) \mathbf{v}=(v_1(1),\ldots,v_r(1)) v = ( v 1 ( 1 ) , … , v r ( 1 ))
g : ( E , ∇ , v ) ⟶ ( E ~ , ∇ ~ , v ~ ) g:(E,\nabla,\mathbf{v})\longrightarrow(\widetilde{E},\widetilde{\nabla},\widetilde{\mathbf{v}}) g : ( E , ∇ , v ) ⟶ ( E , ∇ , v ) is a holomorphic gauge transformation g ∈ G L r ( O ) g\in\mathrm{GL}_r(\mathcal{O}) g ∈ GL r ( O )
∇ ~ = ∇ ⋅ g , v ~ = v g ( 1 ) . \widetilde{\nabla}=\nabla\cdot g,\qquad\widetilde{\mathbf{v}}=\mathbf{v}g(1). ∇ = ∇ ⋅ g , v = v g ( 1 ) . This is the convention used in [19 ] : both the connection matrix and the row of framed basis vectors carry the same right action. Applying first g g g h h h g h gh g h g ( 1 ) = I r g(1)=I_r g ( 1 ) = I r
All quotient calculations below take place after the ordered frame has been changed to the standard row e ( 1 ) \mathbf{e}(1) e ( 1 ) g ( 1 ) = I r g(1)=I_r g ( 1 ) = I r [19 ] : every module occurring there is free of rank r r r z = 1 z=1 z = 1
Lemma 3.2. Every T ∈ B T\in B T ∈ B R ∈ b R\in\mathfrak{b} R ∈ b e R = T e^R=T e R = T
Proof. Write the set of distinct eigenvalues of T T T { σ 1 , … , σ s } \{\sigma_1,\ldots,\sigma_s\} { σ 1 , … , σ s } D 1 , … , D s ⊂ C × D_1,\ldots,D_s\subset\mathbb{C}^{\times} D 1 , … , D s ⊂ C × σ ν ∈ D ν \sigma_\nu\in D_\nu σ ν ∈ D ν ℓ ν \ell_\nu ℓ ν D ν D_\nu D ν D = ⋃ ν = 1 s D ν D=\bigcup_{\nu=1}^sD_\nu D = ⋃ ν = 1 s D ν ℓ \ell ℓ D D D
R = ℓ ( T ) = 1 2 π i ∫ Γ ℓ ( ζ ) ( ζ I r − T ) − 1 d ζ , R=\ell(T)=\frac{1}{2\pi i}\int_{\Gamma}\ell(\zeta)(\zeta I_r-T)^{-1}\,d\zeta, R = ℓ ( T ) = 2 π i 1 ∫ Γ ℓ ( ζ ) ( ζ I r − T ) − 1 d ζ , where Γ ⊂ D \Gamma\subset D Γ ⊂ D T T T ( ζ I r − T ) − 1 (\zeta I_r-T)^{-1} ( ζ I r − T ) − 1 R ∈ b R\in\mathfrak{b} R ∈ b
exp ( ℓ ( T ) ) = ( exp ∘ ℓ ) ( T ) = T \exp(\ell(T))=(\exp\circ\ell)(T)=T exp ( ℓ ( T )) = ( exp ∘ ℓ ) ( T ) = T gives e R = T e^R=T e R = T ℓ \ell ℓ R R R C [ T ] ⊂ b \mathbb{C}[T]\subset\mathfrak{b} C [ T ] ⊂ b [20 , 9 , 26 ] .
arise from the resulting lattice modification and are not defined by arbitrarily prescribing a logarithm of the full monodromy.
Framed Borel rigidity and triangular reduction
We begin with a lattice-theoretic triangular reduction which is intrinsic to the monodromy flag and the prescribed lattice.
Proposition 4.1. Every object of N S Q C c f r \mathrm{NSQC}_{\mathfrak c}^{\mathrm{fr}} NSQC c fr
d + ( ∑ n ≥ − 2 D n z n ) d z , D − 2 ∈ n , D n ∈ b ( n ≥ − 1 ) . (4) d+\left(\sum_{n\geq-2}D_nz^n\right)dz,\qquad D_{-2}\in\mathfrak n,\qquad D_n\in\mathfrak b\quad(n\geq-1). \tag*{(4)} d + ( n ≥ − 2 ∑ D n z n ) d z , D − 2 ∈ n , D n ∈ b ( n ≥ − 1 ) . (4) where the Laurent series converges on Δ \Delta Δ c i c_i c i 1 1 1
Proof. Let
0 = F 0 ( 1 ) ⊂ F 1 ( 1 ) ⊂ ⋯ ⊂ F r ( 1 ) = E ∣ z = 1 0=F_0(1)\subset F_1(1)\subset\cdots\subset F_r(1)=E|_{z=1} 0 = F 0 ( 1 ) ⊂ F 1 ( 1 ) ⊂ ⋯ ⊂ F r ( 1 ) = E ∣ z = 1 be the full flag determined by the ordered frame. Its monodromy invariance gives, by parallel transport, a filtration by flat subbundles
F i ∘ ⊂ E ∣ Δ × . F_i^\circ\subset E|_{\Delta^\times}. F i ∘ ⊂ E ∣ Δ × . Since the connection is regular singular, each F i ∘ F_i^\circ F i ∘ ∇ \nabla ∇
F i ⊂ E ( ∗ 0 ) . F_i\subset E(*0). F i ⊂ E ( ∗ 0 ) . After choosing a logarithmic model, the corresponding sub-local system extends to a meromorphic subconnection; uniqueness follows because two such extensions agree on the punctured disc. Equivalently, one may choose an upper triangular logarithmic model of the monodromy and transport its standard flag. This step uses neither distinctness of the eigenvalues nor exclusion of the eigenvalue 1 1 1
E i = ( E ∩ F i ) s a t ⊂ E , E_i=(E\cap F_i)^{\mathrm{sat}}\subset E, E i = ( E ∩ F i ) sat ⊂ E , where saturation is taken in E E E F i F_i F i z z z E E E z N F i z^N F_i z N F i O Δ , 0 \mathcal O_{\Delta,0} O Δ , 0 Δ \Delta Δ E i / E i − 1 E_i/E_{i-1} E i / E i − 1 Δ × \Delta^\times Δ × E i ( 1 ) = F i ( 1 ) E_i(1)=F_i(1) E i ( 1 ) = F i ( 1 )
0 ⟶ E i − 1 ⟶ E i ⟶ E i / E i − 1 ⟶ 0 0\longrightarrow E_{i-1}\longrightarrow E_i\longrightarrow E_i/E_{i-1}\longrightarrow0 0 ⟶ E i − 1 ⟶ E i ⟶ E i / E i − 1 ⟶ 0 splits holomorphically because
Ext 1 ( E i / E i − 1 , E i − 1 ) ≅ H 1 ( Δ , E i − 1 ⊗ ( E i / E i − 1 ) − 1 ) = 0. \operatorname{Ext}^{1}(E_i/E_{i-1},E_{i-1})\cong H^1\left(\Delta,E_{i-1}\otimes(E_i/E_{i-1})^{-1}\right)=0. Ext 1 ( E i / E i − 1 , E i − 1 ) ≅ H 1 ( Δ , E i − 1 ⊗ ( E i / E i − 1 ) − 1 ) = 0. Since Δ \Delta Δ Pic ( Δ ) = 0 \operatorname{Pic}(\Delta)=0 Pic ( Δ ) = 0 E i / E i − 1 E_i/E_{i-1} E i / E i − 1 i i i E i / E i − 1 E_i/E_{i-1} E i / E i − 1 1 1 1 v i ( 1 ) v_i(1) v i ( 1 ) E i − 1 E_{i-1} E i − 1 1 1 1 v i ( 1 ) v_i(1) v i ( 1 ) E i − 1 E_{i-1} E i − 1 1 1 1 w 1 , … , w r w_1,\ldots,w_r w 1 , … , w r E E E E ∙ E_\bullet E ∙ 1 1 1
The matrix of ∇ \nabla ∇ w 1 , … , w r w_1,\ldots,w_r w 1 , … , w r Δ × \Delta^\times Δ × F i F_i F i ∇ \nabla ∇
It remains to examine the diagonal. The i i i F i / F i − 1 F_i/F_{i-1} F i / F i − 1 E i / E i − 1 E_i/E_{i-1} E i / E i − 1 [26 Chap. II] . A rank-one regular-singular connection has at most a simple pole in every lattice, since any two rank-one lattices differ by multiplication by an integral power of z z z d z / z dz/z d z / z z − 2 z^{-2} z − 2 D − 2 ∈ n D_{-2}\in\mathfrak n D − 2 ∈ n D n ∈ b D_n\in\mathfrak b D n ∈ b n ≥ − 1 n\geq-1 n ≥ − 1 □ \square □
Lemma 4.2. Let
∇ = d + A ( z ) d z , ∇ ~ = d + A ~ ( z ) d z \nabla= d + A(z)\,dz,\qquad\widetilde{\nabla} = d + \widetilde{A}(z)\,dz ∇ = d + A ( z ) d z , ∇ = d + A ( z ) d z be meromorphic connections whose matrices take values in b \mathfrak{b} b g ∈ GL r ( O ) g \in\operatorname{GL}_{r}(\mathcal{O}) g ∈ GL r ( O ) ∇ ~ = ∇ ⋅ g \widetilde{\nabla} = \nabla\cdot g ∇ = ∇ ⋅ g g ( 1 ) = I r g(1) = I_{r} g ( 1 ) = I r g g g B B B
Proof. The gauge identity is equivalent to
g ′ = g A ~ − A g . (5) g' = g\widetilde{A} - Ag. \tag*{(5)} g ′ = g A − A g . (5) For each z ≠ 0 z \ne0 z = 0
X ⟼ X A ~ ( z ) − A ( z ) X X \longmapsto X\widetilde{A}(z) - A(z)X X ⟼ X A ( z ) − A ( z ) X preserves b \mathfrak{b} b L ‾ ( z ) \overline{L}(z) L ( z ) g l r / b \mathfrak{gl}_{r}/\mathfrak{b} gl r / b g ‾ \overline{g} g g g g
g ‾ ′ = L ‾ ( z ) g ‾ , g ‾ ( 1 ) = 0. \overline{g}' = \overline{L}(z)\overline{g},\qquad\overline{g}(1) = 0. g ′ = L ( z ) g , g ( 1 ) = 0. Uniqueness for this homogeneous linear initial-value problem implies g ‾ = 0 \overline{g} = 0 g = 0 g ‾ = 0 \overline{g} = 0 g = 0 Δ × \Delta^{\times} Δ × g g g Δ × \Delta^{\times} Δ × Δ \Delta Δ g g g g : Δ → B g:\Delta\to B g : Δ → B
Corollary 4.3. The full subgroupoid of standard-framed upper triangular connections of the form (4) is essentially surjective in N S Q C c f r \mathrm{NSQC}_{c}^{\mathrm{fr}} NSQC c fr B B B I r I_{r} I r z = 1 z=1 z = 1
Proof. Essential surjectivity is Proposition 4.1. A morphism between standard-framed representatives satisfies g ( 1 ) = I r g(1)=I_{r} g ( 1 ) = I r B B B Lemma 4.2 . □
At this point the exclusion of the eigenvalue 1 becomes unnecessary in the fully framed problem. The proof of [19 Lemma 4.2] uses a residue obstruction to show triangularity of the transported gauge. The framed initial condition already forces the same conclusion without any restriction on the spectrum.
Proposition 4.4. There is a canonical essentially surjective functor
res Δ : N S Q C c f r ⟶ ( ∏ i = 1 r Log − ( c i ) ) d i s c , \operatorname{res}_{\Delta}:\mathrm{NSQC}_{c}^{\mathrm{fr}} \longrightarrow\left(\prod_{i=1}^{r}\operatorname{Log}_{-}(c_{i})\right)_{\mathrm{disc}}, res Δ : NSQC c fr ⟶ ( i = 1 ∏ r Log − ( c i ) ) disc , defined on objects by the ordered diagonal residues of any standard-framed upper triangular representative and on every morphism by the identity of that residue vector.
Proof. Existence of a standard-framed upper triangular representative follows from Corollary 4.3 . Suppose that two such representatives are isomorphic in the original groupoid. The induced morphism is B B B Lemma 4.2 . If its diagonal entries are g i i g_{ii} g ii i i i d log g i i d\log g_{ii} d log g ii g i i g_{ii} g ii
For an upper triangular regular-singular system, the diagonal scalar equation shows that a diagonal residue μ i \mu_i μ i e − 2 π i μ i e^{-2\pi i\mu_i} e − 2 π i μ i ∏ i Log − ( c i ) \prod_i\operatorname{Log}_{-}(c_i) ∏ i Log − ( c i )
d + M μ z − 1 d z . d+M_{\mu}z^{-1}\,dz. d + M μ z − 1 d z . Thus the functor is essentially surjective. □
The functor res Δ \operatorname{res}_{\Delta} res Δ Proposition 4.4 asserts that this datum is independent of every choice made in the reduction.
Normal forms for arbitrary spectra
Once framed Borel rigidity has been established, the remaining normalisation is the rootwise scalar quotient and height-filtered unipotent induction of Konstevich and Szabó [19 Proposition 4.3 and Lemma 4.4] . We give the analytic calculation in full because the resonant monomial, the framed condition at z = 1 z=1 z = 1 [19 ] ; here the same slice is shown to compute the original fully framed groupoid for every spectrum.
Diagonal normalisation and residue strata
Let
∇ = d + A ( z ) d z \nabla= d + A(z)\,dz ∇ = d + A ( z ) d z be a standard-framed upper triangular representative. Denote the diagonal entries of A A A a i ( z ) a_i(z) a i ( z )
μ i = res 0 ( a i ( z ) d z ) , M μ = diag ( μ 1 , … , μ r ) . \mu_i = \operatorname{res}_0\left(a_i(z)\,dz\right),\qquad M_\mu= \operatorname{diag}(\mu_1,\ldots,\mu_r). μ i = res 0 ( a i ( z ) d z ) , M μ = diag ( μ 1 , … , μ r ) . Lemma 5.1. There is a unique diagonal holomorphic gauge h : Δ → ( C × ) r h:\Delta\to(\mathbb{C}^{\times})^r h : Δ → ( C × ) r h ( 1 ) = I r h(1)=I_r h ( 1 ) = I r
M μ z d z . \frac{M_\mu}{z}\,dz. z M μ d z . Moreover, every framed B B B U U U
Proof. By Proposition 4.1, each a i ( z ) d z a_i(z)\,dz a i ( z ) d z
a i ( z ) − μ i z a_i(z)-\frac{\mu_i}{z} a i ( z ) − z μ i is holomorphic on Δ \Delta Δ
h i ′ h i = μ i z − a i ( z ) , h i ( 1 ) = 1 , \frac{h_i'}{h_i}=\frac{\mu_i}{z}-a_i(z),\qquad h_i(1)=1, h i h i ′ = z μ i − a i ( z ) , h i ( 1 ) = 1 , has a unique nowhere-vanishing holomorphic solution, since the integrand below is holomorphic on the simply connected disc, so the integral is path-independent and
h i ( z ) = exp ( ∫ 1 z ( μ i ζ − a i ( ζ ) ) d ζ ) . (6) h_i(z)=\exp\left(\int_1^z\left(\frac{\mu_i}{\zeta}-a_i(\zeta)\right)\,d\zeta\right). \tag*{(6)} h i ( z ) = exp ( ∫ 1 z ( ζ μ i − a i ( ζ ) ) d ζ ) . (6) Taking h = diag ( h 1 , … , h r ) h=\operatorname{diag}(h_1,\ldots,h_r) h = diag ( h 1 , … , h r ) h ~ \widetilde h h h ~ h − 1 \widetilde h h^{-1} h h − 1 1 1 1 z = 1 z=1 z = 1 h ~ = h \widetilde h=h h = h
Now let g g g B B B M μ d z / z M_\mu dz/z M μ d z / z i i i
g i i ′ g i i = 0. \frac{g_{ii}'}{g_{ii}}=0. g ii g ii ′ = 0. Thus each g i i g_{ii} g ii g ( 1 ) = I r g(1)=I_r g ( 1 ) = I r g i i = 1 g_{ii}=1 g ii = 1 g g g U U U
The diagonal gauge need not be polynomial, even when the initial matrix has a convergent Laurent expansion of the form (4) . The natural analytic coefficient space after diagonal normalisation is therefore
A μ = { d + ( M μ z + Q ( z ) ) d z : Q ∈ n ⊗ z − 2 O } . (7) \mathcal{A}_\mu=\left\{d+\left(\frac{M_\mu}{z}+Q(z)\right)\,dz:Q\in\mathfrak{n}\otimes z^{-2}\mathcal{O}\right\}. \tag*{(7)} A μ = { d + ( z M μ + Q ( z ) ) d z : Q ∈ n ⊗ z − 2 O } . (7) The analytic unipotent group
U 1 = { g ∈ U ( O ) : g ( 1 ) = I r } (8) U_1=\{g\in U(\mathcal{O}):g(1)=I_r\} \tag*{(8)} U 1 = { g ∈ U ( O ) : g ( 1 ) = I r } (8) acts on A μ \mathcal{A}_\mu A μ Lemmas 4.2 and 5.1 , it is the full framed gauge group relating elements of this fixed-diagonal space.
Lemma 5.2. Every connection in A μ \mathcal{A}_\mu A μ
Proof. Fix a closed subsector S S S z − μ i z^{-\mu_i} z − μ i z − μ r z^{-\mu_r} z − μ r S S S
Assume inductively that the components with index greater than j j j j j j
y j ′ + μ j z − 1 y j = − ∑ i > j q j i ( z ) y i , q j i ∈ z − 2 O . y_j' + \mu_j z^{-1}y_j=-\sum_{i>j}q_{ji}(z)y_i,\qquad q_{ji}\in z^{-2}\mathcal{O}. y j ′ + μ j z − 1 y j = − i > j ∑ q j i ( z ) y i , q j i ∈ z − 2 O . After setting y j = z − μ j u j y_j=z^{-\mu_j}u_j y j = z − μ j u j
u j ′ = − ∑ i > j z μ j q j i ( z ) y i . u_j'=-\sum_{i>j}z^{\mu_j}q_{ji}(z)y_i. u j ′ = − i > j ∑ z μ j q j i ( z ) y i . On the fixed subsector, the right-hand side is bounded by C ∣ z ∣ − N C|z|^{-N} C ∣ z ∣ − N C , N > 0 C,N>0 C , N > 0 z z z ∣ z ∣ − 1 |z|^{-1} ∣ z ∣ − 1 N = 1 N=1 N = 1 z − μ j z^{-\mu_j} z − μ j S S S z − μ i z^{-\mu_i} z − μ i z z z [21 (1.3)] , therefore applies. □
Proposition 5.3. Let μ ∈ ∏ i Log − ( c i ) \mu\in\prod_i \operatorname{Log}_{-}(c_i) μ ∈ ∏ i Log − ( c i ) A μ A_\mu A μ z = 1 z = 1 z = 1 NSQC c f r \operatorname{NSQC}^{\mathrm{fr}}_c NSQC c fr (1.1) is admissible.
Proof. The matrix has a pole of order at most two by definition. Regular singularity is Lemma 5.2 . Since the matrix is upper triangular, the standard flag is preserved by parallel transport. The i i i
y i ′ + μ i z − 1 y i = 0 , y_i' + \mu_i z^{-1} y_i = 0, y i ′ + μ i z − 1 y i = 0 , so the corresponding diagonal monodromy entry is e − 2 π i μ i = c i e^{-2\pi i\mu_i} = c_i e − 2 π i μ i = c i
Proposition 5.4. Fix μ ∈ ∏ i Log − ( c i ) \mu\in\prod_i \operatorname{Log}_{-}(c_i) μ ∈ ∏ i Log − ( c i ) res Δ − 1 ( μ ) \operatorname{res}_{\Delta}^{-1}(\mu) res Δ − 1 ( μ ) [ A μ ( C ) / U 1 ( C ) ] [A_\mu(\mathbb{C})/U_1(\mathbb{C})] [ A μ ( C ) / U 1 ( C )] A μ A_\mu A μ U 1 U_1 U 1
Proof. Apply the unique framed diagonal gauge of Lemma 5.1 to any standard-framed triangular representative in the residue fibre. This gives an element of A μ A_\mu A μ A μ A_\mu A μ Lemma 4.2 , the relating gauge is B B B I r I_r I r z = 1 z = 1 z = 1
g i i ′ g i i = 0 , \frac{g_{ii}'}{g_{ii}} = 0, g ii g ii ′ = 0 , because both diagonal connection coefficients are μ i z − 1 \mu_i z^{-1} μ i z − 1 g i i g_{ii} g ii g i i = 1 g_{ii} = 1 g ii = 1 U 1 U_1 U 1 U 1 U_1 U 1
The residue fibres are all non-empty by Proposition 4.4 , although Remark 3.3 shows that a fixed non-semisimple monodromy matrix need not occur in every fibre.
The scalar analytic quotient
Set V = { P ∈ O : P ( 1 ) = 0 } V = \{P \in\mathcal{O} : P(1) = 0\} V = { P ∈ O : P ( 1 ) = 0 } W = z − 2 O W = z^{-2}\mathcal{O} W = z − 2 O λ ∈ C \lambda\in\mathbb{C} λ ∈ C
L λ : V ⟶ W , L λ ( P ) = P ′ − λ z − 1 P . (9) L_\lambda: V \longrightarrow W,\qquad L_\lambda(P) = P' - \lambda z^{-1}P. \tag*{(9)} L λ : V ⟶ W , L λ ( P ) = P ′ − λ z − 1 P . (9) Lemma 5.5. For every λ ∈ C \lambda\in\mathbb{C} λ ∈ C L λ L_\lambda L λ
W = L λ ( V ) ⊕ C z − 2 ⊕ C ρ λ , (10) W = L_\lambda(V) \oplus\mathbb{C}z^{-2} \oplus\mathbb{C}\rho_\lambda, \tag*{(10)} W = L λ ( V ) ⊕ C z − 2 ⊕ C ρ λ , (10) where
ρ λ ( z ) = { z λ − 1 , λ ∈ Z ≥ 0 , z − 1 , λ ∉ Z ≥ 0 . \rho_\lambda(z) =
\begin{cases}
z^{\lambda-1}, & \lambda\in\mathbb{Z}_{\geq0},\\
z^{-1}, & \lambda\notin\mathbb{Z}_{\geq0}.
