Introduction
Let X ⊂ P N − 1 X \subset\mathbb{P}^{N-1} X ⊂ P N − 1 d ≥ 2 d \ge2 d ≥ 2 N > d N>d N > d D b ( X ) \mathrm{D}^{\mathrm{b}}(X) D b ( X ) O X , O X ( 1 ) , … , O X ( N − d − 1 ) \mathcal{O}_X, \mathcal{O}_X(1), \ldots, \mathcal{O}_X(N-d-1) O X , O X ( 1 ) , … , O X ( N − d − 1 ) Kuznetsov component of the Fano hypersurface:
D b ( X ) = ⟨ K u ( X ) , O X , O X ( 1 ) , … , O X ( N − d − 1 ) ⟩ . (1) \mathrm{D}^{\mathrm{b}}(X)=\langle\mathrm{Ku}(X),\mathcal{O}_X,\mathcal{O}_X(1),\ldots,\mathcal{O}_X(N-d-1)\rangle. \tag*{(1)} D b ( X ) = ⟨ Ku ( X ) , O X , O X ( 1 ) , … , O X ( N − d − 1 )⟩ . (1) When N = d N=d N = d D b ( X ) \mathrm{D}^{\mathrm{b}}(X) D b ( X ) W W W X X X [34 ] identifies them with the category M F g r ( W ) \mathrm{MF}^{\mathrm{gr}}(W) MF gr ( W )
We write [ n ] [n] [ n ] ( a ) (a) ( a ) M ( a ) j = M j + a M(a)_j=M_{j+a} M ( a ) j = M j + a τ \tau τ τ a E = E ( a ) \tau^a E=E(a) τ a E = E ( a ) τ d ≃ [ 2 ] \tau^d\simeq[2] τ d ≃ [ 2 ]
With a suitable normalization of Orlov’s equivalence, the same autoequivalence on K u ( X ) \mathrm{Ku}(X) Ku ( X ) N > d N>d N > d D b ( X ) \mathrm{D}^{\mathrm{b}}(X) D b ( X ) N = d N=d N = d
τ ( E ) ≃ Cone ( RHom ( O X , E ( 1 ) ) ⊗ C O X → e v E ( 1 ) ) , E ( 1 ) = E ⊗ O X ( 1 ) , \tau(E)\simeq\operatorname{Cone}\left(\operatorname{RHom}(\mathcal{O}_X,E(1))\otimes_{\mathbb{C}}\mathcal{O}_X \xrightarrow{\mathrm{ev}} E(1)\right),\qquad E(1)=E\otimes\mathcal{O}_X(1), τ ( E ) ≃ Cone ( RHom ( O X , E ( 1 )) ⊗ C O X ev E ( 1 ) ) , E ( 1 ) = E ⊗ O X ( 1 ) , where the arrow is the evaluation map [2 Definition 6.11 and Theorem 6.13] . In the Fano case, this is the rotation functor : tensoring by O X ( 1 ) \mathcal{O}_X(1) O X ( 1 ) O X \mathcal{O}_X O X O X ( 1 ) \mathcal{O}_X(1) O X ( 1 ) O X \mathcal{O}_X O X
It is natural to look for a Bridgeland stability condition in which this fractional shift acts by a constant rotation of phases. Since [ 1 ] [1] [ 1 ] 1 1 1 2 / d 2/d 2/ d
Definition 1.1. Let D \mathcal{D} D σ = ( Z , P ) \sigma=(Z,\mathcal{P}) σ = ( Z , P ) Φ : D → ∼ D \Phi:\mathcal{D}\xrightarrow{\sim}\mathcal{D} Φ : D ∼ D s ∈ R s\in\mathbb{R} s ∈ R [40 Definition 1.1] , we say that the stability condition σ \sigma σ Gepner type ( Φ , s ) (\Phi,s) ( Φ , s ) ϕ ∈ R \phi\in\mathbb{R} ϕ ∈ R E ∈ D E\in\mathcal{D} E ∈ D
Φ P ( ϕ ) = P ( ϕ + s ) and Z ( Φ E ) = e i π s Z ( E ) . \Phi\mathcal{P}(\phi)=\mathcal{P}(\phi+s)
\qquad\text{and}\qquad
Z(\Phi E)=e^{i\pi s}Z(E). Φ P ( ϕ ) = P ( ϕ + s ) and Z ( Φ E ) = e iπ s Z ( E ) . In particular, Φ \Phi Φ s s s
Theorem 1.2. Let d ≥ 2 d\ge2 d ≥ 2 X = { W = 0 } ⊂ P N − 1 X=\{W=0\}\subset\mathbb{P}^{N-1} X = { W = 0 } ⊂ P N − 1 N ≥ d N\ge d N ≥ d d d d M F g r ( W ) \mathrm{MF}^{\mathrm{gr}}(W) MF gr ( W ) σ = ( Z , P ) \sigma=(Z,\mathcal{P}) σ = ( Z , P ) K n u m ( M F g r ( W ) ) K_{\mathrm{num}}(\mathrm{MF}^{\mathrm{gr}}(W)) K num ( MF gr ( W )) ( τ , 2 / d ) (\tau,2/d) ( τ , 2/ d ) τ \tau τ
Z ( τ E ) = e 2 π i / d Z ( E ) , τ P ( ϕ ) = P ( ϕ + 2 d ) . (2) Z(\tau E)=e^{2\pi i/d}Z(E),\qquad\tau\mathcal{P}(\phi)=\mathcal{P}\left(\phi+\frac{2}{d}\right). \tag*{(2)} Z ( τ E ) = e 2 π i / d Z ( E ) , τ P ( ϕ ) = P ( ϕ + d 2 ) . (2) For every fixed object A A A C A > 0 C_A > 0 C A > 0
∑ n ∈ Z dim C Hom ( A , E [ n ] ) ≤ C A m σ ( E ) for every E . (3) \sum_{n \in\mathbb{Z}} \dim_{\mathbb{C}} \operatorname{Hom}(A,E[n]) \le C_A m_\sigma(E) \qquad\text{for every } E. \tag*{(3)} n ∈ Z ∑ dim C Hom ( A , E [ n ]) ≤ C A m σ ( E ) for every E . (3) Here m σ ( E ) m_\sigma(E) m σ ( E ) [34 ] , MF g r ( W ) \operatorname{MF}^{\mathrm{gr}}(W) MF gr ( W ) Ku ( X ) \operatorname{Ku}(X) Ku ( X ) N > d N>d N > d D b ( X ) D^{\mathrm{b}}(X) D b ( X ) N = d N=d N = d
Theorem 5.6 also covers the case N < d N<d N < d MF g r ( W ) \operatorname{MF}^{\mathrm{gr}}(W) MF gr ( W ) D b ( X ) D^{\mathrm{b}}(X) D b ( X )
Corollary 1.3. Let σ = ( Z , P ) \sigma=(Z,\mathcal{P}) σ = ( Z , P ) Theorem 1.2 on C = Ku ( X ) \mathcal{C}=\operatorname{Ku}(X) C = Ku ( X ) N > d N>d N > d C = D b ( X ) \mathcal{C}=D^{\mathrm{b}}(X) C = D b ( X ) N = d N=d N = d v ∈ K n u m ( C ) v\in K_{\mathrm{num}}(\mathcal{C}) v ∈ K num ( C ) ϕ ∈ R \phi\in\mathbb{R} ϕ ∈ R Z ( v ) ∈ R > 0 e i π ϕ Z(v)\in\mathbb{R}_{>0}e^{i\pi\phi} Z ( v ) ∈ R > 0 e iπ ϕ v v v ϕ \phi ϕ C \mathbb{C} C M σ ( v , ϕ ) M_\sigma(v,\phi) M σ ( v , ϕ )
The closed points of this space classify S-equivalence classes of semistable objects. The corollary follows from the mass–Hom bound by the theorem of Halpern-Leistner and Robotis [21 Theorem 2.35] ; we give the precise application at the end of § 5 .
For Fano hypersurfaces, even the existence of any stability condition on the Kuznetsov component is already a new result. Building on his joint work with Macrì, Perry, Stellari, and Zhao [30 ] on rigidity of stability conditions for products of curves, Li proves that D b ( Y ) D^{\mathrm{b}}(Y) D b ( Y ) Y Y Y [28 ] . This theorem does not settle existence of stability conditions on admissible subcategories. Indeed, there can be no existence theorem in this generality, since nonzero phantom subcategories have vanishing Grothendieck group and hence admit no stability conditions [16 Definition 1.8 and Corollary 1.15] . For cubic hypersurfaces, several constructions were already available. Bayer–Lahoz–Macrì–Stellari constructed numerical stability conditions on the Kuznetsov component of every smooth cubic fourfold [5 ] , without asserting Gepner symmetry. The stronger requirement of Gepner symmetry was established by Toda for cubic surfaces, cubic threefolds, and a general cubic fourfold containing a plane [41 Theorems 1.2 and 1.3] . More recently, Liu constructed stability conditions with the Gepner property on the Kuznetsov component of every smooth cubic fivefold [31 Corollary 5.12] .
For Calabi–Yau hypersurfaces, the category in question is the full derived category, so Li’s theorem supplies stability conditions in every dimension. The additional challenge is to construct one with Gepner symmetry. Toda did this for plane cubics and quartic K3 surfaces [40 Theorem 1.3] . Li proved existence of numerical Bridgeland stability conditions on smooth quintic threefolds by proving a suitable Bogomolov–Gieseker inequality [27 ] . The search for Gepner symmetry led Toda to formulate a stronger inequality [42 ] , subsequently proved by Xu for slope-semistable sheaves [46 ] . The recent OpenAI release includes a manuscript asserting the existence of Gepner-type stability conditions on every smooth quintic threefold [33 ] .
Our theorem gives a uniform construction of stability conditions with Gepner symmetry on the Kuznetsov components of all smooth Fano hypersurfaces and on the derived categories of all smooth Calabi–Yau hypersurfaces.