\end{cases} ρ λ ( z ) = { z λ − 1 , z − 1 , λ ∈ Z ≥ 0 , λ ∈ / Z ≥ 0 . In particular, dim C coker L λ = 2 \dim_{\mathbb{C}}\operatorname{coker}L_\lambda= 2 dim C coker L λ = 2
Proof. Write
P ( z ) = ∑ n ≥ 0 p n z n , Q ( z ) = ∑ k ≥ − 2 q k z k . P(z) = \sum_{n\geq0} p_n z^n,\qquad Q(z) = \sum_{k\geq-2} q_k z^k. P ( z ) = n ≥ 0 ∑ p n z n , Q ( z ) = k ≥ − 2 ∑ q k z k . A direct calculation gives
L λ ( P ) = ∑ n ≥ 0 ( n − λ ) p n z n − 1 . (11) L_\lambda(P) = \sum_{n\geq0}(n-\lambda)p_n z^{n-1}. \tag*{(11)} L λ ( P ) = n ≥ 0 ∑ ( n − λ ) p n z n − 1 . (11) In particular, no element of the image has a z − 2 z^{-2} z − 2 L λ L_\lambda L λ z − 1 O z^{-1}\mathcal{O} z − 1 O λ ∉ Z ≥ 0 \lambda\notin\mathbb{Z}_{\geq0} λ ∈ / Z ≥ 0 Q 1 = ∑ k ≥ − 1 q k z k Q_1 = \sum_{k\geq-1} q_k z^k Q 1 = ∑ k ≥ − 1 q k z k
P 0 ( z ) = ∑ n ≥ 0 q n − 1 n − λ z n . P_0(z) = \sum_{n\geq0}\frac{q_{n-1}}{n-\lambda}z^n. P 0 ( z ) = n ≥ 0 ∑ n − λ q n − 1 z n . Choose radii 1 < r 0 < R 0 1 < r_0 < R_0 1 < r 0 < R 0 { ∣ z ∣ ≤ R 0 } \{|z| \leq R_0\} { ∣ z ∣ ≤ R 0 } Δ \Delta Δ C R 0 C_{R_0} C R 0 ∣ q n − 1 ∣ ≤ C R 0 R 0 − n |q_{n-1}| \leq C_{R_0}R_0^{-n} ∣ q n − 1 ∣ ≤ C R 0 R 0 − n ∣ n − λ ∣ − 1 = O ( n − 1 ) |n-\lambda|^{-1} = O(n^{-1}) ∣ n − λ ∣ − 1 = O ( n − 1 ) P 0 P_0 P 0 ∣ z ∣ ≤ r 0 |z| \leq r_0 ∣ z ∣ ≤ r 0 Δ \Delta Δ P 0 ∈ O P_0 \in\mathcal{O} P 0 ∈ O L λ ( P 0 ) = Q 1 L_\lambda(P_0) = Q_1 L λ ( P 0 ) = Q 1 P 0 ∈ V P_0 \in V P 0 ∈ V 1 1 1
L λ ( c ) = − λ c z − 1 , L_\lambda(c) = -\lambda c z^{-1}, L λ ( c ) = − λ c z − 1 , the function P = P 0 − P 0 ( 1 ) P=P_0-P_0(1) P = P 0 − P 0 ( 1 ) P ( 1 ) = 0 P(1)=0 P ( 1 ) = 0
Q 1 = L λ ( P ) − λ P 0 ( 1 ) z − 1 . Q_1=L_\lambda(P)-\lambda P_0(1)z^{-1}. Q 1 = L λ ( P ) − λ P 0 ( 1 ) z − 1 . Thus z − 1 O = L λ ( V ) + C z − 1 z^{-1}\mathcal{O}=L_\lambda(V)+\mathbb{C}z^{-1} z − 1 O = L λ ( V ) + C z − 1 L λ ( P ) = β z − 1 L_\lambda(P)=\beta z^{-1} L λ ( P ) = β z − 1 (11) forces P = − β / λ P=-\beta/\lambda P = − β / λ P ( 1 ) = 0 P(1)=0 P ( 1 ) = 0 P = β = 0 P=\beta=0 P = β = 0
Now, let λ = m ∈ Z ≥ 0 \lambda=m\in\mathbb{Z}_{\geq0} λ = m ∈ Z ≥ 0 z m − 1 z^{m-1} z m − 1 (11) is identically zero. Given Q 1 Q_1 Q 1 q m − 1 z m − 1 q_{m-1}z^{m-1} q m − 1 z m − 1
p n = q n − 1 n − m ( n ≠ m ) . p_n=\frac{q_{n-1}}{n-m}\qquad(n\ne m). p n = n − m q n − 1 ( n = m ) . The same exhaustion and Cauchy estimate prove normal convergence of ∑ n ≠ m p n z n \sum_{n\ne m}p_nz^n ∑ n = m p n z n Δ \Delta Δ p m p_m p m L m ( z m ) = 0 L_m(z^m)=0 L m ( z m ) = 0
p m = − ∑ n ≠ m p n , p_m=-\sum_{n\ne m}p_n, p m = − n = m ∑ p n , which is the condition P ( 1 ) = 0 P(1)=0 P ( 1 ) = 0
z − 1 O = L m ( V ) ⊕ C z m − 1 . z^{-1}\mathcal{O}=L_m(V)\oplus\mathbb{C}z^{m-1}. z − 1 O = L m ( V ) ⊕ C z m − 1 . Directness follows by inspecting the coefficient of z m − 1 z^{m-1} z m − 1
Finally, if L λ ( P ) = 0 L_\lambda(P)=0 L λ ( P ) = 0 (11) gives P = 0 P=0 P = 0 λ = m ∈ Z ≥ 0 \lambda=m\in\mathbb{Z}_{\geq0} λ = m ∈ Z ≥ 0 P = c z m P=cz^m P = c z m P ( 1 ) = 0 P(1)=0 P ( 1 ) = 0 c = 0 c=0 c = 0 z − 2 z^{-2} z − 2 [19 Lemma 4.4] . □
Lemma 5.6. Let E j i E_{ji} E j i j j j i i i j < i j<i j < i i − j i-j i − j λ j i = μ i − μ j \lambda_{ji}=\mu_i-\mu_j λ j i = μ i − μ j p p p p ( 1 ) = 0 p(1)=0 p ( 1 ) = 0 g = I r + p E j i g=I_r+pE_{ji} g = I r + p E j i ( j , i ) (j,i) ( j , i ) i − j i-j i − j
q j i ⟼ q j i + p ′ + ( μ j − μ i ) z − 1 p = q j i + L λ j i ( p ) . q_{ji}\longmapsto q_{ji}+p'+(\mu_j-\mu_i)z^{-1}p=q_{ji}+L_{\lambda_{ji}}(p). q j i ⟼ q j i + p ′ + ( μ j − μ i ) z − 1 p = q j i + L λ j i ( p ) . Proof. Since E j i 2 = 0 E_{ji}^2=0 E j i 2 = 0 g − 1 = I r − p E j i g^{-1}=I_r-pE_{ji} g − 1 = I r − p E j i g − 1 ( M μ / z ) g g^{-1}(M_\mu/z)g g − 1 ( M μ / z ) g
1 z [ M μ , p E j i ] = μ j − μ i z p E j i , \frac{1}{z}[M_\mu,pE_{ji}]=\frac{\mu_j-\mu_i}{z}pE_{ji}, z 1 [ M μ , p E j i ] = z μ j − μ i p E j i , while g − 1 g ′ g^{-1}g' g − 1 g ′ p ′ E j i p'E_{ji} p ′ E j i i − j i-j i − j
For the root E j i E_{ji} E j i μ i − μ j \mu_i-\mu_j μ i − μ j ( μ 1 , μ 2 ) = ( 0 , 1 ) (\mu_1,\mu_2)=(0,1) ( μ 1 , μ 2 ) = ( 0 , 1 )
The non-abelian unipotent normal form
For 1 ≤ h ≤ r 1\leq h\leq r 1 ≤ h ≤ r
n ( h ) = span { E j i : j < i , i − j ≥ h } . \mathfrak{n}^{(h)}=\operatorname{span}\{E_{ji}:j<i,\ i-j\geq h\}. n ( h ) = span { E j i : j < i , i − j ≥ h } . Then
n = n ( 1 ) ⊃ n ( 2 ) ⊃ ⋯ ⊃ n ( r ) = 0 \mathfrak{n}=\mathfrak{n}^{(1)}\supset\mathfrak{n}^{(2)}\supset\cdots\supset\mathfrak{n}^{(r)}=0 n = n ( 1 ) ⊃ n ( 2 ) ⊃ ⋯ ⊃ n ( r ) = 0 and
[ n ( a ) , n ( b ) ] ⊆ n ( a + b ) . (12) [\mathfrak{n}^{(a)},\mathfrak{n}^{(b)}]\subseteq\mathfrak{n}^{(a+b)}. \tag*{(12)} [ n ( a ) , n ( b ) ] ⊆ n ( a + b ) . (12) Define the affine slice
S μ = { d + ( M μ z + ∑ j < i ( α j i z − 2 + β j i ρ λ j i ) E j i ) d z } . (13) S_\mu=\left\{d+\left(\frac{M_\mu}{z}+\sum_{j<i}(\alpha_{ji}z^{-2}+\beta_{ji}\rho_{\lambda_{ji}})E_{ji}\right)dz\right\}. \tag*{(13)} S μ = { d + ( z M μ + j < i ∑ ( α j i z − 2 + β j i ρ λ j i ) E j i ) d z } . (13) Proposition 5.7. Every element of A μ A_\mu A μ U 1 U_1 U 1 S μ S_\mu S μ
Proof. We construct the normalising gauge by induction on the height of a positive root. Let A ( 0 ) A^{(0)} A ( 0 ) 1 , … , h − 1 1,\ldots,h-1 1 , … , h − 1 A ( h − 1 ) A^{(h-1)} A ( h − 1 ) h h h
For each pair j < i j<i j < i i − j = h i-j=h i − j = h h h h A ( h − 1 ) A^{(h-1)} A ( h − 1 ) q j i ∈ W q_{ji}\in W q j i ∈ W α j i , β j i ∈ C \alpha_{ji},\beta_{ji}\in\mathbb{C} α j i , β j i ∈ C p j i ∈ V p_{ji}\in V p j i ∈ V
q j i + L λ j i ( p j i ) = α j i z − 2 + β j i ρ λ j i . q_{ji}+L_{\lambda_{ji}}(p_{ji})=\alpha_{ji}z^{-2}+\beta_{ji}\rho_{\lambda_{ji}}. q j i + L λ j i ( p j i ) = α j i z − 2 + β j i ρ λ j i . Set
P h = ∑ i − j = h p j i E j i , g h = I r + P h . P_h=\sum_{i-j=h}p_{ji}E_{ji},\qquad g_h=I_r+P_h. P h = i − j = h ∑ p j i E j i , g h = I r + P h . Because P h P_h P h g h g_h g h P h ( 1 ) = 0 P_h(1)=0 P h ( 1 ) = 0 g h ( 1 ) = I r g_h(1)=I_r g h ( 1 ) = I r
Reduce the gauge action modulo n ( h + 1 ) \mathfrak{n}^{(h+1)} n ( h + 1 ) h h h 2 h ≥ h + 1 2h\ge h+1 2 h ≥ h + 1 (12) shows that commutators of P h P_h P h h h h h h h Lemma 5.6 . Thus g h g_h g h h h h
Define recursively A ( h ) = A ( h − 1 ) ⋅ g h A^{(h)}=A^{(h-1)}\cdot g_h A ( h ) = A ( h − 1 ) ⋅ g h r − 1 r-1 r − 1
g = g 1 g 2 ⋯ g r − 1 ∈ U 1 g=g_1g_2\cdots g_{r-1}\in U_1 g = g 1 g 2 ⋯ g r − 1 ∈ U 1 normalises every positive-root coefficient. No convergence issue occurs: the number of heights is finite, and each factor is a holomorphic unipotent matrix on Δ \Delta Δ S μ S_\mu S μ
Proposition 5.8. If two elements of S μ S_\mu S μ U 1 U_1 U 1
Proof. Let A A A A ~ \widetilde{A} A g ∈ U 1 g\in U_1 g ∈ U 1 A ⋅ g = A ~ A\cdot g=\widetilde{A} A ⋅ g = A g g g g ≠ I r g\ne I_r g = I r h h h g − I r g-I_r g − I r
n ( h ) / n ( h + 1 ) \mathfrak{n}^{(h)}/\mathfrak{n}^{(h+1)} n ( h ) / n ( h + 1 ) is non-zero. Write this image as
P h = ∑ i − j = h p j i E j i . P_h=\sum_{i-j=h}p_{ji}E_{ji}. P h = i − j = h ∑ p j i E j i . Since g ( 1 ) = I r g(1)=I_r g ( 1 ) = I r p j i p_{ji} p j i V V V n ( h ) \mathfrak{n}^{(h)} n ( h ) A A A A ~ \widetilde{A} A n ( h + 1 ) \mathfrak{n}^{(h+1)} n ( h + 1 ) (12) and yields
q ~ j i − q j i = L λ j i ( p j i ) ( i − j = h ) . \widetilde{q}_{ji}-q_{ji}=L_{\lambda_{ji}}(p_{ji})\qquad(i-j=h). q j i − q j i = L λ j i ( p j i ) ( i − j = h ) . The left-hand side lies in C z − 2 ⊕ C ρ λ j i \mathbb{C}z^{-2}\oplus\mathbb{C}\rho_{\lambda_{ji}} C z − 2 ⊕ C ρ λ j i L λ j i ( V ) L_{\lambda_{ji}}(V) L λ j i ( V ) L λ j i L_{\lambda_{ji}} L λ j i p j i = 0 p_{ji}=0 p j i = 0 h h h h h h g = I r g=I_r g = I r A = A ~ A=\widetilde{A} A = A
Theorem 5.9. For every residue vector μ ∈ C r \mu\in\mathbb{C}^r μ ∈ C r
Θ μ : S μ ( C ) × U 1 ( C ) ⟶ A μ ( C ) , ( σ , g ) ⟼ σ ⋅ g − 1 , \Theta_\mu:S_\mu(\mathbb{C})\times U_1(\mathbb{C})\longrightarrow A_\mu(\mathbb{C}),\qquad(\sigma,g)\longmapsto\sigma\cdot g^{-1}, Θ μ : S μ ( C ) × U 1 ( C ) ⟶ A μ ( C ) , ( σ , g ) ⟼ σ ⋅ g − 1 , is a bijection. Equivalently, the action groupoid is equivalent to a discrete affine groupoid:
[ A μ ( C ) / U 1 ( C ) ] ≃ S μ ( C ) d i s c ≃ ( n ⊕ n ) d i s c . [A_\mu(\mathbb{C})/U_1(\mathbb{C})]\simeq S_\mu(\mathbb{C})_{\mathrm{disc}}\simeq(\mathfrak{n}\oplus\mathfrak{n})_{\mathrm{disc}}. [ A μ ( C ) / U 1 ( C )] ≃ S μ ( C ) disc ≃ ( n ⊕ n ) disc . Thus every orbit meets the slice in a single point and every stabiliser is trivial.
Proof. Existence and uniqueness are Propositions 5.7 and 5.8. If A ⋅ g = σ A\cdot g=\sigma A ⋅ g = σ A A A A = σ ⋅ g − 1 A=\sigma\cdot g^{-1} A = σ ⋅ g − 1 ( α j i ) j < i (\alpha_{ji})_{j<i} ( α j i ) j < i ( β j i ) j < i (\beta_{ji})_{j<i} ( β j i ) j < i n \mathfrak{n} n
Proof of the orbit classification
Proof of Theorem 1.1 . The residue functor and its essential surjectivity are established in Proposition 4.4 . For a fixed residue vector μ \mu μ [ A μ ( C ) / U 1 ( C ) ] [A_\mu(\mathbb{C})/U_1(\mathbb{C})] [ A μ ( C ) / U 1 ( C )] n ⊕ n \mathfrak{n}\oplus\mathfrak{n} n ⊕ n
Corollary 5.10. Every fully framed residue fibre is a discrete groupoid equivalent to the complex points of A C r ( r − 1 ) A_{\mathbb{C}}^{r(r-1)} A C r ( r − 1 )
Proof. This follows from Theorem 1.1 , since 2 dim n = r ( r − 1 ) 2\dim\mathfrak{n}=r(r-1) 2 dim n = r ( r − 1 )
The conclusion does not identify different strata as moduli problems. Their affine parameter spaces are abstractly isomorphic, but the universal normal-form connections and their monodromy maps depend on the residue differences. This distinction is illustrated in rank two below.
Rank-two degenerate geometry
Assume r = 2 r=2 r = 2 c c c μ 1 , μ 2 ∈ Log − ( c ) \mu_1,\mu_2\in\operatorname{Log}_{-}(c) μ 1 , μ 2 ∈ Log − ( c )
λ = μ 2 − μ 1 ∈ Z . \lambda=\mu_2-\mu_1\in\mathbb{Z}. λ = μ 2 − μ 1 ∈ Z . The normal form is
∇ α , β = d + ( μ 1 z − 1 q α , β ( z ) 0 μ 2 z − 1 ) d z , q α , β = α z − 2 + β ρ λ . (14) \nabla_{\alpha,\beta}=d+\begin{pmatrix}\mu_1z^{-1}&q_{\alpha,\beta}(z)\\0&\mu_2z^{-1}\end{pmatrix}dz,\qquad q_{\alpha,\beta}=\alpha z^{-2}+\beta\rho_\lambda. \tag*{(14)} ∇ α , β = d + ( μ 1 z − 1 0 q α , β ( z ) μ 2 z − 1 ) d z , q α , β = α z − 2 + β ρ λ . (14) Proposition 6.1. With the horizontal convention fixed in the introduction, the monodromy of (14) with respect to the standard frame at z = 1 z=1 z = 1
T α , β = c ( 1 − 2 π i ν λ ( α , β ) 0 1 ) . T_{\alpha,\beta}=c\begin{pmatrix}1&-2\pi i\nu_\lambda(\alpha,\beta)\\0&1\end{pmatrix}. T α , β = c ( 1 0 − 2 π i ν λ ( α , β ) 1 ) . where
ν λ ( α , β ) = res 0 ( q α , β ( z ) z − λ d z ) = { β , λ ≥ 0 , α , λ = − 1 , 0 , λ ≤ − 2. (15) \nu_\lambda(\alpha,\beta)=\operatorname{res}_{0}\bigl(q_{\alpha,\beta}(z)z^{-\lambda}\,dz\bigr)=
\begin{cases}
\beta,&\lambda\geq0,\\
\alpha,&\lambda=-1,\\
0,&\lambda\leq-2.
\end{cases} \tag*{(15)} ν λ ( α , β ) = res 0 ( q α , β ( z ) z − λ d z ) = ⎩ ⎨ ⎧ β , α , 0 , λ ≥ 0 , λ = − 1 , λ ≤ − 2. (15) Proof. On a simply connected sector containing the base point z = 1 z=1 z = 1 z − μ 1 z^{-\mu_1} z − μ 1 z − μ 2 z^{-\mu_2} z − μ 2 1 1 1 z = 1 z=1 z = 1 f ( 1 ) = 0 f(1)=0 f ( 1 ) = 0 Y ( 1 ) = I 2 Y(1)=I_2 Y ( 1 ) = I 2
Y ( z ) = ( z − μ 1 z − μ 1 f ( z ) 0 z − μ 2 ) . Y(z)=\begin{pmatrix}z^{-\mu_1}&z^{-\mu_1}f(z)\\0&z^{-\mu_2}\end{pmatrix}. Y ( z ) = ( z − μ 1 0 z − μ 1 f ( z ) z − μ 2 ) . The equation d Y + A Y d z = 0 dY+AY\,dz=0 d Y + A Y d z = 0
f ′ ( z ) = − q α , β ( z ) z − λ . f'(z)=-q_{\alpha,\beta}(z)z^{-\lambda}. f ′ ( z ) = − q α , β ( z ) z − λ . Since λ ∈ Z \lambda\in\mathbb{Z} λ ∈ Z f f f
− ∮ q α , β ( z ) z − λ d z = − 2 π i ν λ ( α , β ) . -\oint q_{\alpha,\beta}(z)z^{-\lambda}\,dz=-2\pi i\nu_\lambda(\alpha,\beta). − ∮ q α , β ( z ) z − λ d z = − 2 π i ν λ ( α , β ) . Both diagonal factors acquire the same multiplier c c c Y Y Y λ = m ≥ 0 \lambda=m\geq0 λ = m ≥ 0
q α , β z − λ = α z − m − 2 + β z − 1 , q_{\alpha,\beta}z^{-\lambda}=\alpha z^{-m-2}+\beta z^{-1}, q α , β z − λ = α z − m − 2 + β z − 1 , so the residue is β \beta β λ = − 1 \lambda=-1 λ = − 1
q α , β z − λ = α z − 1 + β , q_{\alpha,\beta}z^{-\lambda}=\alpha z^{-1}+\beta, q α , β z − λ = α z − 1 + β , and the residue is α \alpha α λ ≤ − 2 \lambda\leq-2 λ ≤ − 2 − 1 -1 − 1
Corollary 6.2. Every repeated-eigenvalue residue stratum is the affine plane A α , β 2 \mathbb{A}^2_{\alpha,\beta} A α , β 2 but its Jordan stratification depends on λ \lambda λ
(i) if λ ≥ 0 \lambda\geq0 λ ≥ 0 { β ≠ 0 } \{\beta\neq0\} { β = 0 }
(ii) if λ = − 1 \lambda=-1 λ = − 1 { α ≠ 0 } \{\alpha\neq0\} { α = 0 }
(iii) if λ ≤ − 2 \lambda\leq-2 λ ≤ − 2
Thus isomorphic affine orbit spaces may carry different universal connections and different monodromy maps.
Proof. The affine-plane statement is Theorem 1.1 in rank two. By Proposition 6.1 , the monodromy is non-scalar if and only if ν λ ( α , β ) ≠ 0 \nu_\lambda(\alpha,\beta)\neq0 ν λ ( α , β ) = 0 (15) identifies this condition with β ≠ 0 \beta\neq0 β = 0 λ ≥ 0 \lambda\geq0 λ ≥ 0 α ≠ 0 \alpha\neq0 α = 0 λ = − 1 \lambda=-1 λ = − 1 λ ≤ − 2 \lambda\leq-2 λ ≤ − 2
The last case does not contradict the existence of Jordan monodromy with eigenvalue c c c λ ≤ − 2 \lambda\leq-2 λ ≤ − 2 Remark 3.3 .
Descent from the ordered frame
The geometry of invariant flags, Borel quotients and Springer fibres used below is classical; see Springer [27 ] , and compare the parahoric Riemann–Hilbert description of Boalch [5 ] . This section identifies what is forgotten in passing from the fully framed Kontsevich–Szabó slice to flagged and unframed objects, especially for repeated spectra and spectra containing the eigenvalue 1.
Re-framing and the obstruction to intrinsic affine coordinates
The affine slice of Theorem 1.1 classifies fully framed objects. We next examine the effect of changing the ordered frame while keeping the underlying meromorphic connection fixed. This is the local analogue of passing from framed to parabolic or parahoric Riemann–Hilbert data [5 , 13 ] ; it is also the first necessary test for attaching the coordinates of the slice to an atom without a chosen framing.
Let ∇ α , β \nabla_{\alpha,\beta} ∇ α , β (1) . Replace the standard row frame e ( 1 ) \mathbf{e}(1) e ( 1 ) e ( 1 ) C \mathbf{e}(1)C e ( 1 ) C C ∈ B C \in B C ∈ B C − 1 C^{-1} C − 1
A ( z ) ⟼ C A ( z ) C − 1 . (16) A(z) \longmapsto C A(z) C^{-1}. \tag*{(16)} A ( z ) ⟼ C A ( z ) C − 1 . (16) The resulting matrix is again upper triangular, and it may then be returned to the distinguished slice by the unique framed unipotent normalisation of Theorem 5.9.
Proposition 7.1. Let D = diag ( d 1 , … , d r ) ∈ B D = \operatorname{diag}(d_1,\ldots,d_r) \in B D = diag ( d 1 , … , d r ) ∈ B e ( 1 ) D \mathbf{e}(1)D e ( 1 ) D
α j i ⟼ d j d i α j i , β j i ⟼ d j d i β j i ( j < i ) . (17) \alpha_{ji} \longmapsto\frac{d_j}{d_i}\alpha_{ji}, \qquad\beta_{ji} \longmapsto\frac{d_j}{d_i}\beta_{ji} \qquad(j < i). \tag*{(17)} α j i ⟼ d i d j α j i , β j i ⟼ d i d j β j i ( j < i ) . (17) Consequently, none of the individual affine coordinates descends to the object obtained by forgetting the ordered framing.
Proof. Since D D D M μ M_\mu M μ
D E j i D − 1 = d j d i E j i , D E_{ji} D^{-1} = \frac{d_j}{d_i} E_{ji}, D E j i D − 1 = d i d j E j i , formula (16) sends the normal form directly to another point of the same distinguished slice, with coordinates given by (17) . No further unipotent normalisation is required. Choosing D D D d j / d i ≠ 1 d_j/d_i \ne1 d j / d i = 1
Let H ≅ ( G m ) r ⊂ B H \cong(\mathbb{G}_m)^r \subset B H ≅ ( G m ) r ⊂ B
Corollary 7.2. The ring of polynomial invariants of the diagonal re-framing torus on the affine slice is trivial:
C [ n ⊕ n ] H = C , \mathbb{C}[\mathfrak{n} \oplus\mathfrak{n}]^H = \mathbb{C}, C [ n ⊕ n ] H = C , where the central scalar subgroup acts trivially and H / G m H/\mathbb{G}_m H / G m
Proof. Consider the one-parameter subgroup
D ( t ) = diag ( t − 1 , t − 2 , … , t − r ) . D(t) = \operatorname{diag}(t^{-1},t^{-2},\ldots,t^{-r}). D ( t ) = diag ( t − 1 , t − 2 , … , t − r ) . For every j < i j < i j < i α j i \alpha_{ji} α j i β j i \beta_{ji} β j i i − j i-j i − j (17) . Every non-constant monomial therefore has strictly positive weight and cannot be invariant. Thus only constants are invariant under this one-parameter subgroup, and hence under the full diagonal torus. □
The additional unipotent ambiguity can be described in rank two. Put
A α , β ( z ) = 1 z ( μ 1 0 0 μ 2 ) + ( α z − 2 + β ρ λ ( z ) ) E 12 , λ = μ 2 − μ 1 . A_{\alpha,\beta}(z) = \frac{1}{z}
\begin{pmatrix}
\mu_1 & 0 \\
0 & \mu_2
\end{pmatrix}
+ \left(\alpha z^{-2} + \beta\rho_\lambda(z)\right)E_{12},
\qquad\lambda= \mu_2 - \mu_1. A α , β ( z ) = z 1 ( μ 1 0 0 μ 2 ) + ( α z − 2 + β ρ λ ( z ) ) E 12 , λ = μ 2 − μ 1 . For s ∈ C s \in\mathbb{C} s ∈ C N s = I 2 + s E 12 N_s = I_2 + sE_{12} N s = I 2 + s E 12
Proposition 7.4. The equivalence relation induced by replacing the standard frame by e ( 1 ) N s \mathbf{e}(1)N_s e ( 1 ) N s
( α , β ) ⟼ { ( α , β ) , λ ∈ Z ≥ 0 , ( α , β + s λ ) , λ ∉ Z ≥ 0 . (\alpha,\beta)\longmapsto
\begin{cases}
(\alpha,\beta), & \lambda\in\mathbb{Z}_{\geq0},\\
(\alpha,\beta+s\lambda), & \lambda\notin\mathbb{Z}_{\geq0}.
\end{cases} ( α , β ) ⟼ { ( α , β ) , ( α , β + s λ ) , λ ∈ Z ≥ 0 , λ ∈ / Z ≥ 0 . In particular, when λ ∉ Z ≥ 0 \lambda\notin\mathbb{Z}_{\geq0} λ ∈ / Z ≥ 0 β \beta β
Proof. A direct calculation gives
N s A α , β ( z ) N s − 1 = A α , β ( z ) + s λ z − 1 E 12 . N_sA_{\alpha,\beta}(z)N_s^{-1}=A_{\alpha,\beta}(z)+s\lambda z^{-1}E_{12}. N s A α , β ( z ) N s − 1 = A α , β ( z ) + s λ z − 1 E 12 . It remains to project s λ z − 1 s\lambda z^{-1} s λ z − 1 λ ∉ Z ≥ 0 \lambda\notin\mathbb{Z}_{\geq0} λ ∈ / Z ≥ 0 ρ λ = z − 1 \rho_\lambda=z^{-1} ρ λ = z − 1 β \beta β s λ s\lambda s λ λ = 0 \lambda=0 λ = 0 λ = m ≥ 1 \lambda=m\geq1 λ = m ≥ 1
z − 1 = L m ( z m − 1 m ) , z^{-1}=L_m\left(\frac{z^m-1}{m}\right), z − 1 = L m ( m z m − 1 ) , with ( z m − 1 ) / m (z^m-1)/m ( z m − 1 ) / m z = 1 z=1 z = 1
Corollary 7.5. In rank two, after allowing upper triangular re-framing, the affine coordinates are never canonical. If λ ∉ Z ≥ 0 \lambda\notin\mathbb{Z}_{\geq0} λ ∈ / Z ≥ 0 β = 0 \beta=0 β = 0 α = 0 \alpha=0 α = 0 α ≠ 0 \alpha\ne0 α = 0 λ ∈ Z ≥ 0 \lambda\in\mathbb{Z}_{\geq0} λ ∈ Z ≥ 0 ( α , β ) (\alpha,\beta) ( α , β )
Proof. Combine the unipotent translation of Proposition 7.4 with the common weight-one diagonal action of Proposition 7.1 . In the non-resonant range the additive parameter sets β \beta β
The results above show that the first descent problem has no coordinatewise solution. The coordinates ( α , β ) (\alpha,\beta) ( α , β )
The flagged quotient and its structural interpretation
Fix a residue vector μ \mu μ
S μ = n ⊕ n S_\mu=\mathfrak{n}\oplus\mathfrak{n} S μ = n ⊕ n for the affine normal-form slice of Theorem 1.1 . The passage from a basis to the full flag it spans is classical, as is the appearance of parabolic or Borel data on the Betti side of the Riemann–Hilbert correspondence; compare [14 , 13 , 5 ] . What is specific here is the induced action on the affine slice and its interaction with resonance. The construction of § 7.1 assigns to each C ∈ B C\in B C ∈ B x ∈ S μ x\in S_\mu x ∈ S μ C C C C C C
B × S μ ⟶ S μ , ( C , x ) ⟼ C ⋆ x . B\times S_\mu\longrightarrow S_\mu,\qquad(C,x)\longmapsto C\star x. B × S μ ⟶ S μ , ( C , x ) ⟼ C ⋆ x . The central scalar subgroup G m ⊂ B G_m\subset B G m ⊂ B
Proposition 7.6. For every fixed residue vector μ \mu μ ⋆ \star ⋆ B B B S μ S_\mu S μ n n n h h h C C C h h h (17) .
Proof. Conjugation by a constant element of B B B C C C S μ S_\mu S μ z − 1 z^{-1} z − 1 μ i − μ j \mu_i-\mu_j μ i − μ j C C C B × S μ B\times S_\mu B × S μ C 1 C 2 C_1C_2 C 1 C 2 C 2 C_2 C 2 C 1 C_1 C 1 ⋆ \star ⋆
[ n ( a ) , n ( b ) ] ⊆ n ( a + b ) , [\mathfrak{n}^{(a)},\mathfrak{n}^{(b)}]\subseteq\mathfrak{n}^{(a+b)}, [ n ( a ) , n ( b ) ] ⊆ n ( a + b ) , and the torus formula is Proposition 7.1 .
Definition 7.7. The intrinsic flagged orbit stack in the residue stratum μ \mu μ
A μ f l a g : = [ S μ / B ] . \mathcal{A}^{\mathrm{flag}}_{\mu}:=[S_{\mu}/B]. A μ flag := [ S μ / B ] . Its effective rigidification with respect to the trivially acting central scalar subgroup is
[ S μ / ( B / G m ) ] . [S_{\mu}/(B/\mathbb{G}_{m})]. [ S μ / ( B / G m )] . Proposition 7.8. The stack A μ f l a g \mathcal{A}^{\mathrm{flag}}_{\mu} A μ flag μ \mu μ
Proof. Two ordered bases induce the same full flag if and only if they differ by right multiplication by an element of B B B S μ S_{\mu} S μ ⋆ \star ⋆
Remark 7.9. The quotient by B B B z = 1 z=1 z = 1 G L r \mathrm{GL}_{r} GL r
G L r × B S μ \mathrm{GL}_{r}\times^{B}S_{\mu} GL r × B S μ and then forget the marked flag in the associated pair. The quotient
[ G L r × B S μ / G L r ] ≃ [ S μ / B ] [\mathrm{GL}_{r}\times^{B}S_{\mu}/\mathrm{GL}_{r}]\simeq[S_{\mu}/B] [ GL r × B S μ / GL r ] ≃ [ S μ / B ] still classifies the connection together with a marked invariant full flag. The scheme of possible marked flags is determined by the monodromy and is described in § 7.3 ; the stack-theoretic fibre also retains the action of the automorphism group of the connection. In particular, after fixing the ordering of distinct eigenvalues, the regular semisimple flag scheme is a point, whereas repeated eigenvalues may produce positive-dimensional Springer fibres.
We can compute the flagged quotient in rank two. Put
B e f f = B / G m ≅ G m ⋉ G a . B_{\mathrm{eff}}=B/\mathbb{G}_{m}\cong\mathbb{G}_{m}\ltimes\mathbb{G}_{a}. B eff = B / G m ≅ G m ⋉ G a . Write an element as D ( t ) N s D(t)N_s D ( t ) N s t = d 1 / d 2 t=d_1/d_2 t = d 1 / d 2 λ = μ 2 − μ 1 \lambda=\mu_2-\mu_1 λ = μ 2 − μ 1
Theorem 7.10. On the rank-two affine slice A α , β 2 A^2_{\alpha,\beta} A α , β 2
( t , s ) ⋆ ( α , β ) = { ( t α , t β ) , λ ∈ Z ≥ 0 , ( t α , t ( β + s λ ) ) , λ ∉ Z ≥ 0 . (t,s)\star(\alpha,\beta)=
\begin{cases}
(t\alpha,t\beta), & \lambda\in\mathbb{Z}_{\geq0},\\
(t\alpha,t(\beta+s\lambda)), & \lambda\notin\mathbb{Z}_{\geq0}.
\end{cases} ( t , s ) ⋆ ( α , β ) = { ( t α , tβ ) , ( t α , t ( β + s λ )) , λ ∈ Z ≥ 0 , λ ∈ / Z ≥ 0 . Consequently:
(i) if λ ∉ Z ≥ 0 \lambda\notin\mathbb{Z}_{\geq0} λ ∈ / Z ≥ 0 [ A 2 / B e f f ] ≃ [ A 1 / G m ] [A^2/B_{\mathrm{eff}}]\simeq[A^1/\mathbb{G}_{m}] [ A 2 / B eff ] ≃ [ A 1 / G m ]
(ii) if λ ∈ Z ≥ 0 \lambda\in\mathbb{Z}_{\geq0} λ ∈ Z ≥ 0 unipotent radical acts trivially, and the rigidification of the quotient by that trivially acting subgroup is
[ A α , β 2 / G m ] , [A^2_{\alpha,\beta}/\mathbb{G}_{m}], [ A α , β 2 / G m ] , with both coordinates of weight one. Its non-zero locus has geometric quotient P 1 \mathbb{P}^{1} P 1
In both cases the affine categorical quotient is a point, but the quotient stack retains non-trivial orbit and stabiliser data.
Proof. The displayed action combines Propositions 7.1 and 7.4 ; the order D ( t ) N s D(t)N_s D ( t ) N s t t t λ ∉ Z ≥ 0 \lambda\notin\mathbb{Z}_{\geq0} λ ∈ / Z ≥ 0 α \alpha α { β = 0 } \{\beta=0\} { β = 0 } α ↦ t α \alpha\mapsto t\alpha α ↦ t α λ ∈ Z ≥ 0 \lambda\in\mathbb{Z}_{\geq0} λ ∈ Z ≥ 0 Proposition 7.4 ; rigidifying it leaves the simultaneous weight-one action of G m \mathbb{G}_{m} G m m a t h b b A 2 ∖ { 0 } mathbb{A}^{2}\setminus\{0\} ma t hbb A 2 ∖ { 0 } P 1 \mathbb{P}^{1} P 1
Corollary 7.11. For the local moduli problem defined by Kontsevich and Szabó, the quotient stack [ S μ / B ] [S_\mu/B] [ S μ / B ]
Proof. The first assertion is Proposition 7.8. The second follows from the description of the further forgetful fibre in Proposition 7.17 : when repeated eigenvalues occur, the scheme of invariant full flags need not be a point, so quotienting only by B B B
Invariant monodromy flags and the unflagged descent
The full flag occurring in the original groupoid is a flag in the fibre at z = 1 z=1 z = 1
Let V V V r r r T ∈ GL ( V ) T \in\operatorname{GL}(V) T ∈ GL ( V )
c = ( c 1 , … , c r ) ∈ ( C × ) r \mathbf{c}=(c_1,\ldots,c_r)\in(\mathbb{C}^{\times})^r c = ( c 1 , … , c r ) ∈ ( C × ) r be an ordered list whose multiset is the spectrum of T T T
Definition 7.12. The ordered invariant-flag functor Flag c ( T ) \operatorname{Flag}_{\mathbf{c}}(T) Flag c ( T ) S S S
0 = F 0 ⊂ F 1 ⊂ ⋯ ⊂ F r = V ⊗ C O S 0=\mathcal{F}_0\subset\mathcal{F}_1\subset\cdots\subset\mathcal{F}_r=V\otimes_{\mathbb{C}}\mathcal{O}_S 0 = F 0 ⊂ F 1 ⊂ ⋯ ⊂ F r = V ⊗ C O S such that
( T ⊗ 1 ) ( F i ) ⊆ F i (T\otimes1)(\mathcal{F}_i)\subseteq\mathcal{F}_i ( T ⊗ 1 ) ( F i ) ⊆ F i for every i i i T ⊗ 1 T\otimes1 T ⊗ 1 F i / F i − 1 \mathcal{F}_i/\mathcal{F}_{i-1} F i / F i − 1 c i c_i c i
Proposition 7.13. The functor Flag c ( T ) \operatorname{Flag}_{\mathbf{c}}(T) Flag c ( T ) Flag ( V ) \operatorname{Flag}(V) Flag ( V )
Proof. Let
0 = F 0 u n i v ⊂ F 1 u n i v ⊂ ⋯ ⊂ F r u n i v = V ⊗ O Flag ( V ) 0=\mathcal{F}^{\mathrm{univ}}_0\subset\mathcal{F}^{\mathrm{univ}}_1\subset\cdots\subset\mathcal{F}^{\mathrm{univ}}_r=V\otimes\mathcal{O}_{\operatorname{Flag}(V)} 0 = F 0 univ ⊂ F 1 univ ⊂ ⋯ ⊂ F r univ = V ⊗ O Flag ( V ) be the universal flag. The condition that T T T F i u n i v \mathcal{F}^{\mathrm{univ}}_i F i univ
F i u n i v → T V ⊗ O Flag ( V ) ⟶ ( V ⊗ O Flag ( V ) ) / F i u n i v . \mathcal{F}^{\mathrm{univ}}_i\xrightarrow{T}V\otimes\mathcal{O}_{\operatorname{Flag}(V)}\longrightarrow\left(V\otimes\mathcal{O}_{\operatorname{Flag}(V)}\right)/\mathcal{F}^{\mathrm{univ}}_i. F i univ T V ⊗ O Flag ( V ) ⟶ ( V ⊗ O Flag ( V ) ) / F i univ . It therefore cuts out a closed subscheme. On that closed subscheme, the requirement that the induced endomorphism of the universal line quotient
L i = F i u n i v / F i − 1 u n i v \mathcal{L}_i=\mathcal{F}^{\mathrm{univ}}_i/\mathcal{F}^{\mathrm{univ}}_{i-1} L i = F i univ / F i − 1 univ be c i Id L i c_i\operatorname{Id}_{\mathcal{L}_i} c i Id L i Flag c ( T ) \operatorname{Flag}_{\mathbf{c}}(T) Flag c ( T )
Let the distinct eigenvalues of T T T a ∈ Σ a\in\Sigma a ∈ Σ
V = ⨁ a ∈ Σ V a , T ∣ V a = a I V a + N a , V=\bigoplus_{a\in\Sigma}V_a,\qquad T|_{V_a}=aI_{V_a}+N_a, V = a ∈ Σ ⨁ V a , T ∣ V a = a I V a + N a , where V a V_a V a a a a N a N_a N a B ( N a ) \mathcal{B}(N_a) B ( N a ) N a N_a N a V a V_a V a N a N_a N a [27 ] .
Theorem 7.14. There is an isomorphism of projective schemes
Flag c ( T ) ≅ ∏ a ∈ Σ B ( N a ) . \operatorname{Flag}_{\mathbf{c}}(T)\cong\prod_{a\in\Sigma}\mathcal{B}(N_a). Flag c ( T ) ≅ a ∈ Σ ∏ B ( N a ) . The isomorphism depends on the prescribed order c \mathbf{c} c
Proof. For each a ∈ Σ a\in\Sigma a ∈ Σ e a = p a ( T ) e_a=p_a(T) e a = p a ( T ) V V V V a V_a V a W ⊂ V ⊗ O S W\subset V\otimes\mathcal{O}_S W ⊂ V ⊗ O S T T T T T T W W W e a ∣ W e_a|_W e a ∣ W O S \mathcal{O}_S O S e a ( W ) e_a(W) e a ( W ) W W W
W = ⨁ a ∈ Σ e a ( W ) . W=\bigoplus_{a\in\Sigma}e_a(W). W = a ∈ Σ ⨁ e a ( W ) . Apply this to every member F i \mathcal{F}_i F i Flag c ( T ) ( S ) \operatorname{Flag}_{c}(T)(S) Flag c ( T ) ( S )
m a ( i ) = # { k ≤ i : c k = a } . m_a(i)=\#\{k\leq i:c_k=a\}. m a ( i ) = # { k ≤ i : c k = a } . On the line quotient F k / F k − 1 \mathcal{F}_k/\mathcal{F}_{k-1} F k / F k − 1 e a = p a ( T ) e_a=p_a(T) e a = p a ( T ) p a ( c k ) p_a(c_k) p a ( c k ) c k = a c_k=a c k = a e a F i e_a\mathcal{F}_i e a F i m a ( i ) m_a(i) m a ( i ) i i i
0 ⊂ G a , 1 ⊂ ⋯ ⊂ G a , dim V a = V a ⊗ O S 0\subset\mathcal{G}_{a,1}\subset\cdots\subset\mathcal{G}_{a,\dim V_a}=V_a\otimes\mathcal{O}_S 0 ⊂ G a , 1 ⊂ ⋯ ⊂ G a , d i m V a = V a ⊗ O S which is preserved by T ∣ V a T|_{V_a} T ∣ V a N a N_a N a
Conversely, given an N a N_a N a ( G a , k ) k (\mathcal{G}_{a,k})_k ( G a , k ) k V a ⊗ O S V_a\otimes\mathcal{O}_S V a ⊗ O S
F i = ⨁ a ∈ Σ G a , m a ( i ) , G a , 0 = 0. \mathcal{F}_i=\bigoplus_{a\in\Sigma}\mathcal{G}_{a,m_a(i)},\qquad\mathcal{G}_{a,0}=0. F i = a ∈ Σ ⨁ G a , m a ( i ) , G a , 0 = 0. This is a full flag, it is preserved by T T T T T T F i / F i − 1 \mathcal{F}_i/\mathcal{F}_{i-1} F i / F i − 1 c i c_i c i S S S
Corollary 7.15. If T T T Flag c ( T ) \operatorname{Flag}_{c}(T) Flag c ( T ) T = c I V T=cI_V T = c I V
Flag ( c , … , c ) ( T ) = Flag ( V ) . \operatorname{Flag}_{(c,\ldots,c)}(T)=\operatorname{Flag}(V). Flag ( c , … , c ) ( T ) = Flag ( V ) . If T T T
Proof. In the regular semisimple case every generalised eigenspace is one-dimensional, so every factor in Theorem 7.14 is a point. For a scalar operator every flag is invariant. A regular nilpotent operator has a unique stable full flag, namely the flag by kernels of its successive powers; its Springer fibre is a reduced point. □
We now describe the fibre of the forgetting morphism.
Definition 7.16. Let A c u n f l a g \mathcal{A}^{\mathrm{unflag}}_c A c unflag ( E , ∇ ) (E,\nabla) ( E , ∇ )
∏ i = 1 r ( t − c i ) , \prod_{i=1}^{r}(t-c_i), i = 1 ∏ r ( t − c i ) , with morphisms given by holomorphic gauges. Let A c f l a g \mathcal{A}^{\mathrm{flag}}_c A c flag ( E , ∇ , F ∙ ) (E,\nabla,\mathcal{F}_\bullet) ( E , ∇ , F ∙ ) ( E , ∇ ) (E,\nabla) ( E , ∇ ) F ∙ \mathcal{F}_\bullet F ∙ E ∣ z = 1 E|_{z=1} E ∣ z = 1 Flag c ( T ∇ ) \operatorname{Flag}_{c}(T_\nabla) Flag c ( T ∇ ) z = 1 z=1 z = 1
π f o r g : A c f l a g ⟶ A c u n f l a g . \pi_{\mathrm{forg}}:\mathcal{A}^{\mathrm{flag}}_c\longrightarrow\mathcal{A}^{\mathrm{unflag}}_c. π forg : A c flag ⟶ A c unflag . Let x = ( E , ∇ ) x=(E,\nabla) x = ( E , ∇ ) T x T_x T x E ∣ z = 1 E|_{z=1} E ∣ z = 1 Aut ( x ) \operatorname{Aut}(x) Aut ( x ) ( E , ∇ ) (E,\nabla) ( E , ∇ ) z = 1 z=1 z = 1
ev 1 : Aut ( x ) ↪ Z GL ( E ∣ 1 ) ( T x ) , \operatorname{ev}_1:\operatorname{Aut}(x)\hookrightarrow Z_{\operatorname{GL}(E|_1)}(T_x), ev 1 : Aut ( x ) ↪ Z GL ( E ∣ 1 ) ( T x ) , because an automorphism is horizontal, is determined by its value at one point, and its value commutes with monodromy.
Proposition 7.17. The 2-fibre of π f o r g \pi_{\mathrm{forg}} π forg x = ( E , ∇ ) x=(E,\nabla) x = ( E , ∇ )
[ Flag c ( T x ) / Aut ( x ) ] , [\operatorname{Flag}_{c}(T_x)/\operatorname{Aut}(x)], [ Flag c ( T x ) / Aut ( x )] , where Aut ( x ) \operatorname{Aut}(x) Aut ( x ) z = 1 z=1 z = 1
Proof. An object of the 2-fibre is a full flag in E ∣ z = 1 E|_{z=1} E ∣ z = 1 T x T_x T x c i c_i c i Flag c ( T x ) \operatorname{Flag}_{c}(T_x) Flag c ( T x ) ( E , ∇ ) (E,\nabla) ( E , ∇ ) z = 1 z=1 z = 1
Corollary 7.18. Let r = 2 r=2 r = 2 T T T ( t − c 1 ) ( t − c 2 ) (t-c_1)(t-c_2) ( t − c 1 ) ( t − c 2 )
If c 1 ≠ c 2 c_1\ne c_2 c 1 = c 2 Flag ( c 1 , c 2 ) ( T ) \operatorname{Flag}_{(c_1,c_2)}(T) Flag ( c 1 , c 2 ) ( T ) P ( ker ( T − c 1 I ) ) \mathbb{P}(\ker(T-c_1I)) P ( ker ( T − c 1 I ))
If c 1 = c 2 = c c_1=c_2=c c 1 = c 2 = c T ≠ c I T\ne cI T = c I T = c I + N T=cI+N T = c I + N N ≠ 0 N\ne0 N = 0 N 2 = 0 N^2=0 N 2 = 0
Flag ( c , c ) ( T ) = P ( ker N ) \operatorname{Flag}_{(c,c)}(T)=\mathbb{P}(\ker N) Flag ( c , c ) ( T ) = P ( ker N ) is a single reduced point.
(iii) If T = c I T=cI T = c I
Flag ( c , c ) ( T ) = P 1 . \operatorname{Flag}_{(c,c)}(T)=\mathbb{P}^{1}. Flag ( c , c ) ( T ) = P 1 . Consequently, in rank two the choice of invariant flag is non-trivial only on the scalar-monodromy locus; there the parameter scheme before dividing by automorphisms is P 1 \mathbb{P}^{1} P 1
Proof. A full flag in dimension two is a line L ⊂ V L\subset V L ⊂ V T ∣ L = c 1 I L T|_L=c_1I_L T ∣ L = c 1 I L c 1 ≠ c 2 c_1\ne c_2 c 1 = c 2 c 1 c_1 c 1 c 1 = c 2 = c c_1=c_2=c c 1 = c 2 = c T ≠ c I T\ne cI T = c I N = T − c I N=T-cI N = T − c I T T T c c c T = c I T=cI T = c I
The invariant-flag scheme is determined by the monodromy on the framed fibre. Requiring a constant flag in a chosen trivialisation to be preserved by every Laurent coefficient of a connection matrix would be a strictly stronger, gauge-dependent condition and is not the forgetting problem attached to the groupoid of Kontsevich and Szabó.
The rigidification torsor of a residual constituent
We next determine the ordered-framing ambiguity before comparing the local connection with a global atom. Let V V V r r r T ∈ GL ( V ) T\in\operatorname{GL}(V) T ∈ GL ( V )
c = ( c 1 , … , c r ) ∈ ( C × ) r \mathbf{c}=(c_1,\ldots,c_r)\in(\mathbb{C}^{\times})^r c = ( c 1 , … , c r ) ∈ ( C × ) r be an ordering of the eigenvalues of T T T
B c = { b ∈ B : b i i = c i for every i } . B_{\mathbf{c}}=\{b\in B:b_{ii}=c_i\text{ for every }i\}. B c = { b ∈ B : b ii = c i for every i } . Definition 7.19. The ordered Kontsevich–Szabó frame scheme of T T T
Fr c ( T ) = { ϕ ∈ Isom ( C r , V ) : ϕ − 1 T ϕ ∈ B c } . \operatorname{Fr}_{\mathbf{c}}(T)=\{\phi\in\operatorname{Isom}(\mathbb{C}^r,V):\phi^{-1}T\phi\in B_{\mathbf{c}}\}. Fr c ( T ) = { ϕ ∈ Isom ( C r , V ) : ϕ − 1 T ϕ ∈ B c } . The morphism
q T : Fr c ( T ) ⟶ Flag c ( T ) q_T:\operatorname{Fr}_{\mathbf{c}}(T)\longrightarrow\operatorname{Flag}_{\mathbf{c}}(T) q T : Fr c ( T ) ⟶ Flag c ( T ) sends an ordered frame to the full flag which it defines.
Proposition 7.20. The scheme Fr c ( T ) \operatorname{Fr}_{\mathbf{c}}(T) Fr c ( T ) Flag c ( T ) \operatorname{Flag}_{\mathbf{c}}(T) Flag c ( T )
Isom ( C r , V ) ⟶ Flag ( V ) . \operatorname{Isom}(\mathbb{C}^r,V)\longrightarrow\operatorname{Flag}(V). Isom ( C r , V ) ⟶ Flag ( V ) . Consequently, Fr c ( T ) \operatorname{Fr}_{\mathbf{c}}(T) Fr c ( T ) Isom ( C r , V ) \operatorname{Isom}(\mathbb{C}^r,V) Isom ( C r , V )
q T : Fr c ( T ) ⟶ Flag c ( T ) q_T:\operatorname{Fr}_{\mathbf{c}}(T)\longrightarrow\operatorname{Flag}_{\mathbf{c}}(T) q T : Fr c ( T ) ⟶ Flag c ( T ) is a principal right B B B
Fr c ( T ) / B ≅ Flag c ( T ) . \operatorname{Fr}_{\mathbf{c}}(T)/B\cong\operatorname{Flag}_{\mathbf{c}}(T). Fr c ( T ) / B ≅ Flag c ( T ) . Proof. An isomorphism ϕ : C r → V \phi:\mathbb{C}^r\to V ϕ : C r → V ϕ ( F ∙ s t d ) \phi(F^{\mathrm{std}}_\bullet) ϕ ( F ∙ std ) ϕ − 1 T ϕ \phi^{-1}T\phi ϕ − 1 T ϕ B c B_{\mathbf{c}} B c T T T c 1 , … , c r c_1,\ldots,c_r c 1 , … , c r Fr c ( T ) \operatorname{Fr}_{\mathbf{c}}(T) Fr c ( T ) Flag c ( T ) \operatorname{Flag}_{\mathbf{c}}(T) Flag c ( T ) B B B B B B
Combining Proposition 7.20 and Theorem 7.14 gives the intrinsic geometry of the rigidification space.
Corollary 7.21. With the notation of Theorem 7.14, there is a principal B B B
Fr c ( T ) ⟶ ∏ a ∈ Σ B ( N a ) . \operatorname{Fr}_{\mathbf{c}}(T)\longrightarrow\prod_{a\in\Sigma}\mathcal{B}(N_a). Fr c ( T ) ⟶ a ∈ Σ ∏ B ( N a ) . It is a B B B B ( N a ) \mathcal{B}(N_a) B ( N a ) N a N_a N a T = c I T=cI T = c I
Fr ( c , … , c ) ( T ) = Isom ( C r , V ) \operatorname{Fr}_{(c,\ldots,c)}(T)=\operatorname{Isom}(\mathbb{C}^r,V) Fr ( c , … , c ) ( T ) = Isom ( C r , V ) and the quotient by B B B V V V
Proof. Pull back the principal B B B
Let x = ( E , ∇ ) x=(E,\nabla) x = ( E , ∇ ) T x T_x T x V = E ∣ z = 1 V=E|_{z=1} V = E ∣ z = 1 Aut ( x ) \operatorname{Aut}(x) Aut ( x ) Fr c ( T x ) \operatorname{Fr}_{\mathbf c}(T_x) Fr c ( T x ) z = 1 z=1 z = 1 B B B
Definition 7.22. The Kontsevich–Szabó rigidification stack of x x x c \mathbf c c
Rig c ( x ) = [ Fr c ( T x ) / Aut ( x ) ] . \operatorname{Rig}_{\mathbf c}(x)=[\operatorname{Fr}_{\mathbf c}(T_x)/\operatorname{Aut}(x)]. Rig c ( x ) = [ Fr c ( T x ) / Aut ( x )] . Proposition 7.23. The 2 2 2 x x x A c u n f l a g \mathcal A^{\mathrm{unflag}}_{\mathbf c} A c unflag Rig c ( x ) \operatorname{Rig}_{\mathbf c}(x) Rig c ( x ) B B B
[ Flag c ( T x ) / Aut ( x ) ] . [\operatorname{Flag}_{\mathbf c}(T_x)/\operatorname{Aut}(x)]. [ Flag c ( T x ) / Aut ( x )] . Proof. A fully framed lift of x x x V V V T x T_x T x c \mathbf c c Fr c ( T x ) \operatorname{Fr}_{\mathbf c}(T_x) Fr c ( T x ) q T x q_{T_x} q T x Aut ( x ) \operatorname{Aut}(x) Aut ( x ) B B B
The rigidification space is not, in general, merely a copy of B B B B B B B B B B B B
Isomonodromic variation of the prescribed lattice
The order-two part of the affine slice records lattice moduli that are invisible to the punctured connection and its monodromy. The rank-two family is the simplest instance of the following general construction.
Let S S S V V V R ∈ End ( V ) R\in\operatorname{End}(V) R ∈ End ( V ) a ⊂ End ( V ) \mathfrak a\subset\operatorname{End}(V) a ⊂ End ( V )
[ R , a ] ⊂ a . [R,\mathfrak a]\subset\mathfrak a. [ R , a ] ⊂ a . The algebra a \mathfrak a a A 1 , … , A m A_1,\ldots,A_m A 1 , … , A m N i ≥ 1 N_i\ge1 N i ≥ 1 A i N i = 0 A_i^{N_i}=0 A i N i = 0 ∑ i ( N i − 1 ) \sum_i(N_i-1) ∑ i ( N i − 1 ) A i N i A_i^{N_i} A i N i F : S → a F:S\to\mathfrak a F : S → a
G ( s , u ) = exp ( F ( s ) u ) G(s,u)=\exp\left(\frac{F(s)}{u}\right) G ( s , u ) = exp ( u F ( s ) ) is a finite sum, with a truncation order independent of s s s S × Δ S\times\Delta S × Δ
Theorem 8.1. The gauge transform of the constant logarithmic connection
∇ 0 = d + R u d u \nabla_0=d+\frac{R}{u}\,du ∇ 0 = d + u R d u by G G G
∇ R , F = d + d S F u + R u d u + [ R , F ] − F u 2 d u . (18) \nabla_{R,F}=d+\frac{d_SF}{u}+\frac{R}{u}\,du+\frac{[R,F]-F}{u^2}\,du. \tag*{(18)} ∇ R , F = d + u d S F + u R d u + u 2 [ R , F ] − F d u . (18) It is flat. Moreover, for every vector field ξ \xi ξ S S S
( ∇ R , F ) u ξ = u ξ + ξ ( F ) , ( ∇ R , F ) u 2 ∂ u = u 2 ∂ u + u R + [ R , F ] − F , (\nabla_{R,F})_{u}\xi=u\xi+\xi(F),\qquad
(\nabla_{R,F})_{u^2\partial_u}=u^2\partial_u+uR+[R,F]-F, ( ∇ R , F ) u ξ = u ξ + ξ ( F ) , ( ∇ R , F ) u 2 ∂ u = u 2 ∂ u + u R + [ R , F ] − F , so the standard lattice satisfies the usual F F F s ∈ S s\in S s ∈ S
∇ s = d + ( R u + ( ad R − I ) F ( s ) u 2 ) d u \nabla_s=d+\left(\frac{R}{u}+\frac{(\operatorname{ad}_R-I)F(s)}{u^2}\right)du ∇ s = d + ( u R + u 2 ( ad R − I ) F ( s ) ) d u is meromorphically gauge equivalent to d + R d u / u d+R\,du/u d + R d u / u d + R d u / u d+R\,du/u d + R d u / u
K − 2 ( s ) = ( ad R − I ) F ( s ) K_{-2}(s)=(\operatorname{ad}_R-I)F(s) K − 2 ( s ) = ( ad R − I ) F ( s ) may vary.
Proof. Put X = F / u X=F/u X = F / u G = e X G=e^X G = e X a \mathfrak{a} a ad R \operatorname{ad}_R ad R F F F d S F \mathrm{d}_S F d S F [ R , F ] [R,F] [ R , F ]
G − 1 d G = d X = d S F u − F u 2 d u . G^{-1}\mathrm{d}G=\mathrm{d}X=\frac{\mathrm{d}_S F}{u}-\frac{F}{u^2}\,\mathrm{d}u. G − 1 d G = d X = u d S F − u 2 F d u . Moreover, the adjoint exponential terminates after the first commutator:
G − 1 R G = e − ad X ( R ) = R + [ R , X ] = R + [ R , F ] u . G^{-1}RG=e^{-\operatorname{ad}_X}(R)=R+[R,X]=R+\frac{[R,F]}{u}. G − 1 R G = e − ad X ( R ) = R + [ R , X ] = R + u [ R , F ] . because [ X , [ R , X ] ] = 0 [X,[R,X]]=0 [ X , [ R , X ]] = 0 (3) to ∇ 0 = d + R d u / u \nabla_0=\mathrm{d}+R\,\mathrm{d}u/u ∇ 0 = d + R d u / u
G − 1 ∇ 0 G = d + G − 1 d G + G − 1 R G u d u , G^{-1}\nabla_0G=\mathrm{d}+G^{-1}\mathrm{d}G+\frac{G^{-1}RG}{u}\,\mathrm{d}u, G − 1 ∇ 0 G = d + G − 1 d G + u G − 1 R G d u , which is (18) .
Gauge invariance gives zero curvature on the punctured product. Since the curvature of (18) is a meromorphic two-form, it vanishes identically. Equivalently, one may verify the mixed component directly from d S 2 F = 0 \mathrm{d}_S^2F=0 d S 2 F = 0 [ F , d S F ] = 0 [F,\mathrm{d}_SF]=0 [ F , d S F ] = 0 [ R , d S F ] = d S [ R , F ] [R,\mathrm{d}_SF]=\mathrm{d}_S[R,F] [ R , d S F ] = d S [ R , F ] ξ \xi ξ S S S u ξ u\xi u ξ
( ∇ R , F ) u ξ = u ξ + ξ ( F ) , (\nabla_{R,F})_{u\xi}=u\xi+\xi(F), ( ∇ R , F ) u ξ = u ξ + ξ ( F ) , which preserves the standard lattice. Contraction with u 2 ∂ u u^2\partial_u u 2 ∂ u
( ∇ R , F ) u 2 ∂ u = u 2 ∂ u + u R + [ R , F ] − F , (\nabla_{R,F})_{u^2\partial_u}=u^2\partial_u+uR+[R,F]-F, ( ∇ R , F ) u 2 ∂ u = u 2 ∂ u + u R + [ R , F ] − F , which also preserves it. These are the pole conditions in the definition of an F F F [12 Definition 2.2] .
Finally, for fixed s s s G ( s , u ) G(s,u) G ( s , u ) d + R d u / u \mathrm{d}+R\,\mathrm{d}u/u d + R d u / u s s s u − 2 d u u^{-2}\mathrm{d}u u − 2 d u
[ R , F ( s ) ] − F ( s ) = ( ad R − I ) F ( s ) , [R,F(s)]-F(s)=(\operatorname{ad}_R-I)F(s), [ R , F ( s )] − F ( s ) = ( ad R − I ) F ( s ) , which may vary holomorphically, as claimed.
Corollary 8.2. Assume that R = M μ R=M_\mu R = M μ a ⊂ n \mathfrak{a}\subset\mathfrak{n} a ⊂ n
F ( s ) = ∑ j < i f j i ( s ) E j i . F(s)=\sum_{j<i}f_{ji}(s)E_{ji}. F ( s ) = j < i ∑ f j i ( s ) E j i . Then every fibre of (18) is already in the distinguished normal form, with
α j i ( s ) = ( μ j − μ i − 1 ) f j i ( s ) = − ( λ j i + 1 ) f j i ( s ) , β j i ( s ) = 0. \alpha_{ji}(s)=(\mu_j-\mu_i-1)f_{ji}(s)=-(\lambda_{ji}+1)f_{ji}(s),\qquad\beta_{ji}(s)=0. α j i ( s ) = ( μ j − μ i − 1 ) f j i ( s ) = − ( λ j i + 1 ) f j i ( s ) , β j i ( s ) = 0. Equivalently, its fully framed classifying morphism is
S ⟶ n ⊕ n , s ⟼ ( ( ad M μ − I ) F ( s ) , 0 ) . S\longrightarrow\mathfrak{n}\oplus\mathfrak{n},\qquad s\longmapsto\bigl((\operatorname{ad}_{M_\mu}-I)F(s),0\bigr). S ⟶ n ⊕ n , s ⟼ ( ( ad M μ − I ) F ( s ) , 0 ) . Proof. For j < i j<i j < i
[ M μ , E j i ] = ( μ j − μ i ) E j i . [M_\mu,E_{ji}]=(\mu_j-\mu_i)E_{ji}. [ M μ , E j i ] = ( μ j − μ i ) E j i . The fibre contains no off-diagonal term other than the displayed coefficient of u − 2 d u u^{-2}\mathrm{d}u u − 2 d u
Corollary 8.3. Let R = μ I V R=\mu I_V R = μ I V
∇ μ , F = d + d S F u + μ I V u d u − F u 2 d u , \nabla_{\mu,F}=\mathrm{d}+\frac{\mathrm{d}_SF}{u}+\frac{\mu I_V}{u}\,\mathrm{d}u-\frac{F}{u^2}\,\mathrm{d}u, ∇ μ , F = d + u d S F + u μ I V d u − u 2 F d u , and its classifying map is s ↦ ( − F ( s ) , 0 ) s\mapsto(-F(s),0) s ↦ ( − F ( s ) , 0 ) e − 2 π i μ e^{-2\pi i\mu} e − 2 π i μ
Proof. For R = μ I V R=\mu I_V R = μ I V [ R , F ] = 0 [R,F]=0 [ R , F ] = 0 (18) and into the classifying map of Corollary 8.2.
In particular, if N 1 , … , N m N_1,\ldots,N_m N 1 , … , N m
F ( t 1 , … , t m ) = ∑ a = 1 m t a N a , F(t_1,\ldots,t_m)=\sum_{a=1}^{m}t_aN_a, F ( t 1 , … , t m ) = a = 1 ∑ m t a N a , then
( t 1 , … , t m ) ⟼ ( − ∑ a = 1 m t a N a , 0 ) (t_1,\ldots,t_m)\longmapsto\left(-\sum_{a=1}^{m}t_aN_a,0\right) ( t 1 , … , t m ) ⟼ ( − a = 1 ∑ m t a N a , 0 ) is the classifying map. The original rank-two example is recovered by taking R = 0 R=0 R = 0 m = 1 m=1 m = 1 N 1 = E 12 N_1=E_{12} N 1 = E 12
∇ = d + E 12 u d t + t E 12 u 2 d u . \nabla=d+\frac{E_{12}}{u}\,dt+\frac{tE_{12}}{u^2}\,du. ∇ = d + u E 12 d t + u 2 t E 12 d u . Its monodromy is the identity for every t t t ( α , β ) = ( − t , 0 ) (\alpha,\beta)=(-t,0) ( α , β ) = ( − t , 0 )
The theorem concerns the prescribed lattice and, in the displayed coordinates, the fully framed normal-form slice. After the frame or the flag is forgotten, distinct values of F ( s ) F(s) F ( s ) ( ad R − I ) F ( s ) (\operatorname{ad}_R-I)F(s) ( ad R − I ) F ( s )
Remark 8.4. If a \mathfrak{a} a ad R \operatorname{ad}_R ad R λ \lambda λ ( λ − 1 ) F λ (\lambda-1)F_\lambda ( λ − 1 ) F λ λ = 1 \lambda=1 λ = 1 R R R E j i E_{ji} E j i ad R \operatorname{ad}_R ad R 1 1 1 λ j i = μ i − μ j = − 1 \lambda_{ji}=\mu_i-\mu_j=-1 λ j i = μ i − μ j = − 1 z − 2 z^{-2} z − 2 Proposition 6.1 .