The proof of Theorem 1.2 has three steps. We start with the category of graded matrix factorizations of x d x^d x d A d − 1 A_{d-1} A d − 1 S − 1 [ 1 ] S^{-1}[1] S − 1 [ 1 ] S S S [39 , 23 ] . Qiu [37 Theorem 4.7] proves that the stability condition of Gepner type ( S − 1 [ 1 ] , 2 / d ) (S^{-1}[1],2/d) ( S − 1 [ 1 ] , 2/ d ) C \mathbb{C} C Q d Q_d Q d A d − 1 A_{d-1} A d − 1
Write Q d = D b ( rep C Q d ) Q_d=D^{\mathrm{b}}(\operatorname{rep}_{\mathbb{C}} Q_d) Q d = D b ( rep C Q d ) C \mathcal{C} C C ⊗ Q d C \otimes\mathcal{Q}_d C ⊗ Q d § 3 . Its objects may be viewed as quiver diagrams, with an object of C C C
r j = sin ( j π / d ) sin ( π / d ) , 1 ≤ j ≤ d − 1 , r_j = \frac{\sin(j\pi/d)}{\sin(\pi/d)}, \qquad1 \leq j \leq d-1, r j = sin ( π / d ) sin ( j π / d ) , 1 ≤ j ≤ d − 1 , which form an eigenvector of the adjacency matrix of the underlying unoriented graph. This eigenvector identity turns reflection functors of the quiver into rotations of the central charge.
We iterate this construction to obtain a stability condition on Q d ⊗ M \mathcal{Q}_d^{\otimes M} Q d ⊗ M M M M Q d \mathcal{Q}_d Q d [3 Corollary 2.40] identifies this tensor product with the category of matrix factorizations of the Fermat polynomial
F = y 1 d + ⋯ + y M d , F = y_1^d + \cdots+ y_M^d, F = y 1 d + ⋯ + y M d , graded by the group L = Z M / ⟨ d ( e i − e j ) ⟩ L = \mathbb{Z}^M/\langle d(e_i-e_j)\rangle L = Z M / ⟨ d ( e i − e j )⟩ deg ( y i ) = e i \deg(y_i)=e_i deg ( y i ) = e i e i − e j e_i-e_j e i − e j H ≅ ( Z / d Z ) M − 1 H \cong(\mathbb{Z}/d\mathbb{Z})^{M-1} H ≅ ( Z / d Z ) M − 1 H H H [15 Definition 3.5] , which can be identified with the category of matrix factorizations MF g r ( F ) \operatorname{MF}^{\mathrm{gr}}(F) MF gr ( F )
Finally, every degree-d d d C \mathbb{C} C d d d [32 Sections 2–4] studied mirror stability conditions on tensor products of type A quiver categories and their interpretation through homological mirror symmetry.
As we were completing this paper, Wu’s preprint [45 Theorem 1.1] appeared on the arXiv. Using stability conditions on products of elliptic curves, Wu constructs numerical Gepner-type stability conditions on the Kuznetsov components of smooth Fano hypersurfaces of degrees 3, 4, and 6 and dimension at least three. We also began with products of elliptic curves, before realizing that tensor products of quiver categories allow the construction to work in every degree. The main work went into constructing stability conditions on these tensor products.
Acknowledgements
Acknowledgements. This work was supported in part by a grant of access to OpenAI models through the ChatGPT for Academic Researchers program. Codex Astra helped us find several steps in the construction and was also used to edit the manuscript. Any remaining errors are, of course, our own. J.T. was partially supported by NSF grant DMS-2401387.
Generalities on Categories
We will use enhanced models of triangulated categories. These models will allow us to construct reflection functors, define tensor products, and study finite group actions on categories. Stability conditions are defined on the underlying triangulated categories.
Stability conditions
For a triangulated category C C C K 0 ( C ) K_0(C) K 0 ( C ) [ B ] = [ A ] + [ C ] [B]=[A]+[C] [ B ] = [ A ] + [ C ] A → B → C → A [ 1 ] A \to B \to C \to A[1] A → B → C → A [ 1 ] C C C ⨁ k Hom ( E , F [ k ] ) \bigoplus_k \operatorname{Hom}(E,F[k]) ⨁ k Hom ( E , F [ k ]) C \mathbb{C} C
χ ( E , F ) = ∑ k ( − 1 ) k dim C Hom ( E , F [ k ] ) \chi(E,F)=\sum_k(-1)^k\dim_{\mathbb{C}}\operatorname{Hom}(E,F[k]) χ ( E , F ) = k ∑ ( − 1 ) k dim C Hom ( E , F [ k ]) is defined. The numerical Grothendieck group K n u m ( C ) K_{\mathrm{num}}(C) K num ( C ) K 0 ( C ) K_0(C) K 0 ( C ) v v v χ ( v , w ) = 0 \chi(v,w)=0 χ ( v , w ) = 0 w w w § 2.2 ; Serre duality identifies the left and right radicals of the Euler pairing. A numerical class map v v v K 0 ( C ) K_0(C) K 0 ( C ) Λ \Lambda Λ K n u m ( C ) K_{\mathrm{num}}(C) K num ( C )
A slicing is a family of full additive subcategories P ( ϕ ) ⊂ C \mathcal{P}(\phi) \subset\mathcal{C} P ( ϕ ) ⊂ C ϕ ∈ R \phi\in\mathbb{R} ϕ ∈ R P ( ϕ + 1 ) = P ( ϕ ) [ 1 ] \mathcal{P}(\phi+ 1) = \mathcal{P}(\phi)[1] P ( ϕ + 1 ) = P ( ϕ ) [ 1 ] P ( ϕ 1 ) , … , P ( ϕ s ) \mathcal{P}(\phi_1), \ldots, \mathcal{P}(\phi_s) P ( ϕ 1 ) , … , P ( ϕ s ) ϕ 1 > ⋯ > ϕ s \phi_1 > \cdots> \phi_s ϕ 1 > ⋯ > ϕ s I ⊂ R I \subset\mathbb{R} I ⊂ R P ( I ) \mathcal{P}(I) P ( I ) P ( ϕ ) \mathcal{P}(\phi) P ( ϕ ) I I I
A stability condition consists of a slicing and a central charge Z : K 0 ( C ) → C Z : K_0(\mathcal{C}) \to\mathbb{C} Z : K 0 ( C ) → C
Z ( E ) ∈ R > 0 e i π ϕ for 0 ≠ E ∈ P ( ϕ ) . Z(E) \in\mathbb{R}_{>0}e^{i\pi\phi} \quad\text{for } 0 \ne E \in\mathcal{P}(\phi). Z ( E ) ∈ R > 0 e iπ ϕ for 0 = E ∈ P ( ϕ ) . Each P ( ϕ ) \mathcal{P}(\phi) P ( ϕ ) semistable , and its simple objects are called stable . Throughout the paper we require the support property with respect to a specified numerical class map v : K 0 ( C ) → Λ v : K_0(\mathcal{C}) \to\Lambda v : K 0 ( C ) → Λ Z = Z ‾ ∘ v Z = \overline{Z} \circ v Z = Z ∘ v
∥ v ( E ) ∥ ≤ C ∣ Z ( E ) ∣ for all semistable E , (4) \lVert v(E)\rVert\le C\lvert Z(E)\rvert\quad\text{for all semistable } E, \tag*{(4)} ∥ v ( E )∥ ≤ C ∣ Z ( E )∣ for all semistable E , (4) for some norm on Λ R \Lambda_{\mathbb{R}} Λ R C > 0 C > 0 C > 0 [5 Definition 2.3 and Proposition 5.5] .
For an object E E E A i A_i A i m σ ( E ) = ∑ i ∣ Z ( A i ) ∣ m_\sigma(E) = \sum_i \lvert Z(A_i)\rvert m σ ( E ) = ∑ i ∣ Z ( A i )∣
Mass is unchanged by shifts and is additive under direct sums. It also satisfies the mass triangle inequality : for every distinguished triangle
A ⟶ B ⟶ C ⟶ A [ 1 ] , A \longrightarrow B \longrightarrow C \longrightarrow A[1], A ⟶ B ⟶ C ⟶ A [ 1 ] , one has [22 Proposition 3.3]
m σ ( B ) ≤ m σ ( A ) + m σ ( C ) . (5) m_\sigma(B) \le m_\sigma(A) + m_\sigma(C). \tag*{(5)} m σ ( B ) ≤ m σ ( A ) + m σ ( C ) . (5) We will prove the following strong boundedness property for the conditions constructed here. We say that σ \sigma σ total mass–Hom bound if, for every fixed A A A C A > 0 C_A > 0 C A > 0
∑ n ∈ Z dim C Hom ( A , E [ n ] ) ≤ C A m σ ( E ) for every E ∈ C . (6) \sum_{n \in\mathbb{Z}} \dim_{\mathbb{C}} \operatorname{Hom}(A, E[n]) \le C_A m_\sigma(E) \quad\text{for every } E \in\mathcal{C}. \tag*{(6)} n ∈ Z ∑ dim C Hom ( A , E [ n ]) ≤ C A m σ ( E ) for every E ∈ C . (6) In particular, dim C Hom ( A , E ) ≤ C A m σ ( E ) \dim_{\mathbb{C}} \operatorname{Hom}(A, E) \le C_A m_\sigma(E) dim C Hom ( A , E ) ≤ C A m σ ( E ) A A A E E E
A t-structure is determined by its aisle, a full additive subcategory U ⊂ C \mathcal{U} \subset\mathcal{C} U ⊂ C U [ 1 ] ⊂ U \mathcal{U}[1] \subset\mathcal{U} U [ 1 ] ⊂ U E E E A → E → B → A [ 1 ] A \to E \to B \to A[1] A → E → B → A [ 1 ] A ∈ U A \in\mathcal{U} A ∈ U B ∈ U ⊥ B \in\mathcal{U}^{\perp} B ∈ U ⊥ U ⊥ = { B : Hom ( A , B ) = 0 for all A ∈ U } \mathcal{U}^{\perp} = \{B : \operatorname{Hom}(A, B) = 0 \text{ for all } A \in\mathcal{U}\} U ⊥ = { B : Hom ( A , B ) = 0 for all A ∈ U }
C ≤ n = U [ − n ] and C ≥ n = U ⊥ [ 1 − n ] . \mathcal{C}^{\le n} = \mathcal{U}[-n] \quad\text{and} \quad\mathcal{C}^{\ge n} = \mathcal{U}^{\perp}[1-n]. C ≤ n = U [ − n ] and C ≥ n = U ⊥ [ 1 − n ] . The t-structure is bounded if every object belongs to C ≥ a ∩ C ≤ b \mathcal{C}^{\ge a} \cap\mathcal{C}^{\le b} C ≥ a ∩ C ≤ b a ≤ b a \le b a ≤ b A = U ∩ U ⊥ [ 1 ] \mathcal{A} = \mathcal{U} \cap\mathcal{U}^{\perp}[1] A = U ∩ U ⊥ [ 1 ] H A j ( E ) ∈ A H^j_{\mathcal{A}}(E) \in\mathcal{A} H A j ( E ) ∈ A
A slicing determines the aisle U = P ( ( 0 , ∞ ) ) \mathcal{U} = \mathcal{P}((0,\infty)) U = P (( 0 , ∞ )) A = P ( ( 0 , 1 ] ) \mathcal{A} = \mathcal{P}((0,1]) A = P (( 0 , 1 ]) { z : Im z > 0 } ∪ R < 0 \{z : \operatorname{Im} z > 0\} \cup\mathbb{R}_{<0} { z : Im z > 0 } ∪ R < 0 [8 Proposition 5.3] .
We will repeatedly use tilting of hearts. A torsion pair ( T , F ) (\mathcal{T}, \mathcal{F}) ( T , F ) A \mathcal{A} A Hom ( T , F ) = 0 \operatorname{Hom}(\mathcal{T}, \mathcal{F}) = 0 Hom ( T , F ) = 0 E ∈ A E \in\mathcal{A} E ∈ A T ∈ T T \in\mathcal{T} T ∈ T F ∈ F F \in\mathcal{F} F ∈ F
A ′ = ⟨ F [ 1 ] , T ⟩ ⊂ C . \mathcal{A}' = \langle\mathcal{F}[1], \mathcal{T} \rangle\subset\mathcal{C}. A ′ = ⟨ F [ 1 ] , T ⟩ ⊂ C . Equivalently, E ∈ A ′ E \in\mathcal{A}' E ∈ A ′ H A j ( E ) = 0 H^j_{\mathcal{A}}(E) = 0 H A j ( E ) = 0 j ≠ − 1 , 0 j \ne-1,0 j = − 1 , 0 H A − 1 ( E ) ∈ F H^{-1}_{\mathcal{A}}(E) \in\mathcal{F} H A − 1 ( E ) ∈ F H A 0 ( E ) ∈ T H^0_{\mathcal{A}}(E) \in\mathcal{T} H A 0 ( E ) ∈ T U ′ = { E ∈ U : H A 0 ( E ) ∈ T } \mathcal{U}' = \{E \in\mathcal{U} : H^0_{\mathcal{A}}(E) \in\mathcal{T}\} U ′ = { E ∈ U : H A 0 ( E ) ∈ T } U [ 1 ] ⊂ U ′ ⊂ U \mathcal{U}[1] \subset\mathcal{U}' \subset\mathcal{U} U [ 1 ] ⊂ U ′ ⊂ U
Enhancements.
We follow [25 Sections 2.2–2.3 and 4.5–4.6] for dg categories and enhancements. As a motivating example, let E , F E,F E , F X X X RHom ( E , F ) \operatorname{RHom}(E,F) RHom ( E , F ) k k k Hom D b ( X ) ( E , F [ k ] ) \operatorname{Hom}_{D^b(X)}(E,F[k]) Hom D b ( X ) ( E , F [ k ]) differential graded category , or dg category, retains these complexes. Thus every pair of objects has a cochain complex of morphisms Hom ∙ ( E , F ) \operatorname{Hom}^{\bullet}(E,F) Hom ∙ ( E , F ) d ( g f ) = d ( g ) f + ( − 1 ) deg g g d ( f ) d(gf) = d(g)f + (-1)^{\deg g}g\,d(f) d ( g f ) = d ( g ) f + ( − 1 ) d e g g g d ( f ) [7 Section 5] . The homotopy category H 0 ( E ) H^0(\mathcal{E}) H 0 ( E ) E \mathcal{E} E H 0 Hom ∙ ( E , F ) H^0\operatorname{Hom}^{\bullet}(E,F) H 0 Hom ∙ ( E , F ) d ( f ) = d F f − ( − 1 ) deg f f d E d(f) = d_Ff - (-1)^{\deg f}f\,d_E d ( f ) = d F f − ( − 1 ) d e g f f d E H 0 H^0 H 0
A dg category is pretriangulated if shifts and cones of closed degree-zero maps are represented, up to homotopy, by objects of the category. Its homotopy category is then triangulated. An enhancement of a triangulated category C \mathcal{C} C E \mathcal{E} E H 0 ( E ) ≃ C H^0(\mathcal{E}) \simeq\mathcal{C} H 0 ( E ) ≃ C idempotent completion : every endomorphism e e e e 2 = e e^2=e e 2 = e [4 Definitions 1.1–1.2 and Theorem 1.5] . Our small triangulated categories are idempotent complete, with additive pretriangulated dg enhancements. Alongside enhancements of the bounded derived category of coherent sheaves, we will use the concrete models supplied by quiver representations and matrix factorizations.
A dg functor acts by chain maps on morphism complexes and preserves identities and composition. It is a quasi-equivalence if these chain maps induce isomorphisms in every cohomological degree and the induced functor on H 0 H^0 H 0 enhanced functor is supplied with a dg lift, after replacing the models by quasi-equivalent ones when necessary.
An exact functor F F F conservative if F ( E ) ≃ 0 F(E) \simeq0 F ( E ) ≃ 0 E ≃ 0 E \simeq0 E ≃ 0 F F F
Let E \mathcal{E} E C \mathcal{C} C right dg module over E \mathcal{E} E E o p \mathcal{E}^{\mathrm{op}} E op [25 Section 3.1] . The basic examples are the representable right dg modules h E = Hom E ∙ ( − , E ) h_E=\operatorname{Hom}_{\mathcal{E}}^{\bullet}(-,E) h E = Hom E ∙ ( − , E ) Ind ( C ) \operatorname{Ind}(\mathcal{C}) Ind ( C ) M → N M\to N M → N M ( E ) → N ( E ) M(E)\to N(E) M ( E ) → N ( E ) E E E [25 Section 3.2] . A dg module is called perfect if its image in Ind ( C ) \operatorname{Ind}(\mathcal{C}) Ind ( C ) [25 Section 3.5] . We denote their category by Perf ( E ) \operatorname{Perf}(\mathcal{E}) Perf ( E ) E ↦ h E E\mapsto h_E E ↦ h E C ≃ Perf ( E ) \mathcal{C}\simeq\operatorname{Perf}(\mathcal{E}) C ≃ Perf ( E ) [25 Section 4.6] .
An object G ∈ C G\in\mathcal{C} G ∈ C split generator if every object of C \mathcal{C} C G G G E E E Ind ( C ) \operatorname{Ind}(\mathcal{C}) Ind ( C ) compact if Hom ( E , − ) \operatorname{Hom}(E,-) Hom ( E , − ) [25 Corollary 3.7] , so the compact objects of Ind ( C ) \operatorname{Ind}(\mathcal{C}) Ind ( C ) C \mathcal{C} C C = D b ( X ) \mathcal{C}=\mathrm{D}^{\mathrm{b}}(X) C = D b ( X ) X X X Ind ( C ) \operatorname{Ind}(\mathcal{C}) Ind ( C ) X X X [25 Section 3.6] .
For dg enhancements E , F \mathcal{E},\mathcal{F} E , F C , D \mathcal{C},\mathcal{D} C , D E ⊗ C F \mathcal{E}\otimes_{\mathbb{C}}\mathcal{F} E ⊗ C F E ⊠ F E\boxtimes F E ⊠ F
Hom ∙ ( E ⊠ F , E ′ ⊠ F ′ ) = Hom E ∙ ( E , E ′ ) ⊗ C Hom F ∙ ( F , F ′ ) . \operatorname{Hom}^{\bullet}(E\boxtimes F,E'\boxtimes F')=\operatorname{Hom}_{\mathcal{E}}^{\bullet}(E,E')\otimes_{\mathbb{C}}\operatorname{Hom}_{\mathcal{F}}^{\bullet}(F,F'). Hom ∙ ( E ⊠ F , E ′ ⊠ F ′ ) = Hom E ∙ ( E , E ′ ) ⊗ C Hom F ∙ ( F , F ′ ) . The differential and composition use the usual signs for tensor products of complexes. Over C \mathbb{C} C [25 Sections 2.3 and 4.3] . The Morita tensor product of the enhanced triangulated categories is, by definition, C ⊗ D : = Perf ( E ⊗ C F ) \mathcal{C}\otimes\mathcal{D}:=\operatorname{Perf}(\mathcal{E}\otimes_{\mathbb{C}}\mathcal{F}) C ⊗ D := Perf ( E ⊗ C F ) C ⊗ D \mathcal{C}\otimes\mathcal{D} C ⊗ D E ⊠ F E\boxtimes F E ⊠ F D b ( X ) ⊗ D b ( Y ) ≃ D b ( X × Y ) \mathrm{D}^{\mathrm{b}}(X)\otimes\mathrm{D}^{\mathrm{b}}(Y)\simeq\mathrm{D}^{\mathrm{b}}(X\times Y) D b ( X ) ⊗ D b ( Y ) ≃ D b ( X × Y ) [7 Theorem 5.5] .