Relative normal forms after triangular reduction
Regular-singular connections with parameters admit a rich relative theory; see, for example, the algebraic Riemann–Hilbert equivalence of Phung Ho Hai, dos Santos and Pham Thanh Tam [11 ] , the relative complex theory of [10 ] , and the logarithmic decomposition results of [28 ] . The result below is more elementary but also more rigid: the triangular lattice reduction, residue vector and ordered frame are fixed in advance, and we represent the remaining unipotent quotient.
The prescribed lattice may vary even in a meromorphically isotrivial family with constant monodromy, as § 8 shows. Once a relative triangular lattice reduction has been produced, however, the normalisation is finite, coefficientwise and compatible with arbitrary base change. We work over an arbitrary complex analytic space, possibly non-reduced.
Let S S S
p : S × Δ ⟶ S p:S\times\Delta\longrightarrow S p : S × Δ ⟶ S be the projection. Write
H S = p ∗ O S × Δ , V S = ker ( ev 1 : H S → O S ) , W S = z − 2 H S . \mathcal{H}_S=p_*\mathcal{O}_{S\times\Delta},\qquad
\mathcal{V}_S=\ker(\operatorname{ev}_1:\mathcal{H}_S\to\mathcal{O}_S),\qquad
\mathcal{W}_S=z^{-2}\mathcal{H}_S. H S = p ∗ O S × Δ , V S = ker ( ev 1 : H S → O S ) , W S = z − 2 H S . Thus a section of V S \mathcal{V}_S V S P ( s , z ) P(s,z) P ( s , z ) P ( s , 1 ) = 0 P(s,1)=0 P ( s , 1 ) = 0 W S \mathcal{W}_S W S S × { 0 } S\times\{0\} S × { 0 } λ ∈ C \lambda\in\mathbb{C} λ ∈ C
L λ , S : V S ⟶ W S , L λ , S ( P ) = ∂ z P − λ z − 1 P . L_{\lambda,S}:\mathcal{V}_S\longrightarrow\mathcal{W}_S,\qquad
L_{\lambda,S}(P)=\partial_zP-\lambda z^{-1}P. L λ , S : V S ⟶ W S , L λ , S ( P ) = ∂ z P − λ z − 1 P . Lemma 9.1. For every complex analytic space S S S λ ∈ C \lambda\in\mathbb{C} λ ∈ C L λ , S L_{\lambda,S} L λ , S O S \mathcal{O}_S O S
W S = L λ , S ( V S ) ⊕ O S z − 2 ⊕ O S z λ . (19) \mathcal{W}_S=L_{\lambda,S}(\mathcal{V}_S)\oplus\mathcal{O}_S z^{-2}\oplus\mathcal{O}_S z^\lambda. \tag*{(19)} W S = L λ , S ( V S ) ⊕ O S z − 2 ⊕ O S z λ . (19) The decomposition is functorial under arbitrary morphisms of complex analytic spaces.
Proof. The assertion is local on S S S Q Q Q W S \mathcal{W}_S W S T ⊂ S T\subset S T ⊂ S
Q = q − 2 z − 2 + ∑ k ≥ − 1 q k z k , q k ∈ O S ( T ) . Q=q_{-2}z^{-2}+\sum_{k\geq-1}q_kz^k,\qquad q_k\in\mathcal{O}_S(T). Q = q − 2 z − 2 + k ≥ − 1 ∑ q k z k , q k ∈ O S ( T ) . The coefficient q − 2 q_{-2} q − 2 λ ∉ Z ≥ 0 \lambda\notin\mathbb{Z}_{\geq0} λ ∈ / Z ≥ 0
P 0 = ∑ n ≥ 0 q n − 1 n − λ z n , β = − λ P 0 ( 1 ) , P = P 0 + β λ . P_0=\sum_{n\geq0}\frac{q_{n-1}}{n-\lambda}z^n,\qquad
\beta=-\lambda P_0(1),\qquad
P=P_0+\frac{\beta}{\lambda}. P 0 = n ≥ 0 ∑ n − λ q n − 1 z n , β = − λ P 0 ( 1 ) , P = P 0 + λ β . To justify convergence when T T T T T T T → Ω T\to\Omega T → Ω Ω \Omega Ω Q Q Q Ω × Δ \Omega\times\Delta Ω × Δ Ω 0 ⋐ Ω \Omega_0 \Subset\Omega Ω 0 ⋐ Ω 1 < r 0 < R 0 1<r_0<R_0 1 < r 0 < R 0 { ∣ z ∣ ≤ R 0 } ⋐ Δ \{|z|\le R_0\}\Subset\Delta { ∣ z ∣ ≤ R 0 } ⋐ Δ Ω 0 × { ∣ z ∣ ≤ R 0 } \Omega_0\times\{|z|\le R_0\} Ω 0 × { ∣ z ∣ ≤ R 0 } C R 0 − n CR_0^{-n} C R 0 − n P 0 P_0 P 0 Ω 0 × { ∣ z ∣ ≤ r 0 } \Omega_0\times\{|z|\le r_0\} Ω 0 × { ∣ z ∣ ≤ r 0 } Q Q Q T T T T T T H S ( T ) \mathcal{H}_S(T) H S ( T ) P ( 1 ) = 0 P(1)=0 P ( 1 ) = 0
Q = q − 2 z − 2 + L λ , S ( P ) + β z − 1 . Q=q_{-2}z^{-2}+L_{\lambda,S}(P)+\beta z^{-1}. Q = q − 2 z − 2 + L λ , S ( P ) + β z − 1 . This expression is unique by the same coefficient calculation as in Lemma 5.5.
Now let λ = m ∈ Z ≥ 0 \lambda=m\in\mathbb{Z}_{\geq0} λ = m ∈ Z ≥ 0
β = q m − 1 , p n = q n − 1 n − m ( n ≠ m ) , p m = − ∑ n ≠ m p n , \beta=q_{m-1},\qquad p_n=\frac{q_{n-1}}{n-m}\quad(n\ne m),\qquad p_m=-\sum_{n\ne m}p_n, β = q m − 1 , p n = n − m q n − 1 ( n = m ) , p m = − n = m ∑ p n , and put P = ∑ n ≥ 0 p n z n P=\sum_{n\geq0}p_nz^n P = ∑ n ≥ 0 p n z n p m p_m p m P ( 1 ) = 0 P(1)=0 P ( 1 ) = 0
Q = q − 2 z − 2 + L m , S ( P ) + β z m − 1 . Q=q_{-2}z^{-2}+L_{m,S}(P)+\beta z^{m-1}. Q = q − 2 z − 2 + L m , S ( P ) + β z m − 1 . The coefficient of z m − 1 z^{m-1} z m − 1 L m , S L_{m,S} L m , S L λ , S ( P ) = 0 L_{\lambda,S}(P)=0 L λ , S ( P ) = 0 λ = m ∈ Z ≥ 0 \lambda=m\in\mathbb{Z}_{\geq0} λ = m ∈ Z ≥ 0 a z m az^m a z m a ∈ O S ( T ) a\in\mathcal{O}_S(T) a ∈ O S ( T ) P ( 1 ) = 0 P(1)=0 P ( 1 ) = 0 a = 0 a=0 a = 0
All these formulae use only addition, multiplication by fixed complex scalars, extraction of Taylor coefficients, and evaluation at z = 1 z=1 z = 1 S ′ → S S'\to S S ′ → S □ \square □
Fix a residue vector μ = ( μ 1 , … , μ r ) ∈ C r \boldsymbol{\mu}=(\mu_1,\ldots,\mu_r)\in\mathbb{C}^r μ = ( μ 1 , … , μ r ) ∈ C r S S S A μ ( S ) \mathcal{A}_{\boldsymbol{\mu}}(S) A μ ( S )
d z + ( M μ z + Q ( s , z ) ) d z , Q ∈ Γ ( S × Δ , z − 2 O S × Δ ⊗ n ) , d_z+\left(\frac{M_{\boldsymbol{\mu}}}{z}+Q(s,z)\right)dz,\qquad Q\in\Gamma(S\times\Delta,z^{-2}\mathcal{O}_{S\times\Delta}\otimes\mathfrak{n}), d z + ( z M μ + Q ( s , z ) ) d z , Q ∈ Γ ( S × Δ , z − 2 O S × Δ ⊗ n ) , where d z d_z d z
U 1 ( S ) = { g : S × Δ → U holomorphic : g ∣ z = 1 = I r } U_1(S)=\{g:S\times\Delta\to U\text{ holomorphic}:g|_{z=1}=I_r\} U 1 ( S ) = { g : S × Δ → U holomorphic : g ∣ z = 1 = I r } act by the relative form of (3.1). Finally, let S μ ( S ) \mathcal{S}_{\boldsymbol{\mu}}(S) S μ ( S )
d z + [ M μ z + ∑ j < i ( α j i ( s ) z − 2 + β j i ( s ) ρ λ j i ( z ) ) E j i ] d z , (20) d_z+\left[\frac{M_{\boldsymbol{\mu}}}{z}+\sum_{j<i}\left(\alpha_{ji}(s)z^{-2}+\beta_{ji}(s)\rho_{\lambda_{ji}}(z)\right)E_{ji}\right]dz, \tag*{(20)} d z + [ z M μ + j < i ∑ ( α j i ( s ) z − 2 + β j i ( s ) ρ λ j i ( z ) ) E j i ] d z , (20) where α j i , β j i ∈ Γ ( S , O S ) \alpha_{ji},\beta_{ji}\in\Gamma(S,\mathcal{O}_S) α j i , β j i ∈ Γ ( S , O S ) λ j i = μ i − μ j \lambda_{ji}=\mu_i-\mu_j λ j i = μ i − μ j
Theorem 9.2. For every complex analytic space S S S A ∈ A μ ( S ) A\in\mathcal{A}_{\boldsymbol{\mu}}(S) A ∈ A μ ( S ) g ∈ U 1 ( S ) g\in U_1(S) g ∈ U 1 ( S ) A ⋅ g A\cdot g A ⋅ g S μ ( S ) \mathcal{S}_{\boldsymbol{\mu}}(S) S μ ( S )
α j i , β j i ∈ Γ ( S , O S ) \alpha_{ji},\beta_{ji}\in\Gamma(S,\mathcal{O}_S) α j i , β j i ∈ Γ ( S , O S ) are functorial under arbitrary base change. Namely, if N = 2 dim n = r ( r − 1 ) N=2\dim\mathfrak{n}=r(r-1) N = 2 dim n = r ( r − 1 )
π μ : A μ ⟶ A C N , \pi_{\boldsymbol{\mu}}:\mathcal{A}_{\boldsymbol{\mu}}\longrightarrow\mathbb{A}_{\mathbb{C}}^N, π μ : A μ ⟶ A C N , and the map
Θ μ , S : A C N ( S ) × U 1 ( S ) ⟶ A μ ( S ) , ( σ , g ) ⟼ σ ⋅ g − 1 , \Theta_{\boldsymbol{\mu},S}:\mathbb{A}_{\mathbb{C}}^N(S)\times U_1(S)\longrightarrow\mathcal{A}_{\boldsymbol{\mu}}(S),\qquad(\sigma,g)\longmapsto\sigma\cdot g^{-1}, Θ μ , S : A C N ( S ) × U 1 ( S ) ⟶ A μ ( S ) , ( σ , g ) ⟼ σ ⋅ g − 1 , is bijective, functorially in S S S π μ \pi_{\boldsymbol{\mu}} π μ U 1 U_1 U 1 [ A μ / U 1 ] [\mathcal{A}_{\boldsymbol{\mu}}/U_1] [ A μ / U 1 ] A C N \mathbb{A}_{\mathbb{C}}^N A C N
Proof. The argument is the absolute height induction carried out sheafwise over the base.
Existence. Work locally on an open subset T ⊂ S T\subset S T ⊂ S h h h j < i j<i j < i i − j = h i-j=h i − j = h q j i q_{ji} q j i
p j i ∈ V S ( T ) , α j i , β j i ∈ O S ( T ) p_{ji}\in\mathcal{V}_S(T),\qquad\alpha_{ji},\beta_{ji}\in\mathcal{O}_S(T) p j i ∈ V S ( T ) , α j i , β j i ∈ O S ( T ) such that
q j i + L λ j i , S ( p j i ) = α j i z − 2 + β j i ρ λ j i . q_{ji}+L_{\lambda_{ji},S}(p_{ji})=\alpha_{ji}z^{-2}+\beta_{ji}\rho_{\lambda_{ji}}. q j i + L λ j i , S ( p j i ) = α j i z − 2 + β j i ρ λ j i . Set
P h = ∑ i − j = h p j i E j i , g h = I r + P h . P_h=\sum_{i-j=h}p_{ji}E_{ji},\qquad g_h=I_r+P_h. P h = i − j = h ∑ p j i E j i , g h = I r + P h . The matrix P h P_h P h g h − 1 g_h^{-1} g h − 1 P h P_h P h T × Δ T\times\Delta T × Δ p j i ( s , 1 ) = 0 p_{ji}(s,1)=0 p j i ( s , 1 ) = 0 g h ∣ z = 1 = I r \left.g_h\right|_{z=1}=I_r g h ∣ z = 1 = I r (12) shows that the height-h h h L λ j i , S L_{\lambda_{ji},S} L λ j i , S g h g_h g h h h h g 1 ⋯ g r − 1 g_1\cdots g_{r-1} g 1 ⋯ g r − 1 T T T
The scalar splitting is unique, hence the gauges constructed on two overlapping open subsets coincide there. They therefore glue to a global element of U 1 ( S ) U_1(S) U 1 ( S ) α j i , β j i \alpha_{ji},\beta_{ji} α j i , β j i O S \mathcal{O}_S O S
Uniqueness. Let g ∈ U 1 ( S ) g\in U_1(S) g ∈ U 1 ( S ) S S S ( n ( 1 ) / n ( 2 ) ) ⊗ O S (\mathfrak{n}^{(1)}/\mathfrak{n}^{(2)})\otimes\mathcal{O}_S ( n ( 1 ) / n ( 2 ) ) ⊗ O S
q ~ j i − q j i = L λ j i , S ( p j i ) . \widetilde{q}_{ji}-q_{ji}=L_{\lambda_{ji},S}(p_{ji}). q j i − q j i = L λ j i , S ( p j i ) . The left-hand side lies in the complementary summand of (19) , whereas the right-hand side lies in the image summand. Directness and injectivity force both sides to vanish. Suppose inductively that the image of g − I r g-I_r g − I r h − 1 h-1 h − 1 n ( h + 1 ) \mathfrak{n}^{(h+1)} n ( h + 1 ) h h h h h h g − I r g-I_r g − I r r − 1 r-1 r − 1 g = I r g=I_r g = I r
Base change and representability. Every operation used above—Taylor coefficient extraction, evaluation at z = 1 z=1 z = 1 n \mathfrak{n} n N = r ( r − 1 ) N=r(r-1) N = r ( r − 1 ) π μ ( A ) ∈ A C N ( S ) \pi_\mu(A)\in\mathbb{A}_{\mathbb{C}}^N(S) π μ ( A ) ∈ A C N ( S )
A = σ ⋅ g − 1 A=\sigma\cdot g^{-1} A = σ ⋅ g − 1 is unique and functorial. Hence Θ μ \Theta_\mu Θ μ
A C N × U 1 → ∼ A μ , \mathbb{A}_{\mathbb{C}}^N\times U_1\xrightarrow{\sim}\mathcal{A}_\mu, A C N × U 1 ∼ A μ , equivariant for right translation on the second factor. This is a functorial trivialisation of the U 1 U_1 U 1 π μ \pi_\mu π μ A C N \mathbb{A}_{\mathbb{C}}^N A C N
Corollary 9.3. Fix a complex normal form σ 0 ∈ S μ ( C ) \sigma_0\in S_\mu(\mathbb{C}) σ 0 ∈ S μ ( C ) A A A C \mathbb{C} C m A \mathfrak{m}_A m A σ 0 \sigma_0 σ 0 Spec a n A \operatorname{Spec}_{\mathrm{an}} A Spec an A
( n ⊕ n ) ⊗ C m A . (\mathfrak{n}\oplus\mathfrak{n})\otimes_{\mathbb{C}}\mathfrak{m}_A. ( n ⊕ n ) ⊗ C m A . Consequently, the framed deformation problem is unobstructed, has tangent space n ⊕ n \mathfrak{n}\oplus\mathfrak{n} n ⊕ n
Proof. By Theorem 9.2 , the deformation groupoid is the fibre over σ 0 \sigma_0 σ 0
A C r ( r − 1 ) ( Spec a n A ) ⟶ A C r ( r − 1 ) ( Spec a n C ) . \mathbb{A}_{\mathbb{C}}^{r(r-1)}(\operatorname{Spec}_{\mathrm{an}} A)\longrightarrow\mathbb{A}_{\mathbb{C}}^{r(r-1)}(\operatorname{Spec}_{\mathrm{an}}\mathbb{C}). A C r ( r − 1 ) ( Spec an A ) ⟶ A C r ( r − 1 ) ( Spec an C ) . Using the affine coordinates ( α , β ) (\alpha,\beta) ( α , β ) ( n ⊕ n ) ⊗ m A (\mathfrak{n}\oplus\mathfrak{n})\otimes\mathfrak{m}_A ( n ⊕ n ) ⊗ m A
Corollary 9.4. Let ( E , ∇ ) (\mathcal{E},\nabla) ( E , ∇ ) S S S S S S
a holomorphic ordered framing at z = 1 z=1 z = 1
a holomorphic triangular reduction in the prescribed lattice; and
a locally constant diagonal-residue vector μ \mu μ
Then the family determines a holomorphic classifying morphism
Φ S , μ f r : S ⟶ A C r ( r − 1 ) , \Phi^{\mathrm{fr}}_{S,\mu}:S\longrightarrow\mathbb{A}_{\mathbb{C}}^{r(r-1)}, Φ S , μ fr : S ⟶ A C r ( r − 1 ) , whose value at every complex point is the affine normal-form coordinate of Theorem 1.1 . The construction commutes with arbitrary base change.
Proof. Apply the functoriality in Theorem 9.2 after the prescribed triangular lattice reduction and diagonal normalisation. □
Theorem 9.2 represents the analytic unipotent quotient on each fixed triangular lattice-and-residue stratum, but it does not construct triangular reductions in families. The discrete relative-position data entering the lattice comparison of Kontsevich–Szabó may jump, and a global family need not admit a single residue vector or a single holomorphic invariant flag. The theorem therefore solves the relative analytic quotient once the lattice reduction has been fixed, but it does not construct that reduction in families.
Wall charts, connection lifts in arbitrary rank, and affine flag compatibility
The relative normal-form theorem of § 9 begins after a holomorphic triangular reduction has been supplied. We now analyse the first change of relative lattice position. The main points are as follows. First, the elementary rank-two wall can be classified for every residue difference: there is an extension chart outside the strictly positive integral resonance range, whereas the distinguished normal slice does not meet the oriented wall in that range. Second, the first rank-three correction couples adjacent and composite roots rather than acting independently root by root. The correction can itself be absorbed into polynomially adapted wall coordinates, but two adjacent lattice modifications differ by a central composite-root cocycle. Third, relative saturation acquires an additional generator over a square-zero thickening.
Let S S S t t t O S × Δ \mathcal{O}_{S \times\Delta} O S × Δ O \mathcal{O} O I ⊂ E \mathcal{I} \subset\mathcal{E} I ⊂ E
I s a t = { s ∈ E ∣ z N s ∈ I for some N ≥ 0 } . \mathcal{I}^{\mathrm{sat}}=\{s\in\mathcal{E}\mid z^N s\in\mathcal{I}\text{ for some }N\geq0\}. I sat = { s ∈ E ∣ z N s ∈ I for some N ≥ 0 } . This operation need not commute with restriction to a parameter fibre.
The scalar projection needed at a wall
The special fibres below contain holomorphic linear terms which must still be projected to the framed normal complement of Lemma 5.5. We record that projection.
Lemma 10.1. For ν ∈ C \nu\in\mathbb{C} ν ∈ C a , b ∈ C a,b\in\mathbb{C} a , b ∈ C
σ ν ( a , b ) = { − ν ( a 1 − ν + b 2 − ν ) , ν ∉ Z ≥ 0 , 0 , ν = 0 , a , ν = 1 , b , ν = 2 , 0 , ν ∈ Z ≥ 3 . (21) \sigma_\nu(a,b)=
\begin{cases}
-\nu\left(\dfrac{a}{1-\nu}+\dfrac{b}{2-\nu}\right), & \nu\notin\mathbb{Z}_{\geq0},\\
0, & \nu=0,\\
a, & \nu=1,\\
b, & \nu=2,\\
0, & \nu\in\mathbb{Z}_{\geq3}.
\end{cases} \tag*{(21)} σ ν ( a , b ) = ⎩ ⎨ ⎧ − ν ( 1 − ν a + 2 − ν b ) , 0 , a , b , 0 , ν ∈ / Z ≥ 0 , ν = 0 , ν = 1 , ν = 2 , ν ∈ Z ≥ 3 . (21) Then the unique framed normal representative of the scalar coefficient a + b z a+bz a + b z ν \nu ν σ ν ( a , b ) ρ ν ( z ) \sigma_\nu(a,b)\rho_\nu(z) σ ν ( a , b ) ρ ν ( z )
Equivalently, there is a unique P ∈ O ( Δ ) P\in\mathcal{O}(\Delta) P ∈ O ( Δ ) P ( 1 ) = 0 P(1)=0 P ( 1 ) = 0 a + b z + L ν ( P ) = σ ν ( a , b ) ρ ν ( z ) a+bz+L_\nu(P)=\sigma_\nu(a,b)\rho_\nu(z) a + b z + L ν ( P ) = σ ν ( a , b ) ρ ν ( z )
Proof. Assume first that ν ∉ Z ≥ 0 \nu\notin\mathbb{Z}_{\geq0} ν ∈ / Z ≥ 0
P 0 ( z ) = a 1 − ν z + b 2 − ν z 2 , P ( z ) = − P 0 ( z ) + P 0 ( 1 ) . P_0(z)=\frac{a}{1-\nu}z+\frac{b}{2-\nu}z^2,\qquad P(z)=-P_0(z)+P_0(1). P 0 ( z ) = 1 − ν a z + 2 − ν b z 2 , P ( z ) = − P 0 ( z ) + P 0 ( 1 ) . Then P ( 1 ) = 0 P(1)=0 P ( 1 ) = 0
a + b z + L ν ( P ) = − ν P 0 ( 1 ) z − 1 = σ ν ( a , b ) z − 1 . a+bz+L_\nu(P)=-\nu P_0(1)z^{-1}=\sigma_\nu(a,b)z^{-1}. a + b z + L ν ( P ) = − ν P 0 ( 1 ) z − 1 = σ ν ( a , b ) z − 1 . If ν = m ∈ Z ≥ 0 \nu=m\in\mathbb{Z}_{\geq0} ν = m ∈ Z ≥ 0 z m − 1 z^{m-1} z m − 1 L m L_m L m m = 0 m=0 m = 0 a + b z a+bz a + b z m = 1 m=1 m = 1 m = 2 m=2 m = 2 m ≥ 3 m\geq3 m ≥ 3 z m − 1 z^{m-1} z m − 1 z m z^m z m P ( 1 ) = 0 P(1)=0 P ( 1 ) = 0
In particular, for ν = − 1 \nu=-1 ν = − 1
K + L − 1 ( − K 2 ( z − 1 ) ) = K 2 z − 1 . (22) K+L_{-1}\left(-\frac{K}{2}(z-1)\right)=\frac{K}{2}z^{-1}. \tag*{(22)} K + L − 1 ( − 2 K ( z − 1 ) ) = 2 K z − 1 . (22) Thus a constant term is not removed without changing the normal z − 1 z^{-1} z − 1
The elementary rank-two lattice jump
Set E = O e 1 ⊕ O e 2 \mathcal{E}=\mathcal{O}e_1\oplus\mathcal{O}e_2 E = O e 1 ⊕ O e 2
∇ = d + t z 2 E 21 d z (23) \nabla=d+\frac{t}{z^2}E_{21}\,dz \tag*{(23)} ∇ = d + z 2 t E 21 d z (23) At z = 1 z=1 z = 1
v 1 = e 1 + t e 2 , v 2 = e 2 , v_1=e_1+te_2,\qquad v_2=e_2, v 1 = e 1 + t e 2 , v 2 = e 2 , and, on the punctured disc, put
F ∘ = O ( ∗ 0 ) ( e 1 + t z e 2 ) . (24) \mathcal{F}^{\circ}=\mathcal{O}(*0)\left(e_1+\frac{t}{z}e_2\right). \tag*{(24)} F ∘ = O ( ∗ 0 ) ( e 1 + z t e 2 ) . (24) Proposition 10.2. The line F ∘ \mathcal{F}^{\circ} F ∘ (23) , every punctured fibre has trivial monodromy, and F ∘ ∣ z = 1 \mathcal{F}^{\circ}|_{z=1} F ∘ ∣ z = 1 ( v 1 , v 2 ) (v_1,v_2) ( v 1 , v 2 )
E ∩ F ∘ = O ( z e 1 + t e 2 ) . (25) \mathcal{E}\cap\mathcal{F}^{\circ}=\mathcal{O}(ze_1+te_2). \tag*{(25)} E ∩ F ∘ = O ( z e 1 + t e 2 ) . (25) This module is z z z S S S
( E ∩ F ∘ ) ⊗ O S C 0 = O Δ z e 1 ≠ E 0 ∩ F 0 ∘ = O Δ e 1 . (26) (\mathcal{E}\cap\mathcal{F}^{\circ})\otimes_{\mathcal{O}_S}\mathbb{C}_0=\mathcal{O}_{\Delta}ze_1\ne\mathcal{E}_0\cap\mathcal{F}^{\circ}_0=\mathcal{O}_{\Delta}e_1. \tag*{(26)} ( E ∩ F ∘ ) ⊗ O S C 0 = O Δ z e 1 = E 0 ∩ F 0 ∘ = O Δ e 1 . (26) Proof. Put f = e 1 + t z − 1 e 2 f=e_1+tz^{-1}e_2 f = e 1 + t z − 1 e 2
d f = − t z − 2 e 2 d z , t z 2 E 21 f d z = t z − 2 e 2 d z , df=-tz^{-2}e_2\,dz,\qquad\frac{t}{z^2}E_{21}f\,dz=tz^{-2}e_2\,dz, df = − t z − 2 e 2 d z , z 2 t E 21 f d z = t z − 2 e 2 d z , one has ∇ f = 0 \nabla f=0 ∇ f = 0 ( f , e 2 ) (f,e_2) ( f , e 2 ) f ( 1 ) = v 1 f(1)=v_1 f ( 1 ) = v 1 R = C { t , z } R=\mathbb{C}\{t,z\} R = C { t , z } h f hf h f R e 1 ⊕ R e 2 Re_1\oplus Re_2 R e 1 ⊕ R e 2 h ∈ R h\in R h ∈ R h t / z ∈ R ht/z\in R h t / z ∈ R t t t z z z z ∣ h z\mid h z ∣ h (25) . If
z ( x 1 e 1 + x 2 e 2 ) = a ( z e 1 + t e 2 ) , z(x_1e_1+x_2e_2)=a(ze_1+te_2), z ( x 1 e 1 + x 2 e 2 ) = a ( z e 1 + t e 2 ) , then x 1 = a x_1=a x 1 = a z x 2 = a t zx_2=at z x 2 = a t a = z b a=zb a = z b x 1 e 1 + x 2 e 2 = b ( z e 1 + t e 2 ) x_1e_1+x_2e_2=b(ze_1+te_2) x 1 e 1 + x 2 e 2 = b ( z e 1 + t e 2 ) z z z t = 0 t=0 t = 0 (26) .
On S × = S ∖ { 0 } S^\times=S\setminus\{0\} S × = S ∖ { 0 }
w 1 = z e 1 + t e 2 , w 2 = z − 1 t e 1 + e 2 . (27) w_1=ze_1+te_2,\qquad w_2=\frac{z-1}{t}e_1+e_2. \tag*{(27)} w 1 = z e 1 + t e 2 , w 2 = t z − 1 e 1 + e 2 . (27) whose matrix is
H t = ( z ( z − 1 ) / t t 1 ) , det H t = 1. (28) H_t=\begin{pmatrix} z & (z-1)/t \\ t & 1 \end{pmatrix},\qquad\det H_t=1. \tag*{(28)} H t = ( z t ( z − 1 ) / t 1 ) , det H t = 1. (28) Theorem 10.3. Let μ 1 , μ 2 ∈ C \mu_1,\mu_2\in\mathbb{C} μ 1 , μ 2 ∈ C λ = μ 2 − μ 1 \lambda=\mu_2-\mu_1 λ = μ 2 − μ 1 S × S^\times S ×
A o p e n = ( μ 1 z − 1 q ( z ) 0 μ 2 z − 1 ) , q ( z ) = α z − 2 + β ρ λ ( z ) , (29) A_{\mathrm{open}}=\begin{pmatrix}\mu_1z^{-1} & q(z) \\ 0 & \mu_2z^{-1}\end{pmatrix},\qquad q(z)=\alpha z^{-2}+\beta\rho_\lambda(z), \tag*{(29)} A open = ( μ 1 z − 1 0 q ( z ) μ 2 z − 1 ) , q ( z ) = α z − 2 + β ρ λ ( z ) , (29) where α \alpha α β \beta β t t t
A f i x = H t A o p e n H t − 1 − H t ′ H t − 1 . A_{\mathrm{fix}}=H_tA_{\mathrm{open}}H_t^{-1}-H_t'H_t^{-1}. A fix = H t A open H t − 1 − H t ′ H t − 1 . Set
N λ ( t , z ) = λ z − λ − 1 + t z 2 q ( z ) . (30) N_\lambda(t,z)=\lambda z-\lambda-1+tz^2q(z). \tag*{(30)} N λ ( t , z ) = λ z − λ − 1 + t z 2 q ( z ) . (30) Then
A f i x = ( − λ + μ 2 z − 1 − t z q N λ / t − t ( λ + t z q ) / z λ + μ 1 z − 1 + t z q ) . (31) A_{\mathrm{fix}}=
\begin{pmatrix}
-\lambda+\mu_2z^{-1}-tzq & N_\lambda/t\\
-t(\lambda+tzq)/z & \lambda+\mu_1z^{-1}+tzq
\end{pmatrix}. \tag*{(31)} A fix = ( − λ + μ 2 z − 1 − t z q − t ( λ + t z q ) / z N λ / t λ + μ 1 z − 1 + t z q ) . (31) The following alternatives hold.
(i) If λ ∈ Z ≥ 1 \lambda\in\mathbb{Z}_{\geq1} λ ∈ Z ≥ 1 (29) extends through this oriented elementary wall.
(ii) If λ ∉ Z ≥ 1 \lambda\notin\mathbb{Z}_{\geq1} λ ∈ / Z ≥ 1 ρ λ = z − 1 \rho_\lambda=z^{-1} ρ λ = z − 1 t = 0 t=0 t = 0 a , b ∈ O S a,b\in\mathcal{O}_S a , b ∈ O S
α = λ + 1 t + a , β = − λ t + b . (32) \alpha=\frac{\lambda+1}{t}+a,\qquad\beta=-\frac{\lambda}{t}+b. \tag*{(32)} α = t λ + 1 + a , β = − t λ + b . (32) Its special fibre has residue vector
( μ 1 − 1 , μ 2 + 1 ) (33) (\mu_1-1,\mu_2+1) \tag*{(33)} ( μ 1 − 1 , μ 2 + 1 ) (33) and, before framed normalisation, upper-right coefficient a ( 0 ) + b ( 0 ) z a(0) + b(0)z a ( 0 ) + b ( 0 ) z
( α ~ , β ~ ) = ( 0 , σ λ + 2 ( a ( 0 ) , b ( 0 ) ) ) . (34) (\widetilde{\alpha},\widetilde{\beta})=(0,\sigma_{\lambda+2}(a(0),b(0))). \tag*{(34)} ( α , β ) = ( 0 , σ λ + 2 ( a ( 0 ) , b ( 0 ))) . (34) Proof. A direct multiplication gives (31) . If λ = m ∈ Z ≥ 1 \lambda=m\in\mathbb{Z}_{\geq1} λ = m ∈ Z ≥ 1 z 2 ρ λ = z m + 1 z^2\rho_\lambda=z^{m+1} z 2 ρ λ = z m + 1
N λ t = t α − m − 1 t + m t z + β z m + 1 . \frac{N_\lambda}{t}=\frac{t\alpha-m-1}{t}+\frac{m}{t}z+\beta z^{m+1}. t N λ = t t α − m − 1 + t m z + β z m + 1 . The monomials 1 1 1 z z z z m + 1 z^{m+1} z m + 1 t t t z z z m / t m/t m / t
Assume now that λ ∉ Z ≥ 1 \lambda\notin\mathbb{Z}_{\geq1} λ ∈ / Z ≥ 1 ρ λ = z − 1 \rho_\lambda=z^{-1} ρ λ = z − 1 λ = 0 \lambda=0 λ = 0
N λ = ( t α − λ − 1 ) + ( t β + λ ) z . N_\lambda=(t\alpha-\lambda-1)+(t\beta+\lambda)z. N λ = ( t α − λ − 1 ) + ( tβ + λ ) z . Thus N λ / t N_\lambda/t N λ / t (32) holds with a , b a,b a , b (31) shows that every entry then extends and that the diagonal special residues are (33) . The special upper-right coefficient is a ( 0 ) + b ( 0 ) z a(0)+b(0)z a ( 0 ) + b ( 0 ) z λ + 2 \lambda+2 λ + 2 (34) . □
Corollary 10.4. For ( μ 1 , μ 2 ) = ( 1 , − 1 ) (\mu_1,\mu_2)=(1,-1) ( μ 1 , μ 2 ) = ( 1 , − 1 )
α = − t − 1 + a , β = 2 t − 1 + b , a , b ∈ O S , \alpha=-t^{-1}+a,\qquad\beta=2t^{-1}+b,\qquad a,b\in\mathcal{O}_S, α = − t − 1 + a , β = 2 t − 1 + b , a , b ∈ O S , and every special normal representative is the origin of the residue stratum ( 0 , 0 ) (0,0) ( 0 , 0 ) a = b = 0 a=b=0 a = b = 0 (23) . The finite wall parameters are
a = 1 + t α t , b = − 2 + t β t ; (35) a=\frac{1+t\alpha}{t},\qquad b=\frac{-2+t\beta}{t}; \tag*{(35)} a = t 1 + t α , b = t − 2 + tβ ; (35) the numerators alone merely express the required divisibility by t t t
Proof. Here λ + 2 = 0 \lambda+2=0 λ + 2 = 0 σ 0 = 0 \sigma_0=0 σ 0 = 0 a = b = 0 a=b=0 a = b = 0 (31) gives t z − 2 E 21 tz^{-2}E_{21} t z − 2 E 21
The theorem exhibits a resonance barrier which is invisible in the fixed-stratum dimension count: the oriented wall meets every distinguished normal slice except those with strictly positive integral residue difference. The reverse orientation is a different chart and must be analysed separately.
Polynomial rank-three wall coordinates
We first cross the wall in the first adjacent pair. Embed H t H_t H t
H ^ t 12 = ( z ( z − 1 ) / t 0 t 1 0 0 0 1 ) . \widehat{H}^{12}_t=
\begin{pmatrix}
z & (z-1)/t & 0\\
t & 1 & 0\\
0 & 0 & 1
\end{pmatrix}. H t 12 = z t 0 ( z − 1 ) / t 1 0 0 0 1 . On S × S^\times S ×
A o p e n 12 = ( z − 1 α z − 2 + β z − 1 η z − 2 + θ z − 1 0 − z − 1 γ z − 2 + δ z − 1 0 0 − z − 1 ) . (36) A^{12}_{\mathrm{open}}=
\begin{pmatrix}
z^{-1} & \alpha z^{-2}+\beta z^{-1} & \eta z^{-2}+\theta z^{-1}\\
0 & -z^{-1} & \gamma z^{-2}+\delta z^{-1}\\
0 & 0 & -z^{-1}
\end{pmatrix}. \tag*{(36)} A open 12 = z − 1 0 0 α z − 2 + β z − 1 − z − 1 0 η z − 2 + θ z − 1 γ z − 2 + δ z − 1 − z − 1 . (36) Theorem 10.5. Transport (36) by H ^ t 12 \widehat{H}^{12}_t H t 12 t = 0 t=0 t = 0 a , b , C , D , U , V a,b,C,D,U,V a , b , C , D , U , V
α = − t − 1 + a , β = 2 t − 1 + b , γ = t C , δ = D , \alpha=-t^{-1}+a,\qquad\beta=2t^{-1}+b,\qquad\gamma=tC,\qquad\delta=D, α = − t − 1 + a , β = 2 t − 1 + b , γ = tC , δ = D , η = U − C + D / t , θ = V − D / t . (37) \eta=U-C+D/t,\qquad\theta=V-D/t. \tag*{(37)} η = U − C + D / t , θ = V − D / t . (37) In these parameters the transported matrix is
\begin{pmatrix}
-t(a+bz)/z & a+bz & -Cz^{-2}+Uz^{-1}+V\\
t(1-t(a+bz))/z^2 & t(a+bz)/z & (D+tU)z^{-2}+tVz^{-1}\\
0 & 0 & -z^{-1}
\end{pmatrix}. \tag*{(38)}\end{pmatrix}
The special residue vector is ( 0 , 0 , − 1 ) (0,0,-1) ( 0 , 0 , − 1 )
A = − C − D ( a + b 2 ) , B = U + D a + V 2 + D b 4 . (39) A=-C-D\left(a+\frac{b}{2}\right),\qquad B=U+Da+\frac{V}{2}+\frac{Db}{4}. \tag*{(39)} A = − C − D ( a + 2 b ) , B = U + D a + 2 V + 4 D b . (39) After unique framed normalisation of the special fibre one has
( α ~ 12 , β ~ 12 ) = ( 0 , 0 ) , (40) (\widetilde{\alpha}_{12},\widetilde{\beta}_{12})=(0,0), \tag*{(40)} ( α 12 , β 12 ) = ( 0 , 0 ) , (40) ( α ~ 23 , β ~ 23 ) = ( D ( 0 ) , 0 ) , (41) (\widetilde{\alpha}_{23},\widetilde{\beta}_{23})=(D(0),0), \tag*{(41)} ( α 23 , β 23 ) = ( D ( 0 ) , 0 ) , (41) ( α ~ 13 , β ~ 13 ) = ( A ( 0 ) , B ( 0 ) ) . (42) (\widetilde{\alpha}_{13},\widetilde{\beta}_{13})=(A(0),B(0)). \tag*{(42)} ( α 13 , β 13 ) = ( A ( 0 ) , B ( 0 )) . (42) Moreover,
( a , b , C , D , U , V ) ⟼ ( a , b , D , V , A , B ) (43) (a,b,C,D,U,V)\longmapsto(a,b,D,V,A,B) \tag*{(43)} ( a , b , C , D , U , V ) ⟼ ( a , b , D , V , A , B ) (43) is a polynomial automorphism of the six-dimensional extension parameter space. Thus the correction in the natural root coordinates is triangular and admits polynomially adapted wall coordinates.
Proof. Direct multiplication gives
H ^ t 12 A o p e n 12 ( H ^ t 12 ) − 1 − ( H ^ t 12 ) ( H ^ t 12 ) − 1 = ( − Ξ / z Ξ / t − C z − 2 + U z − 1 + V t ( 1 − Ξ ) / z 2 Ξ / z ( D + t U ) z − 2 + t V z − 1 0 0 − z − 1 ) . \widehat{H}_t^{12}A_{\mathrm{open}}^{12}(\widehat{H}_t^{12})^{-1}-(\widehat{H}_t^{12})(\widehat{H}_t^{12})^{-1}=
\begin{pmatrix}
-\Xi/z & \Xi/t & -Cz^{-2}+Uz^{-1}+V\\
t(1-\Xi)/z^2 & \Xi/z & (D+tU)z^{-2}+tVz^{-1}\\
0 & 0 & -z^{-1}
\end{pmatrix}. H t 12 A open 12 ( H t 12 ) − 1 − ( H t 12 ) ( H t 12 ) − 1 = − Ξ/ z t ( 1 − Ξ ) / z 2 0 Ξ/ t Ξ/ z 0 − C z − 2 + U z − 1 + V ( D + t U ) z − 2 + t V z − 1 − z − 1 . where
Ξ = 1 − 2 z + t α + t β z , C = γ / t , U = ( γ − δ ) / t + η , V = δ / t + θ , D = δ . \Xi=1-2z+t\alpha+t\beta z,\qquad C=\gamma/t,\qquad U=(\gamma-\delta)/t+\eta,\qquad V=\delta/t+\theta,\qquad D=\delta. Ξ = 1 − 2 z + t α + tβ z , C = γ / t , U = ( γ − δ ) / t + η , V = δ / t + θ , D = δ . Coefficientwise holomorphicity is therefore equivalent to (37) , and then Ξ = t ( a + b z ) \Xi=t(a+bz) Ξ = t ( a + b z ) (38) .
At t = 0 t=0 t = 0
q 12 = a + b z , q 23 = D z − 2 . q_{12}=a+bz,\qquad q_{23}=Dz^{-2}. q 12 = a + b z , q 23 = D z − 2 . The unique framed gauge removing q 12 q_{12} q 12
g = I 3 + p ( z ) E 12 , p ( z ) = − a ( z − 1 ) − b 2 ( z 2 − 1 ) . g=I_3+p(z)E_{12},\qquad p(z)=-a(z-1)-\frac{b}{2}(z^2-1). g = I 3 + p ( z ) E 12 , p ( z ) = − a ( z − 1 ) − 2 b ( z 2 − 1 ) . It changes the composite coefficient to
[ − C − D ( a + b 2 ) ] z − 2 + ( U + D a ) z − 1 + V + D b 2 . \left[-C-D\left(a+\frac{b}{2}\right)\right]z^{-2}+(U+Da)z^{-1}+V+\frac{Db}{2}. [ − C − D ( a + 2 b ) ] z − 2 + ( U + D a ) z − 1 + V + 2 D b . The special residue difference for the root 13 is − 1 -1 − 1 (22) , the constant term contributes half of itself to the normal z − 1 z^{-1} z − 1 (39) and (40) –(42) .
Finally, (43) is triangular, with inverse
C = − A − D ( a + b 2 ) , U = B − D a − V 2 − D b 4 . C=-A-D\left(a+\frac{b}{2}\right),\qquad U=B-Da-\frac{V}{2}-\frac{Db}{4}. C = − A − D ( a + 2 b ) , U = B − D a − 2 V − 4 D b . Hence it is a polynomial automorphism. □
The opposite simple-root chart is described in the same way. Embed H t H_t H t
H ^ t 23 = ( 1 0 0 0 z ( z − 1 ) / t 0 t 1 ) . \widehat{H}_t^{23}=
\begin{pmatrix}
1&0&0\\
0&z&(z-1)/t\\
0&t&1
\end{pmatrix}. H t 23 = 1 0 0 0 z t 0 ( z − 1 ) / t 1 . On S × S^\times S ×
A o p e n 23 = ( z − 1 γ z − 2 + δ z − 1 η z − 2 + θ z − 1 0 z − 1 α z − 2 + β z − 1 0 0 − z − 1 ) . A_{\mathrm{open}}^{23}=
\begin{pmatrix}
z^{-1}&\gamma z^{-2}+\delta z^{-1}&\eta z^{-2}+\theta z^{-1}\\
0&z^{-1}&\alpha z^{-2}+\beta z^{-1}\\
0&0&-z^{-1}
\end{pmatrix}. A open 23 = z − 1 0 0 γ z − 2 + δ z − 1 z − 1 0 η z − 2 + θ z − 1 α z − 2 + β z − 1 − z − 1 . Proposition 10.6. The transport of (10.24) by H ^ t 23 \widehat{H}_t^{23} H t 23 a , b , C , D , U , V a,b,C,D,U,V a , b , C , D , U , V
α = − t − 1 + a , β = 2 t − 1 + b , γ = t C , δ = D , \alpha=-t^{-1}+a,\qquad\beta=2t^{-1}+b,\qquad\gamma=tC,\qquad\delta=D, α = − t − 1 + a , β = 2 t − 1 + b , γ = tC , δ = D , η = U + C − D / t , θ = V + D / t . (44) \eta=U+C-D/t,\qquad\theta=V+D/t. \tag*{(44)} η = U + C − D / t , θ = V + D / t . (44) The special residue vector is ( 1 , 0 , 0 ) (1,0,0) ( 1 , 0 , 0 )
A + = C + D ( a + b 2 ) , B + = U − D a + V 2 − D b 4 , (45) A^+=C+D\left(a+\frac{b}{2}\right),\qquad B^+=U-Da+\frac{V}{2}-\frac{Db}{4}, \tag*{(45)} A + = C + D ( a + 2 b ) , B + = U − D a + 2 V − 4 D b , (45) then its special normal coordinates are
( α ~ 23 , β ~ 23 ) = ( 0 , 0 ) , (\widetilde{\alpha}_{23},\widetilde{\beta}_{23})=(0,0), ( α 23 , β 23 ) = ( 0 , 0 ) , ( α ~ 12 , β ~ 12 ) = ( D ( 0 ) , 0 ) , (\widetilde{\alpha}_{12},\widetilde{\beta}_{12})=(D(0),0), ( α 12 , β 12 ) = ( D ( 0 ) , 0 ) , ( α ~ 13 , β ~ 13 ) = ( A + ( 0 ) , B + ( 0 ) ) . (\widetilde{\alpha}_{13},\widetilde{\beta}_{13})=(A^+(0),B^+(0)). ( α 13 , β 13 ) = ( A + ( 0 ) , B + ( 0 )) . The map
( a , b , C , D , U , V ) ⟼ ( a , b , D , V , A + , B + ) (a,b,C,D,U,V)\longmapsto(a,b,D,V,A^+,B^+) ( a , b , C , D , U , V ) ⟼ ( a , b , D , V , A + , B + ) is again a polynomial automorphism.
Proof. A direct multiplication yields the extended root coefficients
q 13 f i x = C z − 2 + U z − 1 + V , q^{\mathrm{fix}}_{13}=Cz^{-2}+Uz^{-1}+V, q 13 fix = C z − 2 + U z − 1 + V , q 12 f i x = ( D − t U ) z − 2 − t V z − 1 , q^{\mathrm{fix}}_{12}=(D-tU)z^{-2}-tVz^{-1}, q 12 fix = ( D − t U ) z − 2 − t V z − 1 , together with the rank-two wall matrix in the ( 2 , 3 ) (2,3) ( 2 , 3 ) (44) . At t = 0 t=0 t = 0
q 23 = a + b z , q 12 = D z − 2 , q 13 = C z − 2 + U z − 1 + V . q_{23}=a+bz,\qquad q_{12}=Dz^{-2},\qquad q_{13}=Cz^{-2}+Uz^{-1}+V. q 23 = a + b z , q 12 = D z − 2 , q 13 = C z − 2 + U z − 1 + V . Removing q 23 q_{23} q 23 I 3 + p ( z ) E 23 I_3+p(z)E_{23} I 3 + p ( z ) E 23 p = − a ( z − 1 ) − b 2 ( z 2 − 1 ) p=-a(z-1)-\frac{b}{2}(z^2-1) p = − a ( z − 1 ) − 2 b ( z 2 − 1 )
[ C + D ( a + b 2 ) ] z − 2 + ( U − D a ) z − 1 + V − D b 2 . \left[C+D\left(a+\frac{b}{2}\right)\right]z^{-2}+(U-Da)z^{-1}+V-\frac{Db}{2}. [ C + D ( a + 2 b ) ] z − 2 + ( U − D a ) z − 1 + V − 2 D b . Applying (22) gives (45) . Triangularity gives the final assertion.
The two propositions show what is intrinsic in the first rank-three calculation. The natural rootwise coordinates are not independent across the wall, but their mixing is polynomially triangular and can be absorbed into adapted wall coordinates. Thus this calculation does not disprove every abstract product chart. What survives coordinate changes is the ordered compatibility at the intersection of two adjacent walls, to which we now turn.
Elementary wall torsors in arbitrary rank
We now pass from the rank-two and rank-three calculations to a single elementary wall in arbitrary rank. The first statement is a block calculation. The second packages all extension equations into a free lattice over the wall parameter and thereby yields a relative affine torsor parametrising connection lifts. For 1 ≤ i < r 1\leq i<r 1 ≤ i < r H i ( t ) H_i(t) H i ( t ) i , i + 1 i,i+1 i , i + 1
H t = ( z ( z − 1 ) / t t 1 ) H_t=\begin{pmatrix}z & (z-1)/t \\ t & 1\end{pmatrix} H t = ( z t ( z − 1 ) / t 1 ) on those coordinates. Write a meromorphic arc in the distinguished normal slice as
A o p e n ( t , z ) = M μ z + ∑ 1 ≤ j < k ≤ r q j k ( t , z ) E j k , q j k = α j k z − 2 + β j k ρ μ k − μ j ( z ) , (46) A_{\mathrm{open}}(t,z)=\frac{M_\mu}{z}+\sum_{1\leq j<k\leq r}q_{jk}(t,z)E_{jk},\qquad q_{jk}=\alpha_{jk}z^{-2}+\beta_{jk}\rho_{\mu_k-\mu_j}(z), \tag*{(46)} A open ( t , z ) = z M μ + 1 ≤ j < k ≤ r ∑ q j k ( t , z ) E j k , q j k = α j k z − 2 + β j k ρ μ k − μ j ( z ) , (46) where α j k \alpha_{jk} α j k β j k \beta_{jk} β j k t t t
Proposition 10.7. Fix i i i λ i = μ i + 1 − μ i \lambda_i=\mu_{i+1}-\mu_i λ i = μ i + 1 − μ i q i = q i , i + 1 q_i=q_{i,i+1} q i = q i , i + 1
A f i x = H i ( t ) A o p e n H i ( t ) − 1 − ( ∂ z H i ( t ) ) H i ( t ) − 1 . A_{\mathrm{fix}}=H_i(t)A_{\mathrm{open}}H_i(t)^{-1}-(\partial_zH_i(t))H_i(t)^{-1}. A fix = H i ( t ) A open H i ( t ) − 1 − ( ∂ z H i ( t )) H i ( t ) − 1 . Then the only entries changed by the raw transport are the direct 2 × 2 2\times2 2 × 2 If
N i ( t , z ) = λ i z − λ i − 1 + t z 2 q i ( t , z ) , N_i(t,z)=\lambda_i z-\lambda_i-1+tz^2q_i(t,z), N i ( t , z ) = λ i z − λ i − 1 + t z 2 q i ( t , z ) , then the direct block is
( A f i x ) i i = − λ i + μ i + 1 z − 1 − t z q i , (47) (A_{\mathrm{fix}})_{ii}=-\lambda_i+\mu_{i+1}z^{-1}-tzq_i, \tag*{(47)} ( A fix ) ii = − λ i + μ i + 1 z − 1 − t z q i , (47) ( A f i x ) i , i + 1 = N i / t , (48) (A_{\mathrm{fix}})_{i,i+1}=N_i/t, \tag*{(48)} ( A fix ) i , i + 1 = N i / t , (48) ( A f i x ) i + 1 , i = − t ( λ i + t z q i ) / z , (49) (A_{\mathrm{fix}})_{i+1,i}=-t(\lambda_i+tzq_i)/z, \tag*{(49)} ( A fix ) i + 1 , i = − t ( λ i + t z q i ) / z , (49) ( A f i x ) i + 1 , i + 1 = λ i + μ i z − 1 + t z q i . (50) (A_{\mathrm{fix}})_{i+1,i+1}=\lambda_i+\mu_i z^{-1}+tzq_i. \tag*{(50)} ( A fix ) i + 1 , i + 1 = λ i + μ i z − 1 + t z q i . (50) For p < i p<i p < i
( A f i x ) p , i = q p , i − t q p , i + 1 , (51) (A_{\mathrm{fix}})_{p,i}=q_{p,i}-tq_{p,i+1}, \tag*{(51)} ( A fix ) p , i = q p , i − t q p , i + 1 , (51) ( A f i x ) p , i + 1 = z q p , i + 1 − z − 1 t q p , i , (52) (A_{\mathrm{fix}})_{p,i+1}=zq_{p,i+1}-\frac{z-1}{t}q_{p,i}, \tag*{(52)} ( A fix ) p , i + 1 = z q p , i + 1 − t z − 1 q p , i , (52) and, for k > i + 1 k>i+1 k > i + 1
( A f i x ) i , k = z q i , k + z − 1 t q i + 1 , k , (53) (A_{\mathrm{fix}})_{i,k}=zq_{i,k}+\frac{z-1}{t}q_{i+1,k}, \tag*{(53)} ( A fix ) i , k = z q i , k + t z − 1 q i + 1 , k , (53) ( A f i x ) i + 1 , k = t q i , k + q i + 1 , k . (54) (A_{\mathrm{fix}})_{i+1,k}=tq_{i,k}+q_{i+1,k}. \tag*{(54)} ( A fix ) i + 1 , k = t q i , k + q i + 1 , k . (54) Every other off-diagonal entry is unchanged. Consequently, extension through t = 0 t=0 t = 0 q i q_i q i t t t (51) –(54) ; there are no further raw extension equations.
Proof. On the active block,
H t − 1 = ( 1 − ( z − 1 ) / t − t z ) . H_t^{-1}=\begin{pmatrix}1&-(z-1)/t\\-t&z\end{pmatrix}. H t − 1 = ( 1 − t − ( z − 1 ) / t z ) . The direct block is the rank-two calculation in Theorem 10.3 . Multiplying the row ( q p , i , q p , i + 1 ) (q_{p,i},q_{p,i+1}) ( q p , i , q p , i + 1 ) H t − 1 H_t^{-1} H t − 1 H t H_t H t ( q i , k , q i + 1 , k ) T (q_{i,k},q_{i+1,k})^{\mathsf T} ( q i , k , q i + 1 , k ) T
Put R = C { t } R=\mathbb{C}\{t\} R = C { t } K t = R [ t − 1 ] K_t=R[t^{-1}] K t = R [ t − 1 ] j < k j<k j < k
V j k μ = C z − 2 ⊕ C ρ μ k − μ j ( z ) , V μ = ⨁ j < k V j k μ E j k . V^\mu_{jk}=\mathbb{C}z^{-2}\oplus\mathbb{C}\rho_{\mu_k-\mu_j}(z),\qquad V_\mu=\bigoplus_{j<k}V^\mu_{jk}E_{jk}. V j k μ = C z − 2 ⊕ C ρ μ k − μ j ( z ) , V μ = j < k ⨁ V j k μ E j k . Thus dim C V μ = N = r ( r − 1 ) \dim_{\mathbb{C}}V_\mu=N=r(r-1) dim C V μ = N = r ( r − 1 ) ν ∈ C \nu\in\mathbb{C} ν ∈ C π ν : z − 2 O ( Δ ) ⟶ C 2 \pi_\nu:z^{-2}\mathcal{O}(\Delta)\longrightarrow\mathbb{C}^2 π ν : z − 2 O ( Δ ) ⟶ C 2 q = L ν ( P ) + α z − 2 + β ρ ν q=L_\nu(P)+\alpha z^{-2}+\beta\rho_\nu q = L ν ( P ) + α z − 2 + β ρ ν π ν ( q ) = ( α , β ) \pi_\nu(q)=(\alpha,\beta) π ν ( q ) = ( α , β )
Assume from now on that λ i ∉ Z ≥ 1 \lambda_i\notin\mathbb{Z}_{\geq1} λ i ∈ / Z ≥ 1 ρ λ i = z − 1 \rho_{\lambda_i}=z^{-1} ρ λ i = z − 1
q i , μ 0 ( t , z ) = ( λ i + 1 t z − 2 − λ i t z − 1 ) E i , i + 1 . (55) q^0_{i,\mu}(t,z)=\left(\frac{\lambda_i+1}{t}z^{-2}-\frac{\lambda_i}{t}z^{-1}\right)E_{i,i+1}. \tag*{(55)} q i , μ 0 ( t , z ) = ( t λ i + 1 z − 2 − t λ i z − 1 ) E i , i + 1 . (55) For p < i p<i p < i
L p , i L = { ( x , y ) ∈ ( V p , i μ ⊕ V p , i + 1 μ ) ⊗ C K t : x − t y ∈ z − 2 R { z } , z y − ( z − 1 ) t − 1 x ∈ z − 2 R { z } } . \mathcal{L}^{L}_{p,i}=\left\{(x,y)\in(V^\mu_{p,i}\oplus V^\mu_{p,i+1})\otimes_{\mathbb{C}}K_t:x-ty\in z^{-2}R\{z\},\ zy-(z-1)t^{-1}x\in z^{-2}R\{z\}\right\}. L p , i L = { ( x , y ) ∈ ( V p , i μ ⊕ V p , i + 1 μ ) ⊗ C K t : x − t y ∈ z − 2 R { z } , z y − ( z − 1 ) t − 1 x ∈ z − 2 R { z } } . and, for k > i + 1 k>i+1 k > i + 1
L i , k R = { ( x , y ) ∈ ( V i , k μ ⊕ V i + 1 , k μ ) ⊗ C K t : z x + ( z − 1 ) t − 1 y ∈ z − 2 R { z } , t x + y ∈ z − 2 R { z } } . \mathcal{L}^{R}_{i,k}=\left\{(x,y)\in(V^\mu_{i,k}\oplus V^\mu_{i+1,k})\otimes_{\mathbb{C}}K_t:zx+(z-1)t^{-1}y\in z^{-2}R\{z\},\ tx+y\in z^{-2}R\{z\}\right\}. L i , k R = { ( x , y ) ∈ ( V i , k μ ⊕ V i + 1 , k μ ) ⊗ C K t : z x + ( z − 1 ) t − 1 y ∈ z − 2 R { z } , t x + y ∈ z − 2 R { z } } . Finally, let V μ o u t V^{\mathrm{out}}_\mu V μ out { i , i + 1 } \{i,i+1\} { i , i + 1 }
L i , μ = ( V i , i + 1 μ ⊗ R ) ⊕ ⨁ p < i L p , i L ⊕ ⨁ k > i + 1 L i , k R ⊕ ( V μ o u t ⊗ R ) . (56) \mathcal{L}_{i,\mu}=(V^\mu_{i,i+1}\otimes R)\oplus\bigoplus_{p<i}\mathcal{L}^{L}_{p,i}\oplus\bigoplus_{k>i+1}\mathcal{L}^{R}_{i,k}\oplus(V^{\mathrm{out}}_\mu\otimes R). \tag*{(56)} L i , μ = ( V i , i + 1 μ ⊗ R ) ⊕ p < i ⨁ L p , i L ⊕ k > i + 1 ⨁ L i , k R ⊕ ( V μ out ⊗ R ) . (56) For a flat local analytic R R R A A A A / t A ≠ 0 A/tA\ne0 A / t A = 0 Wall i , μ ( A ) \operatorname{Wall}_{i,\mu}(A) Wall i , μ ( A ) Q ∈ V μ ⊗ C A [ t − 1 ] Q\in V_\mu\otimes_{\mathbb{C}}A[t^{-1}] Q ∈ V μ ⊗ C A [ t − 1 ] M μ / z + Q M_\mu/z+Q M μ / z + Q H i ( t ) H_i(t) H i ( t ) z − 2 A { z } z^{-2}A\{z\} z − 2 A { z }
Theorem 10.8. (i) If λ i ∈ Z ≥ 1 \lambda_i\in\mathbb{Z}_{\geq1} λ i ∈ Z ≥ 1 Wall i , μ ( A ) = ∅ \operatorname{Wall}_{i,\mu}(A)=\varnothing Wall i , μ ( A ) = ∅ R R R A A A
(ii) If λ i ∉ Z ≥ 1 \lambda_i\notin\mathbb{Z}_{\geq1} λ i ∈ / Z ≥ 1 A A A
Wall i , μ ( A ) = q i , μ 0 + L i , μ ⊗ R A . (57) \operatorname{Wall}_{i,\mu}(A)=q^0_{i,\mu}+\mathcal{L}_{i,\mu}\otimes_R A. \tag*{(57)} Wall i , μ ( A ) = q i , μ 0 + L i , μ ⊗ R A . (57) Every lateral module in (56) is a free R R R 4 4 4 L i , μ \mathcal{L}_{i,\mu} L i , μ
2 + 4 ( r − 2 ) + ( r − 2 ) ( r − 3 ) = r ( r − 1 ) = N . 2+4(r-2)+(r-2)(r-3)=r(r-1)=N. 2 + 4 ( r − 2 ) + ( r − 2 ) ( r − 3 ) = r ( r − 1 ) = N . Thus the elementary connection-lift functor is represented by the affine S S S
q i , μ 0 + V S ( L i , μ ) , q^0_{i,\mu}+V_S(\mathcal{L}_{i,\mu}), q i , μ 0 + V S ( L i , μ ) , a torsor under the analytic vector bundle V S ( L i , μ ) V_S(\mathcal{L}_{i,\mu}) V S ( L i , μ ) R R R S × A C N S\times\mathbb{A}^N_{\mathbb{C}} S × A C N
(iii) Put
μ ′ = μ − e i + e i + 1 . \mu'=\mu-e_i+e_{i+1}. μ ′ = μ − e i + e i + 1 . Evaluation of the transported universal connection at t = 0 t=0 t = 0
sp i , μ : V S ( L i , μ ) 0 ⟶ S μ ′ ≃ A C N . (58) \operatorname{sp}_{i,\mu}:V_S(\mathcal{L}_{i,\mu})_0\longrightarrow S_{\mu'}\simeq\mathbb{A}^N_{\mathbb{C}}. \tag*{(58)} sp i , μ : V S ( L i , μ ) 0 ⟶ S μ ′ ≃ A C N . (58) If q j k r a w q^{\mathrm{raw}}_{jk} q j k raw h = k − j h=k-j h = k − j
( α ~ j k , β ~ j k ) = π μ k ′ − μ j ′ ( q j k r a w ) + P j k ( q a b r a w : b − a < h ) , (59) (\widetilde{\alpha}_{jk},\widetilde{\beta}_{jk})=\pi_{\mu'_k-\mu'_j}(q^{\mathrm{raw}}_{jk})+P_{jk}(q^{\mathrm{raw}}_{ab}:b-a<h), \tag*{(59)} ( α j k , β j k ) = π μ k ′ − μ j ′ ( q j k raw ) + P j k ( q ab raw : b − a < h ) , (59) where P j k P_{jk} P j k C 2 \mathbb{C}^2 C 2 h h h h = 1 h=1 h = 1 P j k = 0 P_{jk}=0 P j k = 0 h ≥ 2 h\geq2 h ≥ 2 P j k P_{jk} P j k 2 2 2
α i , i + 1 = λ i + 1 t + a i , β i , i + 1 = − λ i t + b i , \alpha_{i,i+1}=\frac{\lambda_i+1}{t}+a_i,\qquad\beta_{i,i+1}=-\frac{\lambda_i}{t}+b_i, α i , i + 1 = t λ i + 1 + a i , β i , i + 1 = − t λ i + b i , it is
( a i , b i ) ⟼ ( 0 , σ λ i + 2 ( a i ( 0 ) , b i ( 0 ) ) ) . (a_i,b_i)\longmapsto(0,\sigma_{\lambda_i+2}(a_i(0),b_i(0))). ( a i , b i ) ⟼ ( 0 , σ λ i + 2 ( a i ( 0 ) , b i ( 0 ))) . (iv) The represented wall functor and the polynomial specialisation map commute with flat local analytic base change. No analogous assertion is made for non-flat thickenings; the defect in Proposition 10.15 shows why additional saturation data are then required.