A dg category E \mathcal{E} E proper if ⨁ k H k Hom ∙ ( E , F ) \bigoplus_k H^k\operatorname{Hom}^{\bullet}(E,F) ⨁ k H k Hom ∙ ( E , F ) diagonal bimodule is the right module over E o p ⊗ E \mathcal{E}^{\mathrm{op}}\otimes\mathcal{E} E op ⊗ E ( E , F ) ↦ Hom ∙ ( F , E ) (E,F)\mapsto\operatorname{Hom}^{\bullet}(F,E) ( E , F ) ↦ Hom ∙ ( F , E ) smooth if this bimodule is perfect [25 Section 4.7] . This is the categorical counterpart of the perfectness of O Δ \mathcal{O}_{\Delta} O Δ X × X X\times X X × X
If E \mathcal{E} E C \mathcal{C} C G G G [14 Lemma 2.5] , and a dg module M M M ⨁ k H k ( M ( G ) ) \bigoplus_k H^k(M(G)) ⨁ k H k ( M ( G )) [44 Lemma 2.8 and Corollary 2.9] . Since G G G C \mathcal{C} C G G G C \mathcal{C} C
Smoothness and properness also give Serre duality: C \mathcal{C} C S C \mathcal{S}_{\mathcal{C}} S C Serre functor , with natural isomorphisms Hom ( E , F ) ∗ ≃ Hom ( F , S C E ) \operatorname{Hom}(E,F)^*\simeq\operatorname{Hom}(F,\mathcal{S}_{\mathcal{C}}E) Hom ( E , F ) ∗ ≃ Hom ( F , S C E ) [6 Propositions 3.3 and 3.4(a)] . For D b ( X ) \mathrm{D}^{\mathrm{b}}(X) D b ( X ) X X X S X = ( − ) ⊗ ω X [ dim X ] \mathcal{S}_X=(-)\otimes\omega_X[\dim X] S X = ( − ) ⊗ ω X [ dim X ] [6 Example 3.2(2)] .
Lemma 2.1. Let G G G C \mathcal{C} C A = G A=G A = G E E E K n u m ( C ) K_{\mathrm{num}}(\mathcal{C}) K num ( C )
Proof. For every fixed A A A b A > 0 b_A>0 b A > 0
∑ n ∈ Z dim C Hom ( A , E [ n ] ) ≤ b A ∑ n ∈ Z dim C Hom ( G , E [ n ] ) \sum_{n\in\mathbb{Z}}\dim_{\mathbb{C}}\operatorname{Hom}(A,E[n])\leq b_A\sum_{n\in\mathbb{Z}}\dim_{\mathbb{C}}\operatorname{Hom}(G,E[n]) n ∈ Z ∑ dim C Hom ( A , E [ n ]) ≤ b A n ∈ Z ∑ dim C Hom ( G , E [ n ]) for all E E E
Indeed, the objects A A A G G G G G G E E E E E E
For the second assertion, choose objects A 1 , … , A s A_1,\ldots,A_s A 1 , … , A s
∥ u ∥ 2 = ∑ i χ ( A i , u ) 2 \lVert u\rVert^2=\sum_i\chi(A_i,u)^2 ∥ u ∥ 2 = i ∑ χ ( A i , u ) 2 is a norm on K n u m ( C ) R K_{\mathrm{num}}(\mathcal{C})_{\mathbb{R}} K num ( C ) R
∥ [ E ] ∥ 2 ≤ ∑ i ( ∑ n ∈ Z dim C Hom ( A i , E [ n ] ) ) 2 ≤ ( ∑ i C A i 2 ) m σ ( E ) 2 . \lVert[E]\rVert^2\leq\sum_i\left(\sum_{n\in\mathbb{Z}}\dim_{\mathbb{C}}\operatorname{Hom}(A_i,E[n])\right)^2\leq\left(\sum_i C_{A_i}^2\right)m_{\sigma}(E)^2. ∥[ E ] ∥ 2 ≤ i ∑ ( n ∈ Z ∑ dim C Hom ( A i , E [ n ]) ) 2 ≤ ( i ∑ C A i 2 ) m σ ( E ) 2 . For semistable E E E ∣ Z ( E ) ∣ \lvert Z(E)\rvert ∣ Z ( E )∣ [9 Theorem 2.1] , in its total-Hom formulation. □
For later use, this finite-rank hypothesis holds for D b ( X ) \mathrm{D}^{\mathrm{b}}(X) D b ( X ) X X X [8 Section 1.3] . Indeed, the Hirzebruch–Riemann–Roch theorem factors the Euler pairing through the finite-dimensional space H e v e n ( X , Q ) H^{\mathrm{even}}(X,\mathbb{Q}) H even ( X , Q ) K n u m ( X ) K_{\mathrm{num}}(X) K num ( X ) Z s \mathbb{Z}^s Z s D b ( X ) \mathrm{D}^{\mathrm{b}}(X) D b ( X ) i i i i ! i^! i ! i ! i ≃ id i^!i\simeq\operatorname{id} i ! i ≃ id [7 Proposition 1.16] . The adjunction identity χ ( i A , B ) = χ ( A , i ! B ) \chi(iA,B)=\chi(A,i^!B) χ ( i A , B ) = χ ( A , i ! B ) [35 Section 1.1] .
Matrix factorizations
We follow the grading and translation conventions of [40 Section 2.2, especially Definition 2.6] . Let S = C [ x 1 , … , x N ] S=\mathbb{C}[x_1,\ldots,x_N] S = C [ x 1 , … , x N ] W ∈ S W\in S W ∈ S d d d M ( a ) j = M j + a M(a)_j=M_{j+a} M ( a ) j = M j + a W W W S S S
P 0 → p 0 P 1 → p 1 P 0 ( d ) (7) P_0\xrightarrow{p_0}P_1\xrightarrow{p_1}P_0(d) \tag*{(7)} P 0 p 0 P 1 p 1 P 0 ( d ) (7) such that p 1 p 0 = W id P 0 p_1p_0=W\operatorname{id}_{P_0} p 1 p 0 = W id P 0 p 0 ( d ) p 1 = W id P 1 p_0(d)p_1=W\operatorname{id}_{P_1} p 0 ( d ) p 1 = W id P 1 W W W d d d
A morphism f : P → Q f:P\to Q f : P → Q f i : P i → Q i f_i:P_i\to Q_i f i : P i → Q i
P 0 → p 0 P 1 → p 1 P 0 ( d ) f 0 ↓ f 1 ↓ f 0 ( d ) ↓ Q 0 → q 0 Q 1 → q 1 Q 0 ( d ) . \begin{CD}
P_0 @>{p_0}>> P_1 @>{p_1}>> P_0(d) \\
@V{f_0}VV @V{f_1}VV @V{f_0(d)}VV \\
Q_0 @>{q_0}>> Q_1 @>{q_1}>> Q_0(d).
\end{CD} P 0 f 0 ↓ ⏐ Q 0 p 0 q 0 P 1 f 1 ↓ ⏐ Q 1 p 1 q 1 P 0 ( d ) f 0 ( d ) ↓ ⏐ Q 0 ( d ) . Two morphisms f , g : P → Q f,g:P\to Q f , g : P → Q homotopic if there are degree-zero maps h 0 : P 0 → Q 1 ( − d ) h_0:P_0\to Q_1(-d) h 0 : P 0 → Q 1 ( − d ) h 1 : P 1 → Q 0 h_1:P_1\to Q_0 h 1 : P 1 → Q 0
P 0 → p 0 P 1 → p 1 P 0 ( d ) h 0 ↓ h 1 ↓ h 0 ( d ) ↓ Q 1 ( − d ) → q 1 ( − d ) Q 0 → q 0 Q 1 → q 1 Q 0 ( d ) \begin{CD}
P_0 @>{p_0}>> P_1 @>{p_1}>> P_0(d) \\
@V{h_0}VV @V{h_1}VV @V{h_0(d)}VV \\
Q_1(-d) @>{q_1(-d)}>> Q_0 @>{q_0}>> Q_1 @>{q_1}>> Q_0(d)
\end{CD} P 0 h 0 ↓ ⏐ Q 1 ( − d ) p 0 q 1 ( − d ) P 1 h 1 ↓ ⏐ Q 0 p 1 q 0 P 0 ( d ) h 0 ( d ) ↓ ⏐ Q 1 q 1 Q 0 ( d ) f 0 − g 0 = q 1 ( − d ) h 0 + h 1 p 0 , f_0-g_0=q_1(-d)h_0+h_1p_0, f 0 − g 0 = q 1 ( − d ) h 0 + h 1 p 0 , f 1 − g 1 = q 0 h 1 + h 0 ( d ) p 1 . f_1-g_1=q_0h_1+h_0(d)p_1. f 1 − g 1 = q 0 h 1 + h 0 ( d ) p 1 . These morphisms and homotopies arise from the standard dg category of matrix factorizations [3 Definitions 2.7 and 2.12] . Its degree-zero cycles are morphisms of factorizations, and its degree-zero boundaries are nullhomotopic morphisms. Thus H 0 H^0 H 0 MF g r ( W ) \operatorname{MF}^{\mathrm{gr}}(W) MF gr ( W )
Its translation functor [ 1 ] [1] [ 1 ] (7) to
P 1 → − p 1 P 0 ( d ) → − p 0 ( d ) P 1 ( d ) . P_1 \xrightarrow{-p_1} P_0(d) \xrightarrow{-p_0(d)} P_1(d). P 1 − p 1 P 0 ( d ) − p 0 ( d ) P 1 ( d ) . Applying this translation twice shifts the internal grading by d d d internal grading shift τ \tau τ P P P P ( 1 ) P(1) P ( 1 ) τ a P = P ( a ) \tau^aP=P(a) τ a P = P ( a )
τ = ( 1 ) , τ d = ( d ) ≃ [ 2 ] . (8) \tau=(1),\qquad\tau^d=(d)\simeq[2]. \tag*{(8)} τ = ( 1 ) , τ d = ( d ) ≃ [ 2 ] . (8) This explains the phase increment 2 / d 2/d 2/ d d d d A = S / ( W ) A=S/(W) A = S / ( W ) D g r b ( A ) D^{\mathrm{b}}_{\mathrm{gr}}(A) D gr b ( A ) A A A graded singularity category is the Verdier quotient
D s g g r ( A ) = D g r b ( A ) / Perf g r ( A ) . D^{\mathrm{gr}}_{\mathrm{sg}}(A)=D^{\mathrm{b}}_{\mathrm{gr}}(A)/\operatorname{Perf}^{\mathrm{gr}}(A). D sg gr ( A ) = D gr b ( A ) / Perf gr ( A ) . Here Perf g r ( A ) \operatorname{Perf}^{\mathrm{gr}}(A) Perf gr ( A ) coker ( p 0 ) \operatorname{coker}(p_0) coker ( p 0 ) W W W A A A MF g r ( W ) ≃ D s g g r ( A ) \operatorname{MF}^{\mathrm{gr}}(W)\simeq D^{\mathrm{gr}}_{\mathrm{sg}}(A) MF gr ( W ) ≃ D sg gr ( A ) [40 Theorem 2.7] .