Proof. Part (i) is the rank-two resonance barrier. The direct entry (48) is independent of every lateral coordinate, so no off-block term can cancel its z z z t t t
Assume λ i ∉ Z ≥ 1 \lambda_i \notin\mathbb{Z}_{\ge1} λ i ∈ / Z ≥ 1 Theorem 10.3 , the direct root of every extending arc is uniquely the minimal crossing (55) plus an element of V i , i + 1 μ ⊗ R V^\mu_{i,i+1}\otimes R V i , i + 1 μ ⊗ R (57) over R R R (56) . Since the defining coefficient maps are maps of finite free R R R A A A T L ( x , y ) T^{L}(x,y) T L ( x , y ) x x x y y y t t t T L ( x , y ) T^{L}(x,y) T L ( x , y ) t t t
t ( V p , i μ ⊕ V p , i + 1 μ ) ⊗ R ⊆ L p , i L . t(V^\mu_{p,i}\oplus V^\mu_{p,i+1})\otimes R\subseteq\mathcal{L}^{L}_{p,i}. t ( V p , i μ ⊕ V p , i + 1 μ ) ⊗ R ⊆ L p , i L . Conversely, writing T L ( x , y ) = ( u , v ) T^{L}(x,y)=(u,v) T L ( x , y ) = ( u , v ) H t H_t H t x = z u + t v x=zu+tv x = z u + t v y = ( z − 1 ) t − 1 u + v y=(z-1)t^{-1}u+v y = ( z − 1 ) t − 1 u + v t t t x x x y y y − 1 -1 − 1
L p , i L ⊆ t − 1 ( V p , i μ ⊕ V p , i + 1 μ ) ⊗ R . \mathcal{L}^{L}_{p,i}\subseteq t^{-1}(V^\mu_{p,i}\oplus V^\mu_{p,i+1})\otimes R. L p , i L ⊆ t − 1 ( V p , i μ ⊕ V p , i + 1 μ ) ⊗ R . The same argument, using H t − 1 H_t^{-1} H t − 1
t − 1 ( V p , i μ ⊕ V p , i + 1 μ ) ⊗ R / ( ( V p , i μ ⊕ V p , i + 1 μ ) ⊗ R ) t^{-1}(V^\mu_{p,i}\oplus V^\mu_{p,i+1})\otimes R\big/\bigl((V^\mu_{p,i}\oplus V^\mu_{p,i+1})\otimes R\bigr) t − 1 ( V p , i μ ⊕ V p , i + 1 μ ) ⊗ R / ( ( V p , i μ ⊕ V p , i + 1 μ ) ⊗ R ) has finite length over the Noetherian ring R R R L p , i L \mathcal{L}^{L}_{p,i} L p , i L R R R t t t R R R L i , μ \mathcal{L}_{i,\mu} L i , μ
For part (iii), the special transported matrix is upper triangular: the only potentially lower entry is (49) , which vanishes at t = 0 t=0 t = 0 M μ ′ / z M'_\mu/z M μ ′ / z z − 2 O z^{-2}\mathcal{O} z − 2 O
At height one the normalising gauge and the scalar projection are linear. Suppose inductively that every normalising gauge coordinate of height less than h h h h h h h h h h h h h h h h h h h h h (59) and the assertion about its degree-one part. The active-root formula is Theorem 10.3 .
Finally, for each lateral pair only the normal monomials and their products with z z z z − 1 z-1 z − 1 z z z R R R Theorem 9.2 . This proves part (iv). □
Corollary 10.9. Fix the elementary wall frame H i ( t ) H_i(t) H i ( t )
t ⟼ G i ( t ) I , G i ( t ) = I r + t z − 1 E i , i + 1 . t\longmapsto G_i(t)I,\qquad G_i(t)=I_r+tz^{-1}E_{i,i+1}. t ⟼ G i ( t ) I , G i ( t ) = I r + t z − 1 E i , i + 1 . The corresponding distinguished fully framed connection-wall problem with residue vector μ \mu μ
μ i + 1 − μ i ∉ Z ≥ 1 . \mu_{i+1}-\mu_i\notin\mathbb{Z}_{\ge1}. μ i + 1 − μ i ∈ / Z ≥ 1 . When this condition holds, the extension functor is a relative affine torsor of relative dimension r ( r − 1 ) r(r-1) r ( r − 1 ) (58) , which is triangular by root height. This is a statement for the chosen elementary local-model datum; it does not assert that the original unsliced moduli functor carries a global position morphism to an affine flag variety.
Proof. The active Levi block is the rank-two curve of Proposition 10.13, while the chosen wall frame fixes the lateral comparison data. The existence criterion and the description of all extending connection arcs are given by Theorem 10.8. □
The composite-root correction at a two-wall intersection
Theorem 10.10. Put
G 12 ( t ) = I 3 + t z E 21 , G 23 ( s ) = I 3 + s z E 32 , K 13 ( u ) = I 3 + u z 2 E 31 . G_{12}(t)=I_3+\frac{t}{z}E_{21},\qquad G_{23}(s)=I_3+\frac{s}{z}E_{32},\qquad K_{13}(u)=I_3+\frac{u}{z^2}E_{31}. G 12 ( t ) = I 3 + z t E 21 , G 23 ( s ) = I 3 + z s E 32 , K 13 ( u ) = I 3 + z 2 u E 31 . Then
G 23 ( s ) G 12 ( t ) = K 13 ( s t ) G 12 ( t ) G 23 ( s ) . (60) G_{23}(s)G_{12}(t)=K_{13}(st)G_{12}(t)G_{23}(s). \tag*{(60)} G 23 ( s ) G 12 ( t ) = K 13 ( s t ) G 12 ( t ) G 23 ( s ) . (60) The factor K 13 ( s t ) K_{13}(st) K 13 ( s t ) E 21 E_{21} E 21 E 32 E_{32} E 32 E 31 E_{31} E 31 ( x , y , w ) (x,y,w) ( x , y , w )
( x , y , w ) ( x ′ , y ′ , w ′ ) = ( x + x ′ , y + y ′ , w + w ′ + y x ′ ) , (x,y,w)(x',y',w')=(x+x',y+y',w+w'+yx'), ( x , y , w ) ( x ′ , y ′ , w ′ ) = ( x + x ′ , y + y ′ , w + w ′ + y x ′ ) , so c ( ( x , y ) , ( x ′ , y ′ ) ) = y x ′ c((x,y),(x',y'))=yx' c (( x , y ) , ( x ′ , y ′ )) = y x ′
Proof. One has E 32 E 21 = E 31 E_{32}E_{21}=E_{31} E 32 E 21 = E 31 E 21 E 32 = 0 E_{21}E_{32}=0 E 21 E 32 = 0 E 31 E_{31} E 31 E 21 E_{21} E 21 E 32 E_{32} E 32
G 23 ( s ) G 12 ( t ) = I 3 + t z E 21 + s z E 32 + s t z 2 E 31 , G_{23}(s)G_{12}(t)=I_3+\frac{t}{z}E_{21}+\frac{s}{z}E_{32}+\frac{st}{z^2}E_{31}, G 23 ( s ) G 12 ( t ) = I 3 + z t E 21 + z s E 32 + z 2 s t E 31 , whereas
G 12 ( t ) G 23 ( s ) = I 3 + t z E 21 + s z E 32 . G_{12}(t)G_{23}(s)=I_3+\frac{t}{z}E_{21}+\frac{s}{z}E_{32}. G 12 ( t ) G 23 ( s ) = I 3 + z t E 21 + z s E 32 . This proves (60) . The displayed multiplication law follows from the same calculation, and the cocycle identity
c ( u , v ) + c ( u + v , w ) = c ( v , w ) + c ( u , v + w ) c(u,v)+c(u+v,w)=c(v,w)+c(u,v+w) c ( u , v ) + c ( u + v , w ) = c ( v , w ) + c ( u , v + w ) follows from bilinearity. Matrix units with disjoint supports multiply to zero in both orders. □
Root-subgroup structure in the loop group
The preceding rank-three identity is the first non-trivial instance of a group law valid in arbitrary rank. Put
D r = diag ( 1 , z , … , z r − 1 ) . (61) D_r=\operatorname{diag}(1,z,\ldots,z^{r-1}). \tag*{(61)} D r = diag ( 1 , z , … , z r − 1 ) . (61) and let U r − U_r^{-} U r −
Theorem 10.11. Conjugation by D r D_r D r
U r − ⟶ ∼ U r w a l l : = D r − 1 U r − D r ⊂ G L r ( C ( ( z ) ) ) , U_r^{-}\overset{\sim}{\longrightarrow}U_r^{\mathrm{wall}}:=D_r^{-1}U_r^{-}D_r\subset GL_r(\mathbb{C}((z))), U r − ⟶ ∼ U r wall := D r − 1 U r − D r ⊂ G L r ( C (( z ))) , where
U r w a l l = { I r + ∑ j > i u j i z − ( j − i ) E j i ∣ u j i ∈ C } . (62) U_r^{\mathrm{wall}}=\left\{I_r+\sum_{j>i}u_{ji}z^{-(j-i)}E_{ji}\mathrel{\Big|}u_{ji}\in\mathbb{C}\right\}. \tag*{(62)} U r wall = { I r + j > i ∑ u j i z − ( j − i ) E j i u j i ∈ C } . (62) The construction is functorial after replacing C \mathbb{C} C j > i j>i j > i
X j i ( a ) = I r + a z − ( j − i ) E j i . X_{ji}(a)=I_r+az^{-(j-i)}E_{ji}. X j i ( a ) = I r + a z − ( j − i ) E j i . Then, for k > j > i k>j>i k > j > i
X k j ( b ) X j i ( a ) = X k i ( a b ) X j i ( a ) X k j ( b ) , (63) X_{kj}(b)X_{ji}(a)=X_{ki}(ab)X_{ji}(a)X_{kj}(b), \tag*{(63)} X k j ( b ) X j i ( a ) = X k i ( ab ) X j i ( a ) X k j ( b ) , (63) and two such root subgroups commute whenever both ordered products of the corresponding matrix units vanish. The simple elements
G i ( a ) = X i + 1 , i ( a ) ( 1 ≤ i < r ) G_i(a)=X_{i+1,i}(a)\qquad(1\le i<r) G i ( a ) = X i + 1 , i ( a ) ( 1 ≤ i < r ) generate U r w a l l U_r^{\mathrm{wall}} U r wall z − ( j − i ) z^{-(j-i)} z − ( j − i )
Proof. For j > i j>i j > i
D r − 1 E j i D r = z − ( j − i ) E j i , D_r^{-1}E_{ji}D_r=z^{-(j-i)}E_{ji}, D r − 1 E j i D r = z − ( j − i ) E j i , which proves (62) and the asserted functorial isomorphism. The relation (63) follows from
E k j E j i = E k i , E j i E k j = 0. E_{kj}E_{ji}=E_{ki},\qquad E_{ji}E_{kj}=0. E k j E j i = E k i , E j i E k j = 0. If both ordered products of the relevant matrix units vanish, the root subgroups commute. Finally, the ordinary lower unitriangular group is generated by its simple-root subgroups; conjugation by D r D_r D r U r w a l l U_r^{\mathrm{wall}} U r wall
Corollary 10.12. For the elementary matrices G i ( a ) = I r + a z − 1 E i + 1 , i G_i(a)=I_r+az^{-1}E_{i+1,i} G i ( a ) = I r + a z − 1 E i + 1 , i
G i ( a ) G j ( b ) = G j ( b ) G i ( a ) if ∣ i − j ∣ > 1 , G_i(a)G_j(b)=G_j(b)G_i(a)\qquad\text{if }|i-j|>1, G i ( a ) G j ( b ) = G j ( b ) G i ( a ) if ∣ i − j ∣ > 1 , whereas
G i + 1 ( b ) G i ( a ) = X i + 2 , i ( a b ) G i ( a ) G i + 1 ( b ) . (64) G_{i+1}(b)G_i(a)=X_{i+2,i}(ab)G_i(a)G_{i+1}(b). \tag*{(64)} G i + 1 ( b ) G i ( a ) = X i + 2 , i ( ab ) G i ( a ) G i + 1 ( b ) . (64) Consequently, the pairwise compatibility of the lattice-modification matrices in every rank is determined by the rank-two and rank-three Levi subgroups. This statement concerns the lattice part only: the scalar normal projections and resonance barriers of the connection still have to be analysed in the relevant residue blocks.
Proposition 10.13. Let G ( t ) = I 2 + t z − 1 E 21 G(t)=I_2+tz^{-1}E_{21} G ( t ) = I 2 + t z − 1 E 21 Λ t = G ( t ) O 2 \Lambda_t=G(t)O^2 Λ t = G ( t ) O 2 I ⊂ G L 2 ( O ) I\subset GL_2(O) I ⊂ G L 2 ( O )
the map t ↦ G ( t ) I t\mapsto G(t)I t ↦ G ( t ) I F l 2 \mathrm{Fl}_2 Fl 2 t = 0 t=0 t = 0 t ≠ 0 t\ne0 t = 0 I s ˙ 0 I / I I\dot{s}_0I/I I s ˙ 0 I / I
s ˙ 0 = ( 0 − z z − 1 0 ) ; \dot{s}_0=\begin{pmatrix}0&-z\\ z^{-1}&0\end{pmatrix}; s ˙ 0 = ( 0 z − 1 − z 0 ) ; for t ≠ 0 t\ne0 t = 0 Λ t \Lambda_t Λ t O 2 O^2 O 2 z − 1 z^{-1} z − 1 z z z Λ 0 = O 2 \Lambda_0=O^2 Λ 0 = O 2
the matrix H t H_t H t (28) lies in G L 2 ( O ) GL_2(O) G L 2 ( O ) t ≠ 0 t\ne0 t = 0 H t O 2 = O 2 H_tO^2=O^2 H t O 2 = O 2
Proof. For t ≠ 0 t\ne0 t = 0 b = t − 1 z b=t^{-1}z b = t − 1 z x + ( b ) = I 2 + b E 12 x_+(b)=I_2+bE_{12} x + ( b ) = I 2 + b E 12
G ( t ) = x + ( b ) ( 0 − b t z − 1 0 ) x + ( b ) = x + ( b ) diag ( t − 1 , t ) s ˙ 0 x + ( b ) G(t)=x_+(b)\begin{pmatrix}0&-b\\ tz^{-1}&0\end{pmatrix}x_+(b)=x_+(b)\operatorname{diag}(t^{-1},t)\dot{s}_0x_+(b) G ( t ) = x + ( b ) ( 0 t z − 1 − b 0 ) x + ( b ) = x + ( b ) diag ( t − 1 , t ) s ˙ 0 x + ( b ) shows that G ( t ) ∈ I s ˙ 0 I G(t)\in I\dot{s}_0I G ( t ) ∈ I s ˙ 0 I b ∈ z O b\in zO b ∈ z O diag ( t − 1 , t ) ∈ I \operatorname{diag}(t^{-1},t)\in I diag ( t − 1 , t ) ∈ I G ( 0 ) = I 2 G(0)=I_2 G ( 0 ) = I 2 [24 ] . For the Grassmannian position, multiply by z z z
z G ( t ) = ( z 0 t z ) . zG(t)=\begin{pmatrix}z&0\\t&z\end{pmatrix}. z G ( t ) = ( z t 0 z ) . When t ≠ 0 t\ne0 t = 0 O O O diag ( 1 , z 2 ) \operatorname{diag}(1,z^2) diag ( 1 , z 2 ) G ( t ) G(t) G ( t ) z − 1 z^{-1} z − 1 z z z t = 0 t=0 t = 0 (28) has entries in O O O 1 1 1 t ≠ 0 t\ne0 t = 0
Remark 10.14. For every 1 ≤ i < r 1\le i<r 1 ≤ i < r G ( t ) G(t) G ( t ) ( i , i + 1 ) (i,i+1) ( i , i + 1 ) 1 1 1
Full saturation over the dual numbers
Proposition 10.15. Let A = C [ ε ] / ( ε 2 ) A=\mathbb{C}[\varepsilon]/(\varepsilon^2) A = C [ ε ] / ( ε 2 ) R = A { z } R=A\{z\} R = A { z } I = R ( z e 1 + ε e 2 ) ⊂ R 2 I=R(ze_1+\varepsilon e_2)\subset R^2 I = R ( z e 1 + ε e 2 ) ⊂ R 2
I s a t = I + R ε e 1 , I z s a t / I ≅ C ε . (65) I^{\mathrm{sat}}=I+R\varepsilon e_1,\qquad I^{\mathrm{sat}}_z/I\cong\mathbb{C}_\varepsilon. \tag*{(65)} I sat = I + R ε e 1 , I z sat / I ≅ C ε . (65) where the quotient is supported at z = 0 z=0 z = 0 ε e 1 ∈ I z s a t ∖ I \varepsilon e_1\in I^{\mathrm{sat}}_z\setminus I ε e 1 ∈ I z sat ∖ I
Proof. The inclusion I + R ε e 1 ⊂ I z s a t I+R\varepsilon e_1\subset I^{\mathrm{sat}}_z I + R ε e 1 ⊂ I z sat
z ( ε e 1 ) = ε ( z e 1 + ε e 2 ) ∈ I . z(\varepsilon e_1)=\varepsilon(ze_1+\varepsilon e_2)\in I. z ( ε e 1 ) = ε ( z e 1 + ε e 2 ) ∈ I . Conversely, suppose
z N ( x e 1 + y e 2 ) = c ( z e 1 + ε e 2 ) . z^N(xe_1+ye_2)=c(ze_1+\varepsilon e_2). z N ( x e 1 + y e 2 ) = c ( z e 1 + ε e 2 ) . Write
x = x 0 + ε x 1 , y = y 0 + ε y 1 , c = c 0 + ε c 1 , x i , y i , c i ∈ C { z } . x = x_0 + \varepsilon x_1,\qquad y = y_0 + \varepsilon y_1,\qquad c = c_0 + \varepsilon c_1,\qquad x_i,y_i,c_i \in\mathbb{C}\{z\}. x = x 0 + ε x 1 , y = y 0 + ε y 1 , c = c 0 + ε c 1 , x i , y i , c i ∈ C { z } . The second coordinate gives z N y 0 = 0 z^N y_0 = 0 z N y 0 = 0 z N y 1 = c 0 z^N y_1 = c_0 z N y 1 = c 0 y 0 = 0 y_0 = 0 y 0 = 0 ε \varepsilon ε z N x 0 = z c 0 = z N + 1 y 1 z^N x_0 = z c_0 = z^{N+1} y_1 z N x 0 = z c 0 = z N + 1 y 1 x 0 = z y 1 x_0 = z y_1 x 0 = z y 1
x e 1 + y e 2 − y 1 ( z e 1 + ε e 2 ) = ε x 1 e 1 ∈ R ε e 1 , x e_1 + y e_2 - y_1(z e_1 + \varepsilon e_2) = \varepsilon x_1 e_1 \in R\varepsilon e_1, x e 1 + y e 2 − y 1 ( z e 1 + ε e 2 ) = ε x 1 e 1 ∈ R ε e 1 , which proves the reverse inclusion. Finally, I ∩ R ε e 1 = R z ε e 1 I \cap R\varepsilon e_1 = Rz\varepsilon e_1 I ∩ R ε e 1 = R z ε e 1 (65) is one-dimensional over C \mathbb{C} C
Loop-group interpretation of the wall atlas
Passing to the completion at the pole, put K = C ( ( z ) ) K = \mathbb{C}((z)) K = C (( z )) O = C [ [ z ] ] O = \mathbb{C}[[z]] O = C [[ z ]]
Gr r = G L r ( K ) / G L r ( O ) , Fl r = G L r ( K ) / I , \operatorname{Gr}_r = GL_r(K)/GL_r(O),\qquad\operatorname{Fl}_r = GL_r(K)/I, Gr r = G L r ( K ) / G L r ( O ) , Fl r = G L r ( K ) / I , with I I I [24 ] . A comparison lattice gives a point of Gr r \operatorname{Gr}_r Gr r Fl r \operatorname{Fl}_r Fl r
Proposition 10.13 identifies the relevant matrix. The meromorphic horizontal frame G ( t ) = I 2 + t z − 1 E 21 G(t) = I_2 + t z^{-1} E_{21} G ( t ) = I 2 + t z − 1 E 21 H t H_t H t G L 2 ( O ) GL_2(O) G L 2 ( O ) λ ∈ Z ≥ 1 \lambda\in\mathbb{Z}_{\ge1} λ ∈ Z ≥ 1
In arbitrary rank, Theorem 10.11 gives the lattice-theoretic structure. The wall matrices form the conjugated finite unipotent group D r − 1 U r D r D_r^{-1} U_r D_r D r − 1 U r D r (64) . The rank-three correction is central only inside its minimal 3 × 3 3 \times3 3 × 3 A 1 A_1 A 1 A 1 × A 1 A_1 \times A_1 A 1 × A 1 A 2 A_2 A 2
Finally, Proposition 10.15 shows that reduced loop-group position data do not represent the relative connection problem: the saturation has an additional length-one infinitesimal defect over the dual numbers. Any global local-model stack must retain both the comparison-lattice position and its nilpotent deformation theory, and must prove independence of the chosen logarithmic or comparison lattice. The present results supply the rank-two affine-position theorem and the root-subgroup compatibility law in arbitrary rank, but do not construct that global stack.
Consequences and the global gluing problem
The preceding results determine the local geometry of a fixed residue stratum, a permitted simple wall in arbitrary rank, and the first compatibility between adjacent walls. Each residue stratum has an affine normal-form slice. A simple wall is either excluded by positive integral resonance or represented by a smooth affine torsor whose specialisation to the shifted residue slice is polynomial and triangular by root height. For adjacent walls, the first non-commuting term lies in the composite-root direction and is already visible in the corresponding rank-three Levi subgroup.
These results provide the local data expected in a global wall-crossing space, but they do not construct such a space. Three issues remain. An affine flag position requires a comparison lattice, which is not part of the original prescribed-lattice functor. The wall torsors and their transition data must be descended from the fully framed slices rather than from their individual coordinates. Finally, the rank-three calculation controls the first adjacent-wall compatibility, whereas higher intersections and non-reduced bases require further descent and saturation arguments.
Question 10.16. After adjoining a comparison lattice, can the elementary wall torsors be glued over the Schubert configurations associated with A 1 A_1 A 1 A 2 A_2 A 2
This question lies beyond the moduli problem treated here. It requires a global descent construction and is not used in any of the preceding results. A positive answer would provide a setting in which to ask whether wall-crossing data descend further to residual atoms.
Residual atoms and residual chemical formulae
The preceding sections separate two distinct issues. The affine normal forms describe complex analytic regular-singular connections after a choice of ordered framing, whereas § 7 identifies the descent from those rigidified charts to flagged and unflagged objects. The atomic factors of quantum cohomology are intrinsically unframed and are constructed in the broader formal and non-Archimedean categories of F F F [12 , 17 ] . After defining the unframed vertical residual target for arbitrary order-two lattices, we show that the scalar branch of a one-branch F F F [17 §5] .
The vertical residual stack
Let k k k C \mathbb{C} C S S S D u \mathbb{D}_u D u u u u
Definition 11.1. The groupoid M ≤ 2 , k u n f r ( S ) \mathcal{M}^{\mathrm{unfr}}_{\leq2,k}(S) M ≤ 2 , k unfr ( S ) ( E , ∇ / S ) (\mathcal{E},\nabla/S) ( E , ∇/ S ) E \mathcal{E} E S × D u S \times\mathbb{D}_u S × D u
∇ / S : E ⟶ E ⊗ Ω ( S × D u ) / S 1 ( ∗ { u = 0 } ) \nabla/S : \mathcal{E} \longrightarrow\mathcal{E} \otimes\Omega^1_{(S\times\mathbb{D}_u)/S}(*\{u=0\}) ∇/ S : E ⟶ E ⊗ Ω ( S × D u ) / S 1 ( ∗ { u = 0 }) is a flat relative meromorphic connection satisfying
∇ / S ( E ) ⊂ E ⊗ u − 2 Ω ( S × D u ) / S 1 . (66) \nabla/S(\mathcal{E}) \subset\mathcal{E} \otimes u^{-2}\Omega^1_{(S\times\mathbb{D}_u)/S}. \tag*{(66)} ∇/ S ( E ) ⊂ E ⊗ u − 2 Ω ( S × D u ) / S 1 . (66) Morphisms are bundle isomorphisms intertwining the relative connections. Neither a framing nor an invariant flag is included. The regular-singular subfunctor is denoted by M r s , k u n f r \mathcal{M}^{\mathrm{unfr}}_{\mathrm{rs},k} M rs , k unfr
Proposition 11.2. In each of the three categories, the assignments S ↦ M ≤ 2 , k u n f r ( S ) S \mapsto\mathcal{M}^{\mathrm{unfr}}_{\leq2,k}(S) S ↦ M ≤ 2 , k unfr ( S ) S ↦ M r s , k u n f r ( S ) S \mapsto\mathcal{M}^{\mathrm{unfr}}_{\mathrm{rs},k}(S) S ↦ M rs , k unfr ( S )
Proof. Vector bundles satisfy effective descent in each of the three categories. Suppose that ( E i , ∇ i ) (\mathcal{E}_i,\nabla_i) ( E i , ∇ i ) { S i → S } \{S_i \to S\} { S i → S } ∇ i \nabla_i ∇ i E \mathcal{E} E S S S (66) is also local and therefore descends.