Suppose now that deg x i = 1 \deg x_i=1 deg x i = 1 i i i X = ( W = 0 ) ⊂ P N − 1 X=(W=0)\subset\mathbb{P}^{N-1} X = ( W = 0 ) ⊂ P N − 1 [34 Theorem 3.11] compares its derived category with MF g r ( W ) \operatorname{MF}^{\mathrm{gr}}(W) MF gr ( W ) N − d N-d N − d N > d N>d N > d MF g r ( W ) \operatorname{MF}^{\mathrm{gr}}(W) MF gr ( W ) Ku ( X ) \operatorname{Ku}(X) Ku ( X )
D b ( X ) = ⟨ Ku ( X ) , O X , O X ( 1 ) , … , O X ( N − d − 1 ) ⟩ . D^{\mathrm{b}}(X)=\langle\operatorname{Ku}(X),\mathcal{O}_X,\mathcal{O}_X(1),\ldots,\mathcal{O}_X(N-d-1)\rangle. D b ( X ) = ⟨ Ku ( X ) , O X , O X ( 1 ) , … , O X ( N − d − 1 )⟩ . Equivalently,
Ku ( X ) = { E ∈ D b ( X ) : Hom ∙ ( O X ( j ) , E ) = 0 for 0 ≤ j < N − d } . \operatorname{Ku}(X)=\{E\in D^{\mathrm{b}}(X):\operatorname{Hom}^{\bullet}(\mathcal{O}_X(j),E)=0\text{ for }0\leq j<N-d\}. Ku ( X ) = { E ∈ D b ( X ) : Hom ∙ ( O X ( j ) , E ) = 0 for 0 ≤ j < N − d } . For N = d N=d N = d MF g r ( W ) \operatorname{MF}^{\mathrm{gr}}(W) MF gr ( W ) D b ( X ) D^{\mathrm{b}}(X) D b ( X ) N < d N<d N < d D b ( X ) D^{\mathrm{b}}(X) D b ( X ) d − N d-N d − N
When N > d N>d N > d τ \tau τ rotation functor : tensoring by O X ( 1 ) \mathcal{O}_X(1) O X ( 1 ) O X \mathcal{O}_X O X [2 Definition 6.11 and Theorem 6.13] .
The weighted version used in § 5 replaces X X X [ ( W − 1 ( 0 ) ∖ { 0 } ) / C ∗ ] [(W^{-1}(0)\setminus\{0\})/\mathbb{C}^*] [( W − 1 ( 0 ) ∖ { 0 }) / C ∗ ] C ∗ \mathbb{C}^* C ∗ ∑ i deg x i − d \sum_i\deg x_i-d ∑ i deg x i − d [34 Theorem 2.5 and Remark 3.12] .
Thom–Sebastiani theorem
Let H H H C \mathcal{C} C coherent left action of H H H C \mathcal{C} C F h : C → C F_h:\mathcal{C}\to\mathcal{C} F h : C → C h ∈ H h\in H h ∈ H α g , h : F g F h → F g h \alpha_{g,h}:F_gF_h\to F_{gh} α g , h : F g F h → F g h F 1 = id F_1=\operatorname{id} F 1 = id α 1 , h = α h , 1 = id \alpha_{1,h}=\alpha_{h,1}=\operatorname{id} α 1 , h = α h , 1 = id α g h , k ∘ ( α g , h F k ) = α g , h k ∘ ( F g α h , k ) \alpha_{gh,k}\circ(\alpha_{g,h}F_k)=\alpha_{g,hk}\circ(F_g\alpha_{h,k}) α g h , k ∘ ( α g , h F k ) = α g , hk ∘ ( F g α h , k ) F g F h F k → F g h k F_gF_hF_k\to F_{ghk} F g F h F k → F g hk g , h , k ∈ H g,h,k\in H g , h , k ∈ H E \mathcal{E} E C \mathcal{C} C H 0 ( E ) ≃ C H^0(\mathcal{E})\simeq\mathcal{C} H 0 ( E ) ≃ C F h F_h F h E \mathcal{E} E α g , h \alpha_{g,h} α g , h [15 Definition 8.1] .
An equivariant object is a pair ( E , ρ ) (E,\rho) ( E , ρ ) E ∈ C E \in\mathcal{C} E ∈ C ρ h : F h E ⟶ ∼ E \rho_h:F_hE \overset{\sim}{\longrightarrow} E ρ h : F h E ⟶ ∼ E
ρ 1 = id E , ρ g h ∘ α g , h , E = ρ g ∘ F g ( ρ h ) . \rho_1=\operatorname{id}_E,\qquad\rho_{gh}\circ\alpha_{g,h,E}=\rho_g\circ F_g(\rho_h). ρ 1 = id E , ρ g h ∘ α g , h , E = ρ g ∘ F g ( ρ h ) . A morphism ( E , ρ ) ⟶ ( E ′ , ρ ′ ) (E,\rho)\longrightarrow(E',\rho') ( E , ρ ) ⟶ ( E ′ , ρ ′ ) f : E ⟶ E ′ f:E\longrightarrow E' f : E ⟶ E ′ C \mathcal{C} C f ∘ ρ h = ρ h ′ ∘ F h ( f ) f\circ\rho_h=\rho'_h\circ F_h(f) f ∘ ρ h = ρ h ′ ∘ F h ( f ) h ∈ H h\in H h ∈ H equivariant category C H \mathcal{C}^H C H [15 Definitions 3.1 and 3.5] , written here for left actions. The dg category E H \mathcal{E}^H E H C H \mathcal{C}^H C H [15 Theorem 8.7] . This uses our standing assumption that C \mathcal{C} C
The forgetful functor U : C H ⟶ C U:\mathcal{C}^H\longrightarrow\mathcal{C} U : C H ⟶ C ( E , ρ ) (E,\rho) ( E , ρ ) E E E X ∈ C X\in\mathcal{C} X ∈ C I ( X ) = ⨁ h ∈ H F h X I(X)=\bigoplus_{h\in H}F_hX I ( X ) = ⨁ h ∈ H F h X α g , h \alpha_{g,h} α g , h H H H I I I U U U [15 Lemma 3.8] . The adjunction map I ( U ( E , ρ ) ) ⟶ ( E , ρ ) I(U(E,\rho))\longrightarrow(E,\rho) I ( U ( E , ρ )) ⟶ ( E , ρ ) ρ h \rho_h ρ h C \mathcal{C} C ∣ H ∣ − 1 ρ h |H|^{-1}\rho_h ∣ H ∣ − 1 ρ h [15 Lemma 3.9] . We say that the action preserves a stability condition σ = ( Z , P ) \sigma=(Z,\mathcal{P}) σ = ( Z , P ) C \mathcal{C} C
Z ∘ ( F h ) ∗ = Z , F h P ( ϕ ) = P ( ϕ ) ( h ∈ H , ϕ ∈ R ) . Z\circ(F_h)_*=Z,\qquad F_h\mathcal{P}(\phi)=\mathcal{P}(\phi)\quad(h\in H,\ \phi\in\mathbb{R}). Z ∘ ( F h ) ∗ = Z , F h P ( ϕ ) = P ( ϕ ) ( h ∈ H , ϕ ∈ R ) . For an equivariant object ( E , ρ ) (E,\rho) ( E , ρ ) E j E_j E j E E E F h F_h F h ρ h \rho_h ρ h ρ h j : F h E j ⟶ E j \rho_h^j:F_hE_j\longrightarrow E_j ρ h j : F h E j ⟶ E j E E E ρ g h j ∘ α g , h , E j \rho_{gh}^j\circ\alpha_{g,h,E_j} ρ g h j ∘ α g , h , E j ρ g j ∘ F g ( ρ h j ) \rho_g^j\circ F_g(\rho_h^j) ρ g j ∘ F g ( ρ h j ) F g F h E ⟶ E F_gF_hE\longrightarrow E F g F h E ⟶ E C H \mathcal{C}^H C H
Z H = Z ∘ U ∗ , P H ( ϕ ) = { ( E , ρ ) : E ∈ P ( ϕ ) } . Z^H=Z\circ U_*,\qquad\mathcal{P}^H(\phi)=\{(E,\rho):E\in\mathcal{P}(\phi)\}. Z H = Z ∘ U ∗ , P H ( ϕ ) = {( E , ρ ) : E ∈ P ( ϕ )} . Lemma 4.3 proves the support property and numerical property; compare [30 Theorem 2.16 and Remark 2.17] .
The Thom–Sebastiani theorem relates a tensor product of matrix-factorization categories to the sum of their polynomials. The sum carries a grading that records the degree in each set of variables. Let f ( x ) f(x) f ( x ) g ( y ) g(y) g ( y ) d d d x i x_i x i e 1 e_1 e 1 y j y_j y j e 2 e_2 e 2
L = ( Z e 1 ⊕ Z e 2 ) / ⟨ d ( e 1 − e 2 ) ⟩ ≅ Z ⊕ Z / d . L=(\mathbb{Z}e_1\oplus\mathbb{Z}e_2)/\langle d(e_1-e_2)\rangle\cong\mathbb{Z}\oplus\mathbb{Z}/d. L = ( Z e 1 ⊕ Z e 2 ) / ⟨ d ( e 1 − e 2 )⟩ ≅ Z ⊕ Z / d . Put S = C [ x 1 , … , x s , y 1 , … , y t ] S=\mathbb{C}[x_1,\ldots,x_s,y_1,\ldots,y_t] S = C [ x 1 , … , x s , y 1 , … , y t ] η = d e 1 = d e 2 ∈ L \eta=de_1=de_2\in L η = d e 1 = d e 2 ∈ L f f f g g g L L L S S S M = ⨁ ℓ ∈ L M ℓ M=\bigoplus_{\ell\in L}M_\ell M = ⨁ ℓ ∈ L M ℓ S a M b ⊆ M a + b S_aM_b\subseteq M_{a+b} S a M b ⊆ M a + b M ( a ) ℓ = M ℓ + a M(a)_\ell=M_{\ell+a} M ( a ) ℓ = M ℓ + a a ∈ L a\in L a ∈ L L L L matrix factorization of f + g f+g f + g P 0 , P 1 P_0,P_1 P 0 , P 1 S ( a ) S(a) S ( a )
P 0 → p 0 P 1 → p 1 P 0 ( η ) P_0\xrightarrow{p_0}P_1\xrightarrow{p_1}P_0(\eta) P 0 p 0 P 1 p 1 P 0 ( η ) such that p 1 p 0 = ( f + g ) id P 0 p_1p_0=(f+g)\operatorname{id}_{P_0} p 1 p 0 = ( f + g ) id P 0 p 0 ( η ) p 1 = ( f + g ) id P 1 p_0(\eta)p_1=(f+g)\operatorname{id}_{P_1} p 0 ( η ) p 1 = ( f + g ) id P 1 L L L η \eta η η \eta η MF L ( f + g ) \operatorname{MF}^{L}(f+g) MF L ( f + g ) [ 2 ] ≃ ( η ) [2]\simeq(\eta) [ 2 ] ≃ ( η )
The Thom–Sebastiani theorem [3 Theorem 1.1] gives an equivalence
MF gr ( f ) ⊗ MF gr ( g ) ≃ MF L ( f + g ) . \operatorname{MF}^{\operatorname{gr}}(f)\otimes\operatorname{MF}^{\operatorname{gr}}(g)\simeq\operatorname{MF}^{L}(f+g). MF gr ( f ) ⊗ MF gr ( g ) ≃ MF L ( f + g ) . In the tensor product, degrees a a a b b b a e 1 + b e 2 ae_1+be_2 a e 1 + b e 2 L L L e 1 e_1 e 1 e 2 e_2 e 2
Stability with quiver coefficients
Throughout this section, C \mathcal{C} C E \mathcal{E} E C \mathcal{C} C C Q \mathcal{C}_{Q} C Q D b ( rep C Q ) \mathrm{D}^{\mathrm{b}}(\operatorname{rep}_{\mathbb{C}} Q) D b ( rep C Q ) Q Q Q
An object of C Q \mathcal{C}_{Q} C Q quiver diagram . It specifies an object of C \mathcal{C} C Q Q Q dg quiver diagram consists of a right dg E \mathcal{E} E M i M_i M i M a : M i → M j M_a:M_i\to M_j M a : M i → M j a : i → j a:i\to j a : i → j E \mathcal{E} E C Q \mathbb{C}Q C Q E ⊗ ( C Q ) o p \mathcal{E}\otimes(\mathbb{C}Q)^{\mathrm{op}} E ⊗ ( C Q ) op [25 Sections 3.1–3.2 and 3.5] .
This construction belongs to the general theory of coherent quiver diagrams studied by Groth–Šťovíček [17 Sections 2–3] ; see also their treatment of type A A A [18 ] . In the literature on stability conditions, Kawatani [24 Section 2.2] treats the case of A 2 A_2 A 2 [19 Sections 5.1 and 5.3] use A n A_n A n
Let P i P_i P i Q Q Q j j j i i i j j j a : j → k a:j\to k a : j → k ( P i ) j ⟶ ( P i ) k (P_i)_j\longrightarrow(P_i)_k ( P i ) j ⟶ ( P i ) k p : i → j p:i\to j p : i → j a ∘ p : i → j → k a\circ p:i\to j\to k a ∘ p : i → j → k M M M [11 Section 1] gives a short exact sequence
0 ⟶ ⨁ a : i → j P j ⊗ C M i ⟶ ⨁ i P i ⊗ C M i ⟶ M ⟶ 0. (9) 0\longrightarrow\bigoplus_{a:i\to j}P_j\otimes_{\mathbb{C}}M_i\longrightarrow\bigoplus_iP_i\otimes_{\mathbb{C}}M_i\longrightarrow M\longrightarrow0. \tag*{(9)} 0 ⟶ a : i → j ⨁ P j ⊗ C M i ⟶ i ⨁ P i ⊗ C M i ⟶ M ⟶ 0. (9) To describe the maps, fix a vertex v v v v v v M i M_i M i p : i → v p:i\to v p : i → v M p : M i → M v M_p:M_i\to M_v M p : M i → M v p p p v v v M v M_v M v v v v M i M_i M i a : i → j a:i\to j a : i → j p : j → v p:j\to v p : j → v M i M_i M i p ∘ a p\circ a p ∘ a − M a -M_a − M a M j M_j M j p p p M v M_v M v M p ∘ a = M p ∘ M a M_{p\circ a}=M_p\circ M_a M p ∘ a = M p ∘ M a
The resolution gives a criterion for perfectness in terms of the vertex modules.
Lemma 3.1. Let Q Q Q M M M Q Q Q M M M E ⊗ ( C Q ) o p \mathcal{E}\otimes(\mathbb{C}Q)^{\mathrm{op}} E ⊗ ( C Q ) op M i M_i M i E \mathcal{E} E
Proof. Perfect modules are obtained from representable modules by taking shifts, cones, finite direct sums, and direct summands. For an object A ∈ E A\in\mathcal{E} A ∈ E h A = Hom E ∙ ( − , A ) h_A=\operatorname{Hom}^{\bullet}_{\mathcal{E}}(-,A) h A = Hom E ∙ ( − , A ) E ⊗ ( C Q ) \mathcal{E}\otimes(\mathbb{C}Q) E ⊗ ( C Q ) ( A , j ) (A,j) ( A , j ) A A A E \mathcal{E} E j j j Q Q Q ( A , j ) (A,j) ( A , j ) ( B , v ) (B,v) ( B , v )
Hom E ⊗ ( C Q ) o p ∙ ( ( B , v ) , ( A , j ) ) = Hom E ∙ ( B , A ) ⊗ C Hom C Q ( j , v ) = ⨁ p : j → v h A ( B ) . \operatorname{Hom}^{\bullet}_{\mathcal{E}\otimes(\mathbb{C}Q)^{\mathrm{op}}}((B,v),(A,j))=\operatorname{Hom}^{\bullet}_{\mathcal{E}}(B,A)\otimes_{\mathbb{C}}\operatorname{Hom}_{\mathbb{C}Q}(j,v)=\bigoplus_{p:j\to v}h_A(B). Hom E ⊗ ( C Q ) op ∙ (( B , v ) , ( A , j )) = Hom E ∙ ( B , A ) ⊗ C Hom C Q ( j , v ) = p : j → v ⨁ h A ( B ) . Thus, viewed as a quiver diagram of E \mathcal{E} E h A h_A h A v v v j → v j\to v j → v h A h_A h A P j ⊗ C h A P_j\otimes_{\mathbb{C}}h_A P j ⊗ C h A
Since P j ⊗ C − P_j\otimes_{\mathbb{C}}- P j ⊗ C − E \mathcal{E} E M i M_i M i P j ⊗ C M i P_j\otimes_{\mathbb{C}}M_i P j ⊗ C M i Q Q Q M M M M M M
Conversely, evaluate a representable diagram at a vertex v v v ( P j ⊗ C Q h A ) v = ⨁ p : j → v h A (P_j \otimes_{\mathbb{C}Q} h_A)_v = \bigoplus_{p:j\to v} h_A ( P j ⊗ C Q h A ) v = ⨁ p : j → v h A j j j v v v Q Q Q E \mathcal{E} E v v v E ⊗ ( C Q ) o p \mathcal{E} \otimes(\mathbb{C}Q)^{\mathrm{op}} E ⊗ ( C Q ) op E \mathcal{E} E M M M M v M_v M v □ \square □
Lemma 3.1 is a special case of Toën–Vaquié’s criterion [44 Definition 2.7 and Lemma 2.8(3)] . For dg categories T T T T ′ T' T ′ T ⊗ L T ′ T \otimes^{\mathbb{L}} T' T ⊗ L T ′ T ′ T' T ′ T T T T ′ T' T ′ T T T T ⊗ L T ′ T \otimes^{\mathbb{L}} T' T ⊗ L T ′ T = ( C Q ) o p T = (\mathbb{C}Q)^{\mathrm{op}} T = ( C Q ) op T ′ = E T' = \mathcal{E} T ′ = E T T T Q Q Q M i M_i M i E \mathcal{E} E
Thus quiver diagrams with perfect vertex modules give the category
C Q = Perf ( E ⊗ ( C Q ) o p ) ≃ C ⊗ D b ( rep C Q ) . \mathcal{C}_Q = \operatorname{Perf}(\mathcal{E} \otimes(\mathbb{C}Q)^{\mathrm{op}}) \simeq\mathcal{C} \otimes\mathcal{D}^{b}(\operatorname{rep}_{\mathbb{C}} Q). C Q = Perf ( E ⊗ ( C Q ) op ) ≃ C ⊗ D b ( rep C Q ) . Lemma 3.2. A bounded t t t C \mathcal{C} C t t t C Q \mathcal{C}_Q C Q
C Q ≤ 0 = { M : M i ∈ C ≤ 0 } , C Q ≥ 0 = { M : M i ∈ C ≥ 0 } . \mathcal{C}_Q^{\leq0} = \{M : M_i \in\mathcal{C}^{\leq0}\}, \qquad\mathcal{C}_Q^{\geq0} = \{M : M_i \in\mathcal{C}^{\geq0}\}. C Q ≤ 0 = { M : M i ∈ C ≤ 0 } , C Q ≥ 0 = { M : M i ∈ C ≥ 0 } . If H \mathcal{H} H C \mathcal{C} C Rep ( Q , H ) \operatorname{Rep}(Q,\mathcal{H}) Rep ( Q , H ) C Q \mathcal{C}_Q C Q H \mathcal{H} H
Proof. Extend the t t t C \mathcal{C} C Ind ( C ) \operatorname{Ind}(\mathcal{C}) Ind ( C ) [36 Lemma 2.1.1] . By [38 Theorem A] , the extended t t t t t t Ind ( C Q ) \operatorname{Ind}(\mathcal{C}_Q) Ind ( C Q ) Ind ( C ) \operatorname{Ind}(\mathcal{C}) Ind ( C ) t t t C \mathcal{C} C t t t C Q \mathcal{C}_Q C Q Q Q Q Rep ( Q , H ) \operatorname{Rep}(Q,\mathcal{H}) Rep ( Q , H ) □ \square □
Proposition 3.3 (Takahashi). Let d ≥ 2 d \geq2 d ≥ 2 Q Q Q A d − 1 A_{d-1} A d − 1 rep C Q \operatorname{rep}_{\mathbb{C}} Q rep C Q
Φ : MF g r ( x d ) → ∼ D b ( rep C Q ) (10) \Phi: \operatorname{MF}^{\mathrm{gr}}(x^d) \xrightarrow{\sim} \mathcal{D}^{b}(\operatorname{rep}_{\mathbb{C}} Q) \tag*{(10)} Φ : MF gr ( x d ) ∼ D b ( rep C Q ) (10) that sends the internal grading shift τ \tau τ MF g r ( x d ) \operatorname{MF}^{\mathrm{gr}}(x^d) MF gr ( x d ) S Q − 1 [ 1 ] S_Q^{-1}[1] S Q − 1 [ 1 ] S Q S_Q S Q D b ( rep C Q ) \mathcal{D}^{b}(\operatorname{rep}_{\mathbb{C}} Q) D b ( rep C Q )
S M F ≃ τ − 1 [ 1 ] , S M F d ≃ [ d − 2 ] . (11) S_{\mathrm{MF}} \simeq\tau^{-1}[1], \qquad S_{\mathrm{MF}}^{d} \simeq[d-2]. \tag*{(11)} S MF ≃ τ − 1 [ 1 ] , S MF d ≃ [ d − 2 ] . (11) Proof. We recall Takahashi’s construction [39 ] for the linear orientation of the quiver, with all arrows pointing in the same direction. Put R = C [ x ] R = \mathbb{C}[x] R = C [ x ] deg x = 1 \deg x = 1 deg x = 1
E ℓ = ( R → x ℓ R ( ℓ ) → x d − ℓ R ( d ) ) , 1 ≤ ℓ ≤ d − 1 , E_\ell= \left(R \xrightarrow{x^\ell} R(\ell) \xrightarrow{x^{d-\ell}} R(d)\right), \qquad1 \leq\ell\leq d-1, E ℓ = ( R x ℓ R ( ℓ ) x d − ℓ R ( d ) ) , 1 ≤ ℓ ≤ d − 1 , split-generate MF g r ( x d ) \operatorname{MF}^{\mathrm{gr}}(x^d) MF gr ( x d )
Hom ( E k , E ℓ [ n ] ) = { C , n = 0 and k ≤ ℓ , 0 , otherwise . (12) \operatorname{Hom}(E_k,E_\ell[n]) =
\begin{cases}
\mathbb{C}, & n=0 \text{ and } k \leq\ell,\\
0, & \text{otherwise}.