For the regular-singular subfunctor, regular singularity is imposed locally through the existence of a logarithmic lattice and is preserved by pullback. Such a lattice is auxiliary and is not part of the object, so the descent problem concerns only the meromorphic connection itself. In the formal category one uses completed tensor products; in the non-Archimedean category one uses effective descent for vector bundles and coherent morphisms in the admissible topology, as in the foundational setup of [12 §2] . Base-change compatibility follows from the same local description. □
The stack M ≤ 2 , k u n f r \mathcal{M}^{\mathrm{unfr}}_{\leq2,k} M ≤ 2 , k unfr F F F F F F
Scalar branches and exponential shifts
For an F F F ( H , ∇ ) / S (\mathcal{H},\nabla)/S ( H , ∇ ) / S
κ ∇ : = ∇ u 2 ∂ u ∣ H / u H . \kappa_\nabla:= \left.\nabla_{u^2\partial_u}\right|_{\mathcal{H}/u\mathcal{H}}. κ ∇ := ∇ u 2 ∂ u ∣ H / u H . This is the residual endomorphism used in the spectral decomposition theory [12 §3] and in the definition of atoms [17 §§3–5] .
Definition 11.3. Let ( H , ∇ ) / S (\mathcal{H},\nabla)/S ( H , ∇ ) / S n n n S S S a ∈ Γ ( S , O S ) a \in\Gamma(S,\mathcal{O}_S) a ∈ Γ ( S , O S ) Fbund a ( S ) \operatorname{Fbund}_a(S) Fbund a ( S ) F F F
det ( T I − κ ∇ ) = ( T + a ) n in Γ ( S , O S ) [ T ] . \det(TI-\kappa_\nabla)=(T+a)^n \qquad\text{in }\Gamma(S,\mathcal{O}_S)[T]. det ( T I − κ ∇ ) = ( T + a ) n in Γ ( S , O S ) [ T ] . Let Fbund 1 b r ( S ) \operatorname{Fbund}_{1\mathrm{br}}(S) Fbund 1 br ( S ) Fbund a ( S ) \operatorname{Fbund}_a(S) Fbund a ( S ) a a a Fbund 0 ( S ) \operatorname{Fbund}_0(S) Fbund 0 ( S ) − a -a − a
The variable exponential line of branch a a a F F F
E a : = ( O S × D u , d + d ( a / u ) ) . E_a := \left(\mathcal{O}_{S\times\mathbb{D}_u},\,d+d(a/u)\right). E a := ( O S × D u , d + d ( a / u ) ) . Its residual endomorphism is multiplication by − a -a − a
Lemma 11.4. For an object of F b u n d 1 b r ( S ) \mathrm{Fbund}_{1\mathrm{br}}(S) Fbund 1 br ( S ) n n n
a = − 1 n tr ( κ ∇ ) . a=-\frac{1}{n}\operatorname{tr}(\kappa_{\nabla}). a = − n 1 tr ( κ ∇ ) . Moreover, N ∇ : = κ ∇ + a I N_{\nabla}:=\kappa_{\nabla}+aI N ∇ := κ ∇ + a I N ∇ n = 0 N_{\nabla}^{n}=0 N ∇ n = 0
Proof. If det ( T I − κ ∇ ) = ( T + a ) n \det(TI-\kappa_{\nabla})=(T+a)^{n} det ( T I − κ ∇ ) = ( T + a ) n T n − 1 T^{n-1} T n − 1 − tr ( κ ∇ ) = n a -\operatorname{tr}(\kappa_{\nabla})=na − tr ( κ ∇ ) = na n n n O S \mathcal{O}_{S} O S a a a N ∇ = κ ∇ + a I N_{\nabla}=\kappa_{\nabla}+aI N ∇ = κ ∇ + a I T n T^{n} T n N ∇ n = 0 N_{\nabla}^{n}=0 N ∇ n = 0
For constant a a a [17 §3.3] ; allowing a a a
Theorem 11.5. For every admissible base S S S
F b u n d 1 b r ( S ) ⟶ ∼ Γ ( S , O S ) d i s c × F b u n d 0 ( S ) , ( H , ∇ ) ⟼ ( a , H , ∇ − d ( a / u ) I ) , (67) \mathrm{Fbund}_{1\mathrm{br}}(S)\overset{\sim}{\longrightarrow}\Gamma(S,\mathcal{O}_{S})_{\mathrm{disc}}\times\mathrm{Fbund}_{0}(S),\qquad(\mathcal{H},\nabla)\longmapsto(a,\mathcal{H},\nabla-d(a/u)I), \tag*{(67)} Fbund 1 br ( S ) ⟶ ∼ Γ ( S , O S ) disc × Fbund 0 ( S ) , ( H , ∇ ) ⟼ ( a , H , ∇ − d ( a / u ) I ) , (67) where, on each connected rank component, a = − ( 1 / rk H ) tr ( κ ∇ ) a=-(1/\operatorname{rk}\mathcal{H})\operatorname{tr}(\kappa_{\nabla}) a = − ( 1/ rk H ) tr ( κ ∇ ) Res a ( H , ∇ ) = ( H , ∇ − d ( a / u ) I ) \operatorname{Res}_{a}(\mathcal{H},\nabla)=(\mathcal{H},\nabla-d(a/u)I) Res a ( H , ∇ ) = ( H , ∇ − d ( a / u ) I )
Res a + b ( H 1 ⊗ H 2 ) ≅ Res a ( H 1 ) ⊗ Res b ( H 2 ) , (68) \operatorname{Res}_{a+b}(\mathcal{H}_{1}\otimes\mathcal{H}_{2})\cong\operatorname{Res}_{a}(\mathcal{H}_{1})\otimes\operatorname{Res}_{b}(\mathcal{H}_{2}), \tag*{(68)} Res a + b ( H 1 ⊗ H 2 ) ≅ Res a ( H 1 ) ⊗ Res b ( H 2 ) , (68) Res − a ( H ∨ ) ≅ Res a ( H ) ∨ . (69) \operatorname{Res}_{-a}(\mathcal{H}^{\vee})\cong\operatorname{Res}_{a}(\mathcal{H})^{\vee}. \tag*{(69)} Res − a ( H ∨ ) ≅ Res a ( H ) ∨ . (69) The same statement holds for external products over different bases.
Proof. The branch function is intrinsic and base-change compatible by Lemma 11.4. Since d ( a / u ) d(a/u) d ( a / u )
d ( a / u ) = d S a u − a u 2 d u . (70) d(a/u)=\frac{d_{S}a}{u}-\frac{a}{u^{2}}\,du. \tag*{(70)} d ( a / u ) = u d S a − u 2 a d u . (70) For a vector field ξ \xi ξ S S S
( ∇ r e s ) u ξ = ∇ u ξ − ξ ( a ) I , (71) (\nabla^{\mathrm{res}})_{u\xi}=\nabla_{u\xi}-\xi(a)I, \tag*{(71)} ( ∇ res ) u ξ = ∇ u ξ − ξ ( a ) I , (71) ( ∇ r e s ) u 2 ∂ u = ∇ u 2 ∂ u + a I . (72) (\nabla^{\mathrm{res}})_{u^{2}\partial_{u}}=\nabla_{u^{2}\partial_{u}}+aI. \tag*{(72)} ( ∇ res ) u 2 ∂ u = ∇ u 2 ∂ u + a I . (72) Both correction terms are regular endomorphisms of the intrinsic lattice. Hence the F F F
κ ∇ r e s = κ ∇ + a I , \kappa_{\nabla^{\mathrm{res}}}=\kappa_{\nabla}+aI, κ ∇ res = κ ∇ + a I , whose characteristic polynomial is T n T^{n} T n F b u n d 0 ( S ) \mathrm{Fbund}_{0}(S) Fbund 0 ( S ) ( a , H , ∇ 0 ) (a,\mathcal{H},\nabla^{0}) ( a , H , ∇ 0 ) d ( a / u ) I d(a/u)I d ( a / u ) I ( T + a ) n (T+a)^{n} ( T + a ) n
For tensor products, the residual endomorphism is
κ ∇ 1 ⊗ ∇ 2 = κ ∇ 1 ⊗ I + I ⊗ κ ∇ 2 . \kappa_{\nabla_{1}\otimes\nabla_{2}}=\kappa_{\nabla_{1}}\otimes I+I\otimes\kappa_{\nabla_{2}}. κ ∇ 1 ⊗ ∇ 2 = κ ∇ 1 ⊗ I + I ⊗ κ ∇ 2 . Writing κ ∇ i = − a i I + N i \kappa_{\nabla_{i}}=-a_{i}I+N_{i} κ ∇ i = − a i I + N i N i N_{i} N i N 1 ⊗ I + I ⊗ N 2 N_{1}\otimes I+I\otimes N_{2} N 1 ⊗ I + I ⊗ N 2 a 1 + a 2 a_{1}+a_{2} a 1 + a 2
d ( ( a 1 + a 2 ) / u ) = d ( a 1 / u ) + d ( a 2 / u ) , d((a_{1}+a_{2})/u)=d(a_{1}/u)+d(a_{2}/u), d (( a 1 + a 2 ) / u ) = d ( a 1 / u ) + d ( a 2 / u ) , this gives (68) . The dual residual endomorphism is − κ ∇ t -\kappa_{\nabla}^{\mathsf{t}} − κ ∇ t − a -a − a (69) . Symmetry and associativity are inherited from the tensor category of F F F [17 §3.2] . □
The operation (11.2) changes the connection on the same intrinsic lattice. It is not a holomorphic gauge transformation at u = 0 u=0 u = 0 e − a / u e^{-a/u} e − a / u
Pairings, duality and enhanced atoms
Let ν : S × D u → S × D u \nu: S \times\mathrm{Du} \to S \times\mathrm{Du} ν : S × Du → S × Du ν ( s , u ) = ( s , − u ) \nu(s,u)=(s,-u) ν ( s , u ) = ( s , − u ) F F F [17 §6.4] .
Proposition 11.6. Let ( H , ∇ ) / S (\mathcal{H},\nabla)/S ( H , ∇ ) / S F F F a a a
ψ : H ⊗ ν ∗ H ⟶ O S × D u \psi:\mathcal{H}\otimes\nu^*\mathcal{H}\longrightarrow\mathcal{O}_{S\times\mathrm{Du}} ψ : H ⊗ ν ∗ H ⟶ O S × Du be a horizontal non-degenerate pairing. Then the same morphism is horizontal for the residualised connection. In the complex analytic category, scalar residualisation leaves the punctured monodromy, and hence every duality automorphism defined from that monodromy, unchanged. Consequently the construction extends to atoms enhanced by such pairings and duality automorphisms.
Proof. The scalar correction on the first factor is − d ( a / u ) -\mathrm{d}(a/u) − d ( a / u )
ν ∗ d ( a / u ) = d ( a / ( − u ) ) = − d ( a / u ) , \nu^*\mathrm{d}(a/u)=\mathrm{d}(a/(-u))=-\mathrm{d}(a/u), ν ∗ d ( a / u ) = d ( a / ( − u )) = − d ( a / u ) , the correction on ν ∗ H \nu^*\mathcal{H} ν ∗ H ψ \psi ψ e − a / u e^{-a/u} e − a / u [17 §6.4] is preserved. □
The proposition does not address integral or Γ \Gamma Γ [17 §6.4] .
Residual atoms
Fix a motivic symmetry pair ( G , ϵ G ) (G,\epsilon_G) ( G , ϵ G ) K K K k k k A A A F F F Atoms G K , F \operatorname{Atoms}^{K,F}_{G} Atoms G K , F CF G F ( X ) \operatorname{CF}^{F}_{G}(X) CF G F ( X ) [17 §5.3] . For a connected component U U U ( A U , ∇ U , a U ) / U (\mathcal{A}_{U},\nabla_{U},a_{U})/U ( A U , ∇ U , a U ) / U G G G F F F
( A U , ∇ U r e s ) = Res a U ( A U , ∇ U ) (\mathcal{A}_{U},\nabla^{\mathrm{res}}_{U})=\operatorname{Res}_{a_{U}}(\mathcal{A}_{U},\nabla_{U}) ( A U , ∇ U res ) = Res a U ( A U , ∇ U ) and a classifying morphism
Φ U : U ⟶ M ≤ 2 , k u n f r . \Phi_{U}:U\longrightarrow\mathcal{M}^{\mathrm{unfr}}_{\leq2,k}. Φ U : U ⟶ M ≤ 2 , k unfr . Definition 11.7. A local residual G G G
r U = ( U , a U , ( A U , ∇ U r e s ) , Φ U , ϵ U , ρ U , T U ) , r_{U}=\left(U,a_{U},(\mathcal{A}_{U},\nabla^{\mathrm{res}}_{U}),\Phi_{U},\epsilon_{U},\rho_{U},\mathcal{T}_{U}\right), r U = ( U , a U , ( A U , ∇ U res ) , Φ U , ϵ U , ρ U , T U ) , where ( A U , ∇ U r e s ) / U (\mathcal{A}_{U},\nabla^{\mathrm{res}}_{U})/U ( A U , ∇ U res ) / U F F F Φ U \Phi_{U} Φ U ϵ U \epsilon_{U} ϵ U ρ U \rho_{U} ρ U G G G T U \mathcal{T}_{U} T U
A residual atomic correspondence is obtained by applying scalar branch splitting to an atomic correspondence. The set RAtoms G K \operatorname{RAtoms}^{K}_{G} RAtoms G K G G G U U U a U a_{U} a U F F F Φ U \Phi_{U} Φ U
Proposition 11.8. Scalar branch splitting induces a bijection
res : Atoms G K , F → ∼ RAtoms G K . \operatorname{res}:\operatorname{Atoms}^{K,F}_{G}\xrightarrow{\sim}\operatorname{RAtoms}^{K}_{G}. res : Atoms G K , F ∼ RAtoms G K . Every atomic correspondence induces the corresponding equivalence of residual atoms and preserves the intrinsic lattice, parity, G G G Proposition 11.6 .
Proof. On a one-branch factor the branch function is intrinsic and is recovered by the trace formula of Lemma 11.4 . By Theorem 11.5 , the pair ( a U , ∇ U r e s ) (a_{U},\nabla^{\mathrm{res}}_{U}) ( a U , ∇ U res ) ∇ U \nabla_{U} ∇ U d ( a U / u ) I \mathrm{d}(a_{U}/u)I d ( a U / u ) I Definition 11.7 makes the reconstruction unique.
An atomic correspondence is represented, after the prescribed base and quantum-variable changes, by an isomorphism
Ψ : f ∗ ( A ′ , ∇ ′ ) → ∼ ( A , ∇ ) . \Psi:f^*(\mathcal{A}',\nabla')\xrightarrow{\sim}(\mathcal{A},\nabla). Ψ : f ∗ ( A ′ , ∇ ′ ) ∼ ( A , ∇ ) . Conjugacy identifies the residual endomorphisms and therefore gives a = f ∗ a ′ a=f^{*}a' a = f ∗ a ′ Ψ ∘ f ∗ ( ∇ ′ − d ( a ′ / u ) I ) = ( ∇ − d ( a / u ) I ) ∘ Ψ \Psi\circ f^{*}(\nabla'-d(a'/u)I)=(\nabla-d(a/u)I)\circ\Psi Ψ ∘ f ∗ ( ∇ ′ − d ( a ′ / u ) I ) = ( ∇ − d ( a / u ) I ) ∘ Ψ
This is the residual correspondence. The remaining structures are preserved because the correction is central, and the assertions about pairings and duality follow from Proposition 11.6 . The inverse reconstruction shows that no further identifications have been introduced.
Residual chemical formulae
Let X X X K K K U ~ X → U X \widetilde{U}_X\to U_X U X → U X A A A F F F m α m_\alpha m α U ~ X , α → U X \widetilde{U}_{X,\alpha}\to U_X U X , α → U X
Definition 11.9. The residual chemical formula of X X X
CF G r e s ( X ) : = ∑ α ∈ π 0 ( U ~ X ) m α δ r α ∈ N ( RAtoms G K ) (73) \operatorname{CF}^{\mathrm{res}}_G(X):=\sum_{\alpha\in\pi_0(\widetilde{U}_X)}m_\alpha\delta_{r_\alpha}\in\mathbb{N}^{(\operatorname{RAtoms}^K_G)} \tag*{(73)} CF G res ( X ) := α ∈ π 0 ( U X ) ∑ m α δ r α ∈ N ( RAtoms G K ) (73) where δ r \delta_r δ r r r r Z ( RAtoms G K ) \mathbb{Z}^{(\operatorname{RAtoms}^K_G)} Z ( RAtoms G K )
Theorem 11.10. The residual chemical formula is independent of all local trivialisations, auxiliary framings, rigid points and decomposition isomorphisms used in its construction. It is compatible with atomic correspondences and satisfies
CF G r e s ( X 1 ⊔ X 2 ) = CF G r e s ( X 1 ) + CF G r e s ( X 2 ) , (74) \operatorname{CF}^{\mathrm{res}}_G(X_1\sqcup X_2)=\operatorname{CF}^{\mathrm{res}}_G(X_1)+\operatorname{CF}^{\mathrm{res}}_G(X_2), \tag*{(74)} CF G res ( X 1 ⊔ X 2 ) = CF G res ( X 1 ) + CF G res ( X 2 ) , (74) CF G r e s ( Bl Z X ) = CF G r e s ( X ) + ( c − 1 ) CF G r e s ( Z ) , (75) \operatorname{CF}^{\mathrm{res}}_G(\operatorname{Bl}_Z X)=\operatorname{CF}^{\mathrm{res}}_G(X)+(c-1)\operatorname{CF}^{\mathrm{res}}_G(Z), \tag*{(75)} CF G res ( Bl Z X ) = CF G res ( X ) + ( c − 1 ) CF G res ( Z ) , (75) CF G r e s ( P X ( V ) ) = m CF G r e s ( X ) , (76) \operatorname{CF}^{\mathrm{res}}_G(\mathbb{P}_X(V))=m\operatorname{CF}^{\mathrm{res}}_G(X), \tag*{(76)} CF G res ( P X ( V )) = m CF G res ( X ) , (76) where Z ⊂ X Z\subset X Z ⊂ X c ≥ 2 c\ge2 c ≥ 2 V V V m m m
Proof. The spectral factors are determined over connected spectral components by the formal and non-Archimedean decomposition theorems [12 Theorems 3.31 and 3.41] ; the extension and reconstruction of framings are unique in the sense of [12 §4] . The equivalence relation defining geometric atomic F F F [17 Definition 5.21 and Proposition 5.22] . Since scalar branch splitting commutes with restriction and base change, the residual families glue and are independent of all such choices.
The disjoint-union identity follows directly from the decomposition into connected components. The blow-up and projective-bundle identities for atomic F F F F F F [17 Theorems 4.5 and 4.11, Proposition 5.22] , using the existence theorems of Iritani [15 ] and Iritani–Koto [16 ] and the uniqueness results of [12 §5] . Applying the bijection of Proposition 11.8 term by term gives (11.9)–(11.11).
The set Atoms G K , F \operatorname{Atoms}^{K,F}_G Atoms G K , F F F F N ( Atoms G K , F ) \mathbb{N}^{(\operatorname{Atoms}^{K,F}_G)} N ( Atoms G K , F ) Z ( Atoms G K , F ) \mathbb{Z}^{(\operatorname{Atoms}^{K,F}_G)} Z ( Atoms G K , F ) [17 §5.3] . Transport this multiplication through the bijection of Proposition 11.8 .
Corollary 11.11. The monoid N ( RAtoms G K ) \mathbb{N}^{(\operatorname{RAtoms}^K_G)} N ( RAtoms G K ) Z ( RAtoms G K ) \mathbb{Z}^{(\operatorname{RAtoms}^K_G)} Z ( RAtoms G K )
[ X ] ⟼ CF G r e s ( X ) [X]\longmapsto\operatorname{CF}^{\mathrm{res}}_G(X) [ X ] ⟼ CF G res ( X ) extends to a ring homomorphism
cf G r e s : K 0 ( Var K ) ⟶ Z ( RAtoms G K ) . \operatorname{cf}^{\mathrm{res}}_G:K_0(\operatorname{Var}_K)\longrightarrow\mathbb{Z}^{(\operatorname{RAtoms}^K_G)}. cf G res : K 0 ( Var K ) ⟶ Z ( RAtoms G K ) . Proof. The branch-residual splitting is a bijection on atomic equivalence classes, so the multiplication on atomic F F F E = P Z ( N Z / X ) E=\mathbb{P}_Z(N_{Z/X}) E = P Z ( N Z / X ) c = codim X Z c=\operatorname{codim}_X Z c = codim X Z (75) and (76) ,
CF G r e s ( Bl Z X ) − CF G r e s ( E ) = CF G r e s ( X ) + ( c − 1 ) CF G r e s ( Z ) − c CF G r e s ( Z ) = CF G r e s ( X ) − CF G r e s ( Z ) . \operatorname{CF}^{\mathrm{res}}_G(\operatorname{Bl}_Z X)-\operatorname{CF}^{\mathrm{res}}_G(E)
=\operatorname{CF}^{\mathrm{res}}_G(X)+(c-1)\operatorname{CF}^{\mathrm{res}}_G(Z)-c\operatorname{CF}^{\mathrm{res}}_G(Z)
=\operatorname{CF}^{\mathrm{res}}_G(X)-\operatorname{CF}^{\mathrm{res}}_G(Z). CF G res ( Bl Z X ) − CF G res ( E ) = CF G res ( X ) + ( c − 1 ) CF G res ( Z ) − c CF G res ( Z ) = CF G res ( X ) − CF G res ( Z ) . Together with additivity for disjoint unions, this is the relation in Bittner’s presentation of K 0 ( Var K ) K_0(\operatorname{Var}_K) K 0 ( Var K ) [4 ] . The transported product gives multiplicativity.
Corollary 11.12. Let X X X d d d CF G r e s ( X ) \operatorname{CF}^{\mathrm{res}}_G(X) CF G res ( X ) d − 2 d-2 d − 2 X X X
Proof. The birational input is the non-rationality criterion of Katzarkov–Kontsevich–Pantev–Yu [17 §6] : if a geometric atomic F F F d d d d − 2 d-2 d − 2 d d d Proposition 11.8 , branch-residual splitting induces a bijection
Atoms G K , F ⟶ ∼ RAtoms G K \operatorname{Atoms}_{G}^{K,F} \overset{\sim}{\longrightarrow} \operatorname{RAtoms}_{G}^{K} Atoms G K , F ⟶ ∼ RAtoms G K which preserves atomic correspondences and hence the dimension filtration. Therefore a residual atom occurs in dimension at most d − 2 d-2 d − 2 [17 §6] to that preimage proves the assertion.
Thus the corollary transports the KKPYY obstruction through the branch-residual correspondence rather than strengthening the birational criterion. The contribution here is the residual presentation of the same obstruction, together with the lattice strata carried functorially by its residual representatives.
Determinantal strata of residual families
Definition 11.13. For ( E , ∇ / S ) ∈ M ≤ 2 , k u n f r ( S ) (\mathcal{E},\nabla/S) \in\mathcal{M}_{\leq2,k}^{\mathrm{unfr}}(S) ( E , ∇/ S ) ∈ M ≤ 2 , k unfr ( S )
κ E r e s : = u 2 ∇ ∂ u ∣ u = 0 ∈ End O S ( E / u E ) . (77) \kappa_{\mathcal{E}}^{\mathrm{res}} := \left.u^{2}\nabla_{\partial_u}\right|_{u=0} \in\operatorname{End}_{\mathcal{O}_S}(\mathcal{E}/u\mathcal{E}). \tag*{(77)} κ E res := u 2 ∇ ∂ u u = 0 ∈ End O S ( E / u E ) . (77) For the vertical connection underlying an object of Fbund 0 ( S ) \operatorname{Fbund}_0(S) Fbund 0 ( S ) T n T^n T n ( κ E r e s ) n = 0 (\kappa_{\mathcal{E}}^{\mathrm{res}})^n=0 ( κ E res ) n = 0 Lemma 11.4 . More generally, the rank-profile construction below applies whenever the endomorphism is fibrewise nilpotent. If n = rk E n=\operatorname{rk}\mathcal{E} n = rk E s ∈ S s\in S s ∈ S
JType l a t ( s ) : = ( rk κ s r e s , rk ( κ s r e s ) 2 , … , rk ( κ s r e s ) n − 1 ) . \operatorname{JType}_{\mathrm{lat}}(s):=\left(\operatorname{rk}\kappa_s^{\mathrm{res}},\operatorname{rk}(\kappa_s^{\mathrm{res}})^2,\ldots,\operatorname{rk}(\kappa_s^{\mathrm{res}})^{n-1}\right). JType lat ( s ) := ( rk κ s res , rk ( κ s res ) 2 , … , rk ( κ s res ) n − 1 ) . This tuple determines, and is determined by, the Jordan partition of the nilpotent endomorphism κ s r e s \kappa_s^{\mathrm{res}} κ s res
Theorem 11.14. The rank loci of the powers of κ E r e s \kappa_{\mathcal{E}}^{\mathrm{res}} κ E res S S S JType l a t \operatorname{JType}_{\mathrm{lat}} JType lat CF G r e s ( X ) \operatorname{CF}_{G}^{\mathrm{res}}(X) CF G res ( X )
Proof. A lattice isomorphism conjugates κ E r e s \kappa_{\mathcal{E}}^{\mathrm{res}} κ E res E / u E \mathcal{E}/u\mathcal{E} E / u E m m m q q q
D m , q : = { rk ( ( κ r e s ) m ) ≤ q } . D_{m,q}:=\{\operatorname{rk}((\kappa^{\mathrm{res}})^m)\leq q\}. D m , q := { rk (( κ res ) m ) ≤ q } . Scheme-theoretically, D m , q D_{m,q} D m , q ( q + 1 ) (q+1) ( q + 1 ) ( κ r e s ) m (\kappa^{\mathrm{res}})^m ( κ res ) m q = ( q 1 , … , q n − 1 ) \mathbf{q}=(q_1,\ldots,q_{n-1}) q = ( q 1 , … , q n − 1 ) rk ( ( κ r e s ) m ) = q m \operatorname{rk}((\kappa^{\mathrm{res}})^m)=q_m rk (( κ res ) m ) = q m m m m
⋂ m = 1 n − 1 ( D m , q m ∖ D m , q m − 1 ) , D m , − 1 = ∅ . \bigcap_{m=1}^{n-1}(D_{m,q_m}\setminus D_{m,q_m-1}),\qquad D_{m,-1}=\varnothing. m = 1 ⋂ n − 1 ( D m , q m ∖ D m , q m − 1 ) , D m , − 1 = ∅ . These locally closed loci are pairwise disjoint and cover S S S n n n n n n Proposition 11.8 .
Proposition 11.15. Let ( E , ∇ / S ) (\mathcal{E},\nabla/S) ( E , ∇/ S ) n n n Fbund 0 ( S ) \operatorname{Fbund}_0(S) Fbund 0 ( S ) N n ⊆ g l n \mathcal{N}_n\subseteq\mathfrak{gl}_n N n ⊆ gl n
J E : S ⟶ [ N n / GL n ] \mathcal{J}_{\mathcal{E}}:S\longrightarrow[\mathcal{N}_n/\operatorname{GL}_n] J E : S ⟶ [ N n / GL n ] in the corresponding analytic, formal or non-Archimedean category. It is functorial under base change and invariant under isomorphism of residual families. The rank-profile stratification of Theorem 11.14 is the pullback of the Jordan-type stratification of [ N n / GL n ] [\mathcal{N}_n/\operatorname{GL}_n] [ N n / GL n ]
Proof. Let V = E / u E V=\mathcal{E}/u\mathcal{E} V = E / u E Fr ( V ) → S \operatorname{Fr}(V)\to S Fr ( V ) → S Lemma 11.4 , the characteristic polynomial of κ E r e s \kappa_{\mathcal{E}}^{\mathrm{res}} κ E res T n T^n T n V V V N n \mathcal{N}_n N n κ E r e s \kappa_{\mathcal{E}}^{\mathrm{res}} κ E res GL n \operatorname{GL}_n GL n
Fr ( V ) ⟶ N n , \operatorname{Fr}(V)\longrightarrow\mathcal{N}_n, Fr ( V ) ⟶ N n , which descends to the stated morphism to the quotient stack. Pullback of V V V
A low-dimensional representative criterion
The preceding non-rationality criterion becomes effective once one can exclude all atoms arising in dimension at most two. The residual endomorphism supplies a simple necessary condition for such a low-dimensional origin.