\end{cases} \tag*{(12)} Hom ( E k , E ℓ [ n ]) = { C , 0 , n = 0 and k ≤ ℓ , otherwise . (12) For k ≤ ℓ k \leq\ell k ≤ ℓ f k ℓ = ( 1 , x ℓ − k ) f_{k\ell} = (1,x^{\ell-k}) f k ℓ = ( 1 , x ℓ − k ) Hom ( E k , E ℓ ) \operatorname{Hom}(E_k,E_\ell) Hom ( E k , E ℓ ) f ℓ m f k ℓ = f k m f_{\ell m}f_{k\ell}=f_{km} f ℓ m f k ℓ = f k m
Thus ( E 1 , … , E d − 1 ) (E_1,\ldots,E_{d-1}) ( E 1 , … , E d − 1 ) T = ⨁ ℓ = 1 d − 1 E ℓ T = \bigoplus_{\ell=1}^{d-1} E_\ell T = ⨁ ℓ = 1 d − 1 E ℓ B = End ( T ) B = \operatorname{End}(T) B = End ( T ) B B B 1 → 2 → ⋯ → d − 1 1 \to2 \to\cdots\to d-1 1 → 2 → ⋯ → d − 1 [35 Corollary 1.9] , with its dg enhancement [25 Theorem 3.8(b) and Section 3.8] , gives an enhanced equivalence
Ψ : MF g r ( x d ) → ∼ Perf ( B ) , E ⟼ RHom ( T , E ) . \Psi: \operatorname{MF}^{\mathrm{gr}}(x^d) \xrightarrow{\sim} \operatorname{Perf}(B), \qquad E \longmapsto\operatorname{RHom}(T,E). Ψ : MF gr ( x d ) ∼ Perf ( B ) , E ⟼ RHom ( T , E ) . The same theorem identifies Perf ( B ) \operatorname{Perf}(B) Perf ( B ) B B B V i = RHom ( E i , E ) V_i=\operatorname{RHom}(E_i,E) V i = RHom ( E i , E ) d = 5 d=5 d = 5
E 1 → f 12 E 2 → f 23 E 3 → f 34 E 4 E_1 \xrightarrow{f_{12}} E_2 \xrightarrow{f_{23}} E_3 \xrightarrow{f_{34}} E_4 E 1 f 12 E 2 f 23 E 3 f 34 E 4 V 1 ← f 12 ∗ V 2 ← f 23 ∗ V 3 ← f 34 ∗ V 4 . V_1 \xleftarrow{f_{12}^{*}} V_2 \xleftarrow{f_{23}^{*}} V_3 \xleftarrow{f_{34}^{*}} V_4. V 1 f 12 ∗ V 2 f 23 ∗ V 3 f 34 ∗ V 4 . Here f i , i + 1 ∗ f_{i,i+1}^{*} f i , i + 1 ∗ f i , i + 1 f_{i,i+1} f i , i + 1 i i i d − i d-i d − i (10) for the linear orientation.
Takahashi proves the triangulated equivalence (10) in [39 Theorem 3.1] . His trace pairing identifies F = τ − 1 [ 1 ] F=\tau^{-1}[1] F = τ − 1 [ 1 ] S M F S_{\mathrm{MF}} S MF [39 Lemma 3.5 and Remark 3.6] . On Perf ( B ) \operatorname{Perf}(B) Perf ( B ) [26 Section 5.5]
N ⟼ N ⊗ B L D ( B ) , D ( B ) = Hom C ( B , C ) . N \longmapsto N\otimes_B^{\mathbb{L}}D(B), \qquad D(B)=\operatorname{Hom}_{\mathbb{C}}(B,\mathbb{C}). N ⟼ N ⊗ B L D ( B ) , D ( B ) = Hom C ( B , C ) . Under Ψ \Psi Ψ F F F N ↦ N ⊗ B L M N\mapsto N\otimes_B^{\mathbb{L}}M N ↦ N ⊗ B L M M = RHom ( T , F T ) M=\operatorname{RHom}(T,FT) M = RHom ( T , F T ) B B B F F F T T T M M M D ( B ) D(B) D ( B ) T [ − n ] T[-n] T [ − n ] T T T
H n ( M ) = Hom ( T , F T [ n ] ) ≃ Hom ( T [ − n ] , F T ) ≃ Hom ( T , T [ − n ] ) ∗ . H^n(M)=\operatorname{Hom}(T,FT[n])\simeq\operatorname{Hom}(T[-n],FT)\simeq\operatorname{Hom}(T,T[-n])^{*}. H n ( M ) = Hom ( T , F T [ n ]) ≃ Hom ( T [ − n ] , F T ) ≃ Hom ( T , T [ − n ] ) ∗ . By (12) , this vanishes for n ≠ 0 n\ne0 n = 0 H 0 ( M ) ≃ End ( T ) ∗ = D ( B ) H^0(M)\simeq\operatorname{End}(T)^{*}=D(B) H 0 ( M ) ≃ End ( T ) ∗ = D ( B ) B B B M ≃ D ( B ) M\simeq D(B) M ≃ D ( B )
To pass to arbitrary orientations of the quiver, we use the derived Bernstein–Gelfand–Ponomarev reflection functors. Let i i i Q Q Q i i i Q ′ Q' Q ′ V V V Q Q Q V i V_i V i Cone ( V i → ⨁ i → j V j ) \operatorname{Cone}(V_i\to\bigoplus_{i\to j}V_j) Cone ( V i → ⨁ i → j V j ) V j V_j V j Q Q Q a a a b b b u = ( a , b ) u=(a,b) u = ( a , b ) C = Cone ( u ) C=\operatorname{Cone}(u) C = Cone ( u )
V i − 1 ← a V i → b V i + 1 → G V i − 1 → j − C ← j + V i + 1 V_{i-1}\xleftarrow{a}V_i\xrightarrow{b}V_{i+1}
\;\xrightarrow{G}\;
V_{i-1}\xrightarrow{j^{-}}C\xleftarrow{j^{+}}V_{i+1} V i − 1 a V i b V i + 1 G V i − 1 j − C j + V i + 1 Both new arrows come from the same distinguished triangle:
V i → u V i − 1 ⊕ V i + 1 → ( j − , j + ) C ⟶ V i [ 1 ] (13) V_i\xrightarrow{u}V_{i-1}\oplus V_{i+1}\xrightarrow{(j^{-},j^{+})}C\longrightarrow V_i[1] \tag*{(13)} V i u V i − 1 ⊕ V i + 1 ( j − , j + ) C ⟶ V i [ 1 ] (13) At an endpoint there is only one neighboring term. In Q ′ Q' Q ′ i i i W W W Q ′ Q' Q ′ i i i
( G − 1 W ) i ≃ Cone ( ⨁ j → i W j ⟶ W i ) [ − 1 ] . (14) (G^{-1}W)_i\simeq\operatorname{Cone}\left(\bigoplus_{j\to i}W_j\longrightarrow W_i\right)[-1]. \tag*{(14)} ( G − 1 W ) i ≃ Cone ( j → i ⨁ W j ⟶ W i ) [ − 1 ] . (14) Rotating the triangle (13) shows that this recovers V i V_i V i
G : D b ( rep C Q ) → ∼ D b ( rep C Q ′ ) . G:\mathrm{D}^{b}(\operatorname{rep}_{\mathbb{C}}Q)\xrightarrow{\sim}\mathrm{D}^{b}(\operatorname{rep}_{\mathbb{C}}Q'). G : D b ( rep C Q ) ∼ D b ( rep C Q ′ ) . For the classical construction and its realization as a derived tensor product with a complex of bimodules, see [26 Section 9.1] . This realization also gives G G G X , Y ∈ D b ( rep C Q ) X,Y\in\mathrm{D}^{b}(\operatorname{rep}_{\mathbb{C}}Q) X , Y ∈ D b ( rep C Q )
RHom Q ′ ( G X , G Y ) ∨ ≃ RHom Q ( X , Y ) ∨ ≃ RHom Q ( Y , S Q X ) ≃ RHom Q ′ ( G Y , G S Q X ) . \operatorname{RHom}_{Q'}(GX,GY)^{\vee}\simeq\operatorname{RHom}_{Q}(X,Y)^{\vee}\simeq\operatorname{RHom}_{Q}(Y,S_QX)\simeq\operatorname{RHom}_{Q'}(GY,GS_QX). RHom Q ′ ( GX , G Y ) ∨ ≃ RHom Q ( X , Y ) ∨ ≃ RHom Q ( Y , S Q X ) ≃ RHom Q ′ ( G Y , G S Q X ) . Here ∨ \vee ∨ G G G G S Q X GS_QX G S Q X S Q ′ G X S_{Q'}GX S Q ′ GX G S Q ≃ S Q ′ G GS_Q\simeq S_{Q'}G G S Q ≃ S Q ′ G