Let
E = ( H , ∇ r e s ) / U \mathcal{E} = (\mathcal{H},\nabla^{\mathrm{res}})/U E = ( H , ∇ res ) / U be a zero-branch residual family. For m ≥ 1 m \ge1 m ≥ 1
Σ m ( E ) : = Z ( ( κ r e s ) m ) ⊂ U . (78) \Sigma_m(\mathcal{E}) := Z\left((\kappa^{\mathrm{res}})^m\right) \subset U. \tag*{(78)} Σ m ( E ) := Z ( ( κ res ) m ) ⊂ U . (78) Its underlying set is { s ∈ U : ( κ s r e s ) m = 0 } \{s \in U : (\kappa_s^{\mathrm{res}})^m = 0\} { s ∈ U : ( κ s res ) m = 0 } Σ m ( E ) \Sigma_m(\mathcal{E}) Σ m ( E )
Theorem 11.16. If a residual atom r \mathbf{r} r r \mathbf{r} r E = ( H , ∇ r e s ) / U \mathcal{E} = (\mathcal{H},\nabla^{\mathrm{res}})/U E = ( H , ∇ res ) / U Σ 3 ( E ) \Sigma_3(\mathcal{E}) Σ 3 ( E )
Proof. For a point, the maximal A A A C C C C ≃ P 1 C \simeq\mathbb{P}^1 C ≃ P 1 g ( C ) ≥ 1 g(C) \ge1 g ( C ) ≥ 1 K C K_C K C [17 Lemma 5.24] shows that the maximal A A A F F F q β = 0 q^\beta= 0 q β = 0 t = 0 t = 0 t = 0 [17 §3.5] identifies the residual endomorphism with − c 1 ( T C ) ⌣ ( − ) -c_1(T_C) \smile(-) − c 1 ( T C ) ⌣ ( − )
Let S S S ( − 1 ) (-1) ( − 1 ) S S S S min S_{\min} S m i n S S S S min S_{\min} S m i n S min S_{\min} S m i n − ∞ -\infty − ∞ P 2 \mathbb{P}^2 P 2
It remains to consider a minimal surface with non-negative Kodaira dimension. Its canonical divisor is nef. By the one-atom statement for varieties with nef canonical class in [17 Lemma 5.24] , the unique atom is represented by the maximal A A A F F F A A A [17 §3.5] shows that the quantum product specialises to the ordinary cup product and the Euler element specialises to c 1 ( T S ) c_1(T_S) c 1 ( T S )
κ 0 r e s = − c 1 ( T S ) ⌣ ( − ) . \kappa^{\mathrm{res}}_0 = -c_1(T_S) \smile(-). κ 0 res = − c 1 ( T S ) ⌣ ( − ) . Three successive multiplications by a class of degree two land in cohomological degree at least six, but the cohomology of a surface vanishes in those degrees. Thus ( κ 0 r e s ) 3 = 0 (\kappa^{\mathrm{res}}_0)^3 = 0 ( κ 0 res ) 3 = 0
Corollary 11.17. Let X X X r \mathbf{r} r C F G r e s ( X ) \mathrm{CF}^{\mathrm{res}}_G(X) CF G res ( X ) r \mathbf{r} r X X X
Proof. By Theorem 11.16, an atom occurring in dimension at most two has at least one representative with non-empty cube-zero locus. The hypothesis therefore excludes every realisation of r \mathbf{r} r Corollary 11.12 applies.
The corollary is deliberately stated uniformly over all representatives. A single empty cube-zero locus is sufficient only when the comparison map to every other representative is known to preserve emptiness, for example under an isomorphism or a surjective base change. The criterion is not asserted to be necessary.
The residual profile of a projective K3 surface
For a projective K 3 K3 K 3 [18 , 23 ] . The ordinary genus-zero quantum product entering the A A A F F F
Let S S S K 3 K3 K 3 1 ∈ H 0 ( S ) 1 \in H^0(S) 1 ∈ H 0 ( S ) p ∈ H 4 ( S ) \mathrm{p} \in H^4(S) p ∈ H 4 ( S ) t 0 t^0 t 0 t 4 t^4 t 4 H 2 ( S ) H^2(S) H 2 ( S ) c 1 ( T S ) = 0 c_1(T_S)=0 c 1 ( T S ) = 0
E u = t 0 1 − t 4 p . (79) \mathrm{Eu}=t^0 1-t^4\mathrm{p}. \tag*{(79)} Eu = t 0 1 − t 4 p . (79) Theorem 11.18. The maximal A A A F F F K 3 K3 K 3
κ r e s = t 4 L p , L p ( x ) = p ⌣ x . (80) \kappa^{\mathrm{res}}=t^4L_{\mathrm{p}},\qquad L_{\mathrm{p}}(x)=\mathrm{p}\smile x. \tag*{(80)} κ res = t 4 L p , L p ( x ) = p ⌣ x . (80) In particular,
( κ r e s ) 2 = 0 , rk κ r e s ≤ 1 \left(\kappa^{\mathrm{res}}\right)^2=0,\qquad\operatorname{rk}\kappa^{\mathrm{res}}\leq1 ( κ res ) 2 = 0 , rk κ res ≤ 1 identically on the Hodge base. The residual lattice stratification has the following two profiles
rk κ r e s = 0 on { t 4 = 0 } , rk κ r e s = 1 on { t 4 ≠ 0 } . \operatorname{rk}\kappa^{\mathrm{res}}=0\quad\text{on }\{t^4=0\},\qquad\operatorname{rk}\kappa^{\mathrm{res}}=1\quad\text{on }\{t^4\ne0\}. rk κ res = 0 on { t 4 = 0 } , rk κ res = 1 on { t 4 = 0 } . Proof. Because the ordinary quantum product is the cup product, quantum multiplication by the Euler element (79) is
L E u = t 0 I − t 4 L p . L_{\mathrm{Eu}}=t^0I-t^4L_{\mathrm{p}}. L Eu = t 0 I − t 4 L p . With the A A A κ ∇ = − L E u \kappa_{\nabla}=-L_{\mathrm{Eu}} κ ∇ = − L Eu
κ ∇ = − t 0 I + t 4 L p . \kappa_{\nabla}=-t^0I+t^4L_{\mathrm{p}}. κ ∇ = − t 0 I + t 4 L p . The operator L p L_{\mathrm{p}} L p 1 1 1 p \mathrm{p} p H 2 ( S ) ⊕ H 4 ( S ) H^2(S)\oplus H^4(S) H 2 ( S ) ⊕ H 4 ( S ) L p 2 = 0 L_{\mathrm{p}}^2=0 L p 2 = 0 rk L p = 1 \operatorname{rk}L_{\mathrm{p}}=1 rk L p = 1 κ ∇ \kappa_{\nabla} κ ∇ ( T + t 0 ) dim H ∙ ( S ) (T+t^0)^{\dim H^\bullet(S)} ( T + t 0 ) d i m H ∙ ( S ) a = t 0 a=t^0 a = t 0 t 0 I t^0I t 0 I κ ∇ \kappa_{\nabla} κ ∇ (80) . If t 4 = 0 t^4=0 t 4 = 0 t 4 ≠ 0 t^4\ne0 t 4 = 0 L p L_{\mathrm{p}} L p
Corollary 11.19. A residual family isomorphic to the standard maximal A A A K 3 K3 K 3
( κ r e s ) 2 = 0 , rk κ r e s ≤ 1 \left(\kappa^{\mathrm{res}}\right)^2=0,\qquad\operatorname{rk}\kappa^{\mathrm{res}}\leq1 ( κ res ) 2 = 0 , rk κ res ≤ 1 throughout its base. Hence a residual family violating either bound is not isomorphic, as a residual family, to that standard K 3 K3 K 3
Proof. An isomorphism of residual families conjugates the residual endomorphisms fibrewise, so the assertion follows directly from Theorem 11.18. □
Corollary 11.20. Let X X X r \mathbf{r} r C F G r e s ( X ) \mathrm{CF}^{\mathrm{res}}_G(X) CF G res ( X ) r \mathbf{r} r K 3 K3 K 3 r \mathbf{r} r
( κ r e s ) 2 = 0 , rk κ r e s ≤ 1 , \left(\kappa^{\mathrm{res}}\right)^2=0,\qquad\operatorname{rk}\kappa^{\mathrm{res}}\leq1, ( κ res ) 2 = 0 , rk κ res ≤ 1 , then X X X
Proof. The hypotheses and Corollary 11.19 exclude every surface realisation of r \mathbf{r} r
Benedetti–Manivel–Perrin compute the small quantum cohomology of Gushel–Mukai fourfolds, deduce that a very general Gushel–Mukai fourfold is not rational, and show that a rational Gushel–Mukai fourfold has the same rational cohomology as a K 3 K3 K 3 [1 ] . Their second conclusion makes the K 3 K3 K 3 K 3 K3 K 3
Unit-direction descent of the A-model residual factors
Scalar branch splitting is already non-trivial along the universal coordinate of the cohomological unit. This direction occurs in the A-model base of every smooth projective variety and provides a test of the variable-branch formalism.
Let t 0 t^{0} t 0 1 ∈ H 0 ( X ) 1 \in H^{0}(X) 1 ∈ H 0 ( X )
B X 0 : = { t 0 = 0 } ⊂ B X , p : B X ⟶ B X 0 B_{X}^{0}:=\{t^{0}=0\}\subset B_{X},\qquad p:B_{X}\longrightarrow B_{X}^{0} B X 0 := { t 0 = 0 } ⊂ B X , p : B X ⟶ B X 0 be the projection which forgets the unit coordinate, and write ( H X , ∇ X ) / B X (\mathcal{H}_{X},\nabla_{X})/B_{X} ( H X , ∇ X ) / B X F F F A t 0 1 × B X 0 \mathbb{A}^{1}_{t^{0}}\times B_{X}^{0} A t 0 1 × B X 0
Theorem 11.21. The quantum product is pulled back from B X 0 B_{X}^{0} B X 0 H X ≅ p ∗ H X 0 \mathcal{H}_{X}\cong p^{*}\mathcal{H}_{X}^{0} H X ≅ p ∗ H X 0
∇ X = p ∗ ∇ X 0 + d ( t 0 / u ) I , (81) \nabla_{X}=p^{*}\nabla_{X}^{0}+d(t^{0}/u)I, \tag*{(81)} ∇ X = p ∗ ∇ X 0 + d ( t 0 / u ) I , (81) where ( H X 0 , ∇ X 0 ) (\mathcal{H}_{X}^{0},\nabla_{X}^{0}) ( H X 0 , ∇ X 0 ) t 0 = 0 t^{0}=0 t 0 = 0 α \alpha α B X 0 B_{X}^{0} B X 0 a α , 0 a_{\alpha,0} a α , 0
a α = p ∗ a α , 0 + t 0 , (82) a_{\alpha}=p^{*}a_{\alpha,0}+t^{0}, \tag*{(82)} a α = p ∗ a α , 0 + t 0 , (82) and its residual factor descends:
Res a α ( H α , ∇ α ) ≅ p ∗ Res a α , 0 ( H α , 0 , ∇ α , 0 ) . (83) \operatorname{Res}_{a_{\alpha}}(\mathcal{H}_{\alpha},\nabla_{\alpha})\cong p^{*}\operatorname{Res}_{a_{\alpha,0}}(\mathcal{H}_{\alpha,0},\nabla_{\alpha,0}). \tag*{(83)} Res a α ( H α , ∇ α ) ≅ p ∗ Res a α , 0 ( H α , 0 , ∇ α , 0 ) . (83) In particular, the classifying morphism of the zero-branch residual factor is constant along the unit direction. Before scalar residualisation, the atomic F F F F F F B X 0 B_{X}^{0} B X 0
Proof. The unit axiom implies that the quantum product is independent of t 0 t^{0} t 0 1 1 1
∇ ∂ t 0 = ∂ t 0 + u − 1 I , ∇ ∂ u = ∂ u − u − 2 ( Eu ⋆ ( − ) ) + u − 1 Gr , \nabla_{\partial_{t^{0}}}=\partial_{t^{0}}+u^{-1}I,\qquad
\nabla_{\partial_{u}}=\partial_{u}-u^{-2}(\operatorname{Eu}\star(-))+u^{-1}\operatorname{Gr}, ∇ ∂ t 0 = ∂ t 0 + u − 1 I , ∇ ∂ u = ∂ u − u − 2 ( Eu ⋆ ( − )) + u − 1 Gr , and
Eu ( t 0 , b ) = Eu ( 0 , b ) + t 0 1 ; \operatorname{Eu}(t^{0},b)=\operatorname{Eu}(0,b)+t^{0}1; Eu ( t 0 , b ) = Eu ( 0 , b ) + t 0 1 ; see [17 ] , Equations (3.30)–(3.31). All connection components tangent to B X 0 B_{X}^{0} B X 0 t 0 t^{0} t 0
d t 0 u I − t 0 u 2 d u I = d ( t 0 / u ) I . \frac{dt^{0}}{u}I-\frac{t^{0}}{u^{2}}\,du\,I=d(t^{0}/u)I. u d t 0 I − u 2 t 0 d u I = d ( t 0 / u ) I . This proves (81) . The residual endomorphism is κ ∇ X = − L Eu \kappa\nabla_{X}=-L_{\operatorname{Eu}} κ ∇ X = − L Eu t 0 1 t^{0}1 t 0 1 κ ∇ X \kappa\nabla_{X} κ ∇ X − t 0 -t^{0} − t 0 α \alpha α − a -a − a (82) . Uniqueness of spectral decomposition identifies the corresponding spectral components along the unit line. Finally,
∇ α − d ( a α / u ) I = p ∗ ∇ α , 0 + d ( t 0 / u ) I − d ( ( p ∗ a α , 0 + t 0 ) / u ) I = p ∗ ( ∇ α , 0 − d ( a α , 0 / u ) I ) , \begin{aligned}
\nabla_{\alpha}-d(a_{\alpha}/u)I=p^{*}\nabla_{\alpha,0}+d(t^{0}/u)I-d\bigl((p^{*}a_{\alpha,0}+t^{0})/u\bigr)I \\
&=p^{*}\bigl(\nabla_{\alpha,0}-d(a_{\alpha,0}/u)I\bigr),
\end{aligned} ∇ α − d ( a α / u ) I = p ∗ ∇ α , 0 + d ( t 0 / u ) I − d ( ( p ∗ a α , 0 + t 0 ) / u ) I = p ∗ ( ∇ α , 0 − d ( a α , 0 / u ) I ) , which gives (83) . After residualisation, the component in the unit direction is ∂ t 0 \partial_{t^{0}} ∂ t 0
∂ t 0 g = − u − 1 g . \partial_{t^{0}}g=-u^{-1}g. ∂ t 0 g = − u − 1 g . Thus g = e − t 0 / u g 0 g=e^{-t^{0}/u}g_{0} g = e − t 0 / u g 0 g 0 g_{0} g 0 t 0 t^{0} t 0 e − t 0 / u e^{-t^{0}/u} e − t 0 / u u = 0 u=0 u = 0 F F F □ \square □
On the punctured u u u d ( t 0 / u ) d(t^{0}/u) d ( t 0 / u ) u = 0 u=0 u = 0 Theorem 11.21 is not a change of holomorphic gauge inside the category of F F F
Corollary 11.22. Let X X X b L R ∈ B X 0 b_{\mathrm{LR}}\in B_{X}^{0} b LR ∈ B X 0 F F F a X , 0 a_{X,0} a X , 0 t 0 = 0 t^{0}=0 t 0 = 0
a X = p ∗ a X , 0 + t 0 , a X , 0 ( b L R ) = 0. a_{X}=p^{*}a_{X,0}+t^{0},\qquad a_{X,0}(b_{\mathrm{LR}})=0. a X = p ∗ a X , 0 + t 0 , a X , 0 ( b LR ) = 0. In particular, the restriction of a X a_{X} a X A t 0 1 × { b L R } \mathbb{A}^{1}_{t^{0}}\times\{b_{\mathrm{LR}}\} A t 0 1 × { b LR } t 0 t^{0} t 0 t 0 = 0 t^{0}=0 t 0 = 0
Proof. The translation formula and the residual pullback statement are (82) and (83) . At the large-radius central point, the Euler element is c 1 ( T X ) c_1(T_X) c 1 ( T X ) [17 §3.5] ; it vanishes when K X ≃ O X K_X \simeq\mathcal{O}_X K X ≃ O X a X , 0 ( b L R ) = 0 a_{X,0}(b_{\mathrm{LR}})=0 a X , 0 ( b LR ) = 0 b L R b_{\mathrm{LR}} b LR a X = t 0 a_X=t^0 a X = t 0
A projective illustration: the Pfaffian–Grassmannian pair
The determinantal strata record the nilpotent part of the vertical connection. At the formal large-radius boundary, the horizontal logarithmic residues retain a complementary piece of information: the classical divisor algebra. The comparison requires only the marking specified below.
Definition 11.23. Let X X X A A A F F F rationally marked large-radius residual germ is the logarithmic formal germ of the branch-residual F F F q β = 0 q^\beta=0 q β = 0 t = 0 t=0 t = 0
(i) the rational Néron–Severi space NS ( X ) Q \operatorname{NS}(X)_{\mathbb{Q}} NS ( X ) Q
(ii) the unit section 1 1 1
(iii) the Poincaré pairing; and
(iv) the rational Z / 2 \mathbb{Z}/2 Z /2
Isomorphisms of such germs are required to preserve these markings and the logarithmic divisor directions.
This is a marked germ attached to a residual atom, rather than a new unmarked atomic equivalence relation or an integral Γ \Gamma Γ A A A [17 §3.5] .
Proposition 11.24. Let X X X smooth projective Calabi–Yau threefold . The rationally marked large-radius residual germ determines the cubic form
C X ( D 1 , D 2 , D 3 ) : = ∫ X D 1 ⌣ D 2 ⌣ D 3 , D i ∈ NS ( X ) Q . C_X(D_1,D_2,D_3):=\int_X D_1\smile D_2\smile D_3,\qquad D_i\in\operatorname{NS}(X)_{\mathbb{Q}}. C X ( D 1 , D 2 , D 3 ) := ∫ X D 1 ⌣ D 2 ⌣ D 3 , D i ∈ NS ( X ) Q . Consequently, an isomorphism of rationally marked large-radius residual germs induces an isomorphism of the corresponding rational cubic spaces which preserves the ample cones.
Proof. For D ∈ NS ( X ) Q D\in\operatorname{NS}(X)_{\mathbb{Q}} D ∈ NS ( X ) Q ∂ D log \partial_D^{\log} ∂ D l o g D D D A A A
∇ ∂ D log = ∂ D log + u − 1 L D ⋆ . \nabla_{\partial_D^{\log}}=\partial_D^{\log}+u^{-1}L_D^\star. ∇ ∂ D l o g = ∂ D l o g + u − 1 L D ⋆ . where L D ⋆ L_D^\star L D ⋆ D D D [17 §3.5] . At the formal boundary q β = 0 q^\beta=0 q β = 0 t = 0 t=0 t = 0 u − 1 u^{-1} u − 1
L D ( x ) = D ⌣ x . L_D(x)=D\smile x. L D ( x ) = D ⌣ x . On the Novikov slice t = 0 t=0 t = 0 A A A c 1 ( T X ) c_1(T_X) c 1 ( T X ) [17 §3.5] ; it therefore vanishes identically when X X X L D L_D L D
C X ( D 1 , D 2 , D 3 ) = ⟨ L D 1 L D 2 L D 3 1 , 1 ⟩ . C_X(D_1,D_2,D_3)=\langle L_{D_1}L_{D_2}L_{D_3}1,1\rangle. C X ( D 1 , D 2 , D 3 ) = ⟨ L D 1 L D 2 L D 3 1 , 1 ⟩ . All constructions are preserved by an isomorphism of marked germs, proving the final assertion. □
Proposition 11.25. Let X X X Y Y Y smooth projective Calabi–Yau threefolds of Picard rank one whose maximal A A A F F F . Let H X H_X H X H Y H_Y H Y
( H X 3 H Y 3 ) ∈ ( Q > 0 ) 3 . \left(\frac{H_X^3}{H_Y^3}\right)\in(\mathbb{Q}_{>0})^3. ( H Y 3 H X 3 ) ∈ ( Q > 0 ) 3 . Equivalently, there exists c ∈ Q > 0 c\in\mathbb{Q}_{>0} c ∈ Q > 0 H X 3 = c 3 H Y 3 H_X^3=c^3H_Y^3 H X 3 = c 3 H Y 3
Proof. By Proposition 11.24, an isomorphism of marked residual germs induces an isomorphism
NS ( X ) Q ⟶ NS ( Y ) Q \operatorname{NS}(X)_{\mathbb{Q}}\longrightarrow\operatorname{NS}(Y)_{\mathbb{Q}} NS ( X ) Q ⟶ NS ( Y ) Q that preserves the cubic forms and the ample cones. Both vector spaces are one-dimensional, so the primitive ample generator H X H_X H X c H Y cH_Y c H Y c ∈ Q > 0 c\in\mathbb{Q}_{>0} c ∈ Q > 0
H X 3 = ( c H Y ) 3 = c 3 H Y 3 . (84) H_X^3=(cH_Y)^3=c^3H_Y^3. \tag*{(84)} H X 3 = ( c H Y ) 3 = c 3 H Y 3 . (84) Corollary 11.26. Let X X X Proposition 11.25 . For any non-zero rational ample class D ∈ N S ( X ) Q \mathrm{D} \in\mathrm{NS}(X)_{\mathbb{Q}} D ∈ NS ( X ) Q
δ r e s ( X ) : = [ D 3 ] ∈ Q > 0 × / ( Q > 0 × ) 3 \delta_{\mathrm{res}}(X) := [\mathrm{D}^{3}] \in\mathbb{Q}_{>0}^{\times}/(\mathbb{Q}_{>0}^{\times})^{3} δ res ( X ) := [ D 3 ] ∈ Q > 0 × / ( Q > 0 × ) 3 is independent of the choice of D \mathrm{D} D
Proof. Any two non-zero rational ample classes are related by D ′ = c D \mathrm{D}' = c\mathrm{D} D ′ = c D c ∈ Q > 0 c \in\mathbb{Q}_{>0} c ∈ Q > 0 ( D ′ ) 3 = c 3 D 3 (\mathrm{D}')^{3} = c^{3}\mathrm{D}^{3} ( D ′ ) 3 = c 3 D 3 Proposition 11.24 ; it therefore preserves this class modulo rational cubes. □
Let V V V W ⊂ ⋀ 2 V ∗ W \subset\bigwedge^{2}V^{*} W ⊂ ⋀ 2 V ∗
X G r : = Gr ( 2 , V ) ∩ P ( Ann ( W ) ) X_{\mathrm{Gr}} := \operatorname{Gr}(2,V) \cap\mathbb{P}(\operatorname{Ann}(W)) X Gr := Gr ( 2 , V ) ∩ P ( Ann ( W )) be the codimension-seven Grassmannian linear section, and let
X P f : = Pf ( V ) ∩ P ( W ) X_{\mathrm{Pf}} := \operatorname{Pf}(V) \cap\mathbb{P}(W) X Pf := Pf ( V ) ∩ P ( W ) be the dual Pfaffian linear section. Borisov and Căldăraru prove that both are smooth Calabi–Yau threefolds with h 1 , 1 = 1 h^{1,1}=1 h 1 , 1 = 1 h 2 , 1 = 50 h^{2,1}=50 h 2 , 1 = 50
D b ( X G r ) ≃ D b ( X P f ) D^{b}(X_{\mathrm{Gr}}) \simeq D^{b}(X_{\mathrm{Pf}}) D b ( X Gr ) ≃ D b ( X Pf ) for dual general sections [6 ] . The cohomological Fourier–Mukai transform identifies the relevant odd rational Hodge structures; together with the common Tate-type even contribution, this yields isomorphic Z / 2 \mathbb{Z}/2 Z /2 [7 ] . Nevertheless, if H G r H_{\mathrm{Gr}} H Gr H P f H_{\mathrm{Pf}} H Pf
H G r 3 = 42 , H P f 3 = 14 (85) H_{\mathrm{Gr}}^{3}=42,\qquad H_{\mathrm{Pf}}^{3}=14 \tag*{(85)} H Gr 3 = 42 , H Pf 3 = 14 (85) by Rødland’s calculation [25 , 6 ] .
Theorem 11.27. Along the unit lines through their marked large-radius points, the unit-extended atomic F F F X G r X_{\mathrm{Gr}} X Gr X P f X_{\mathrm{Pf}} X Pf a = t 0 a=t^{0} a = t 0 Z / 2 \mathbb{Z}/2 Z /2
Proof. Because the canonical classes are trivial, each maximal A-model F F F [17 Lemma 5.24] . By Corollary 11.22, the restrictions of the two branches to the unit lines through the marked large-radius points are both t 0 t^{0} t 0 t 0 = 0 t^{0}=0 t 0 = 0 Corollary 11.26 would imply
H G r 3 H P f 3 = 42 14 = 3 ∈ ( Q > 0 × ) 3 . \frac{H_{\mathrm{Gr}}^{3}}{H_{\mathrm{Pf}}^{3}}=\frac{42}{14}=3\in(\mathbb{Q}_{>0}^{\times})^{3}. H Pf 3 H Gr 3 = 14 42 = 3 ∈ ( Q > 0 × ) 3 . Thus there would be c ∈ Q > 0 c\in\mathbb{Q}_{>0} c ∈ Q > 0 c 3 = 3 c^{3}=3 c 3 = 3 c = p / q c=p/q c = p / q p , q p,q p , q p 3 = 3 q 3 p^{3}=3q^{3} p 3 = 3 q 3 3 ∣ p 3\mid p 3 ∣ p p = 3 p 1 p=3p_{1} p = 3 p 1 9 p 1 3 = q 3 9p_{1}^{3}=q^{3} 9 p 1 3 = q 3 3 ∣ q 3\mid q 3 ∣ q 3 3 3
Corollary 11.28. The forgetful assignment from unit-extended, branch-labelled, rationally marked residual germs to their Z / 2 \mathbb{Z}/2 Z /2 t 0 t^{0} t 0
Proof. The Pfaffian–Grassmannian pair in Theorem 11.27 has isomorphic Z / 2 \mathbb{Z}/2 Z /2 t 0 t^{0} t 0
This gives a projective example in which scalar residualisation acts non-trivially as a descent operation. The geometric distinction used in the proof is not new: the degrees in (85) , the derived equivalence and the non-birationality of the pair are already contained in [25 , 6 ] . What is new here is the structural interpretation. Before residualisation, the unit-extended connection contains the universal non-constant scalar contribution t 0 t^{0} t 0 t 0 = 0 t^{0}=0 t 0 = 0 d ( t 0 / u ) d(t^{0}/u) d ( t 0 / u ) t 0 t^0 t 0 F F F [17 ] ; rather, scalar residualisation supplies the decomposition used to isolate and compare the projective part of that information.
Residual lattices beyond monodromy
On the complex regular-singular locus, Deligne’s Riemann–Hilbert correspondence [9 ] gives a forgetful morphism
RH : M r s , C u n f r ⟶ LocSys \operatorname{RH}: \mathcal{M}^{\mathrm{unfr}}_{\mathrm{rs},\mathbb{C}} \longrightarrow\operatorname{LocSys} RH : M rs , C unfr ⟶ LocSys which remembers the punctured local system and forgets the prescribed lattice.
Theorem 11.29. The isomorphism class of a prescribed residual lattice, and even its lattice Jordan profile, is not determined by its image under RH. There exists a one-parameter family in M r s , C u n f r \mathcal{M}^{\mathrm{unfr}}_{\mathrm{rs},\mathbb{C}} M rs , C unfr
Proof. Let N ≠ 0 N \ne0 N = 0 N 2 = 0 N^2=0 N 2 = 0 t t t
∇ = d + N u d t − t N u 2 d u . \nabla= d + \frac{N}{u}\,dt - \frac{tN}{u^2}\,du. ∇ = d + u N d t − u 2 tN d u . Its vertical part defines a family in M r s , C u n f r \mathcal{M}^{\mathrm{unfr}}_{\mathrm{rs},\mathbb{C}} M rs , C unfr Theorem 8.1 with R = 0 R=0 R = 0 F = t N F=tN F = tN t t t G t = exp ( t N / u ) = I + t N / u G_t=\exp(tN/u)=I+tN/u G t = exp ( tN / u ) = I + tN / u N 2 = 0 N^2=0 N 2 = 0
G t − 1 d G t = − t N u 2 d u , G_t^{-1}dG_t=-\frac{tN}{u^2}\,du, G t − 1 d G t = − u 2 tN d u , and therefore ∇ t = G t − 1 d G t \nabla_t=G_t^{-1}dG_t ∇ t = G t − 1 d G t
κ t r e s = u 2 ( ∇ t ) ∂ u ∣ u = 0 = − t N . \kappa_t^{\mathrm{res}}=u^2(\nabla_t)_{\partial_u}\big|_{u=0}=-tN. κ t res = u 2 ( ∇ t ) ∂ u u = 0 = − tN . Thus its rank is zero at t = 0 t=0 t = 0 rk N \operatorname{rk}N rk N t ≠ 0 t\ne0 t = 0
The residual moduli target therefore retains lattice information forgotten by the Betti local system. Together with Theorems 11.21 and 11.27 , this gives two complementary illustrations of the residual presentation. In the first, scalar residualisation removes the variable universal scalar branch and exposes a descended factor not determined by the Hodge realisation. In the second, a flat isomonodromic family has a constant local system but a varying prescribed lattice.
Rigidified charts and intrinsic residual atoms
Every geometric atomic factor has an intrinsic branch-residual presentation. For a complex analytic regular-singular residual germ, choosing an invariant full flag, an ordered frame, a triangular reduction and a locally constant ordered residue vector yields, by Theorem 9.2 , a classifying map
S ⟶ n ⊕ n . S \longrightarrow\mathfrak{n}\oplus\mathfrak{n}. S ⟶ n ⊕ n . The descent results of § 7 give the transformation laws under re-framing, identify the flagged quotient, and determine the further fibre over the unflagged constituent. Thus the normal form for arbitrary spectra provides coordinates on rigidified regular-singular strata of the global residual target, while the residual atom itself is represented only after the corresponding frame-and-flag data have been forgotten. The definition of residual atoms and residual chemical formulae requires none of these rigidifications and does not require regular singularity. The unresolved global step is the descent problem formulated in Question 10.16 . It asks for an enhanced local model that retains comparison-lattice data, recovers the fixed-stratum quotient and the elementary wall charts, and is independent of the auxiliary comparison lattice. The present paper does not construct this stack, and none of the definitions or results on residual atoms depends on its existence. Dipartimento di Matematica, Università degli Studi di Bari Aldo Moro, Bari, Italy
Email address: mauricio.barros@uniba.it