Every orientation of a path can be made linear: move the leftmost left-pointing arrow to the left endpoint by sink reflections, reverse it by reflecting at that endpoint, and repeat. For example, consider a path on four vertices. In the following diagrams, a label i i i i i i
( 1 → 2 ← 3 ← 4 ) → 2 ( 1 ← 2 → 3 ← 4 ) → 1 ( 1 → 2 → 3 ← 4 ) . (1 \to2 \leftarrow3 \leftarrow4) \xrightarrow{2} (1 \leftarrow2 \to3 \leftarrow4) \xrightarrow{1} (1 \to2 \to3 \leftarrow4). ( 1 → 2 ← 3 ← 4 ) 2 ( 1 ← 2 → 3 ← 4 ) 1 ( 1 → 2 → 3 ← 4 ) . Repeat with the remaining left-pointing arrow:
( 1 → 2 → 3 ← 4 ) → 3 ( 1 → 2 ← 3 → 4 ) → 2 ( 1 ← 2 → 3 → 4 ) → 1 ( 1 → 2 → 3 → 4 ) . (1 \to2 \to3 \leftarrow4) \xrightarrow{3} (1 \to2 \leftarrow3 \to4) \xrightarrow{2} (1 \leftarrow2 \to3 \to4) \xrightarrow{1} (1 \to2 \to3 \to4). ( 1 → 2 → 3 ← 4 ) 3 ( 1 → 2 ← 3 → 4 ) 2 ( 1 ← 2 → 3 → 4 ) 1 ( 1 → 2 → 3 → 4 ) . Each pass removes one left-pointing arrow, so the procedure terminates. Consequently, τ \tau τ S Q − 1 [ 1 ] S_Q^{-1}[1] S Q − 1 [ 1 ] τ d ≃ [ 2 ] \tau^d \simeq[2] τ d ≃ [ 2 ] (11) . □
Definition 3.4. Fix d ≥ 2 d \ge2 d ≥ 2 Q d Q_d Q d A d − 1 A_{d-1} A d − 1 1 , … , d − 1 1,\ldots,d-1 1 , … , d − 1 Figure 1 . For d = 2 d=2 d = 2 Q d o p Q_d^{\mathrm{op}} Q d op
Q d = D b ( rep C Q d ) , Q ^ d = D b ( rep C ( Q d □ Q d o p ) ) . \mathcal{Q}_d = \mathrm{D}^{b}(\operatorname{rep}_{\mathbb{C}} Q_d), \qquad\widehat{\mathcal{Q}}_d = \mathrm{D}^{b}\left(\operatorname{rep}_{\mathbb{C}}(Q_d \mathbin{\square} Q_d^{\mathrm{op}})\right). Q d = D b ( rep C Q d ) , Q d = D b ( rep C ( Q d □ Q d op ) ) . In Q d Q_d Q d x i x_i x i i i i i i i y i y_i y i i i i Q d o p Q_d^{\mathrm{op}} Q d op x i → y j x_i \to y_j x i → y j ∣ i − j ∣ = 1 |i-j|=1 ∣ i − j ∣ = 1 d = 2 d=2 d = 2 x 1 x_1 x 1 y 1 y_1 y 1
Figure 1. Alternating quiver Q d Q_d Q d
Let B = C ⊕ ( d − 1 ) \mathcal{B}=\mathbb{C}^{\oplus(d-1)} B = C ⊕ ( d − 1 ) adjacency functor F : B → B F:\mathcal{B}\to\mathcal{B} F : B → B
( F B ) i = B i − 1 ⊕ B i + 1 , where B 0 = B d = 0. (15) (FB)_i=B_{i-1}\oplus B_{i+1}, \qquad\text{where } B_0=B_d=0. \tag*{(15)} ( F B ) i = B i − 1 ⊕ B i + 1 , where B 0 = B d = 0. (15) The functor F F F Hom B ( F B , B ′ ) ≃ Hom B ( B , F B ′ ) \operatorname{Hom}_{\mathcal{B}}(F B,B')\simeq\operatorname{Hom}_{\mathcal{B}}(B,F B') Hom B ( F B , B ′ ) ≃ Hom B ( B , F B ′ )
For a quiver diagram, write X i X_i X i x i x_i x i Y i Y_i Y i y i y_i y i X = ( X i ) X=(X_i) X = ( X i ) Y = ( Y i ) Y=(Y_i) Y = ( Y i ) B \mathcal{B} B x i x_i x i y i − 1 y_{i-1} y i − 1 y i + 1 y_{i+1} y i + 1 m i : X i → ( F Y ) i m_i:X_i\to(FY)_i m i : X i → ( F Y ) i
E = ( X → m F Y ) , m i : X i → Y i − 1 ⊕ Y i + 1 . E=(X \xrightarrow{m} FY), \qquad m_i:X_i\to Y_{i-1}\oplus Y_{i+1}. E = ( X m F Y ) , m i : X i → Y i − 1 ⊕ Y i + 1 . Figure 2 illustrates this description for d = 5 d=5 d = 5 m m m
Figure 2. A quiver diagram for d = 5 d=5 d = 5
Definition 3.5. For a quiver diagram E = ( X → m F Y ) E=(X\xrightarrow{m}FY) E = ( X m F Y )
C i = Cone ( X i → m i Y i − 1 ⊕ Y i + 1 ) , C = ( C i ) = Cone ( m ) . C_i=\operatorname{Cone}\left(X_i\xrightarrow{m_i}Y_{i-1}\oplus Y_{i+1}\right), \qquad C=(C_i)=\operatorname{Cone}(m). C i = Cone ( X i m i Y i − 1 ⊕ Y i + 1 ) , C = ( C i ) = Cone ( m ) . The reflection functor R R R C ⊗ Q ^ d \mathcal{C}\otimes\widehat{\mathcal{Q}}_d C ⊗ Q d x i x_i x i
the old target y i y_i y i x i x_i x i C i C_i C i y i y_i y i Y Y Y C C C
R E = ( Y → m ′ F C ) . RE=(Y\xrightarrow{m'}FC). R E = ( Y m ′ F C ) . For ∣ i − j ∣ = 1 |i-j|=1 ∣ i − j ∣ = 1 x i → y j x_i\to y_j x i → y j Y i → Y j − 1 ⊕ Y j + 1 → C j Y_i\to Y_{j-1}\oplus Y_{j+1}\to C_j Y i → Y j − 1 ⊕ Y j + 1 → C j R R R d ≥ 3 d\ge3 d ≥ 3
The path-reversal functor Π \Pi Π ( J B ) i = B d − i (JB)_i=B_{d-i} ( J B ) i = B d − i Π E = ( J X → J m J Y ≃ F J Y ) \Pi E=(JX\xrightarrow{Jm}JY\simeq FJY) Π E = ( J X J m J Y ≃ F J Y ) J F Y ≃ F J Y JFY\simeq FJY J F Y ≃ F J Y Π E \Pi E Π E X d − i X_{d-i} X d − i x i x_i x i Y d − i Y_{d-i} Y d − i y i y_i y i x i → y j x_i\to y_j x i → y j X d − i → Y d − j X_{d-i}\to Y_{d-j} X d − i → Y d − j Π \Pi Π d d d d d d
For d = 2 d=2 d = 2 R ( X , Y ) = ( Y , X [ 1 ] ) R(X,Y)=(Y,X[1]) R ( X , Y ) = ( Y , X [ 1 ]) Π = id \Pi=\operatorname{id} Π = id C = Perf ( C ) \mathcal{C}=\operatorname{Perf}(\mathbb{C}) C = Perf ( C ) Q ^ d \widehat{\mathcal{Q}}_d Q d
Figure 3 illustrates the two functors for d = 5 d=5 d = 5
Figure 3. Quivers for d = 5 d=5 d = 5 R R R Π \Pi Π
Lemma 3.6. The functor R R R Π \Pi Π R R R
R 2 ≃ S Q ^ d − 1 [ 1 ] , R d ≃ Π [ 1 ] , R^2\simeq S_{\widehat{\mathcal{Q}}_d}^{-1}[1],\qquad R^d\simeq\Pi[1], R 2 ≃ S Q d − 1 [ 1 ] , R d ≃ Π [ 1 ] , on Q ^ d \widehat{\mathcal{Q}}_d Q d C ⊗ Q ^ d \mathcal{C}\otimes\widehat{\mathcal{Q}}_d C ⊗ Q d
Proof. Reversing the path twice fixes every vertex, and reversal commutes with the cone construction. Thus Π 2 ≃ id \Pi^2\simeq\operatorname{id} Π 2 ≃ id Π R ≃ R Π \Pi R\simeq R\Pi Π R ≃ R Π [17 Theorem 9.19 and Definition 9.20] ; see also [13 Sections 1–2] . The inverse sink reflection recovers the source object from its cone triangle, as explained in the proof of Proposition 3.3. Hence R R R
The Coxeter functor is the composition of one derived sink reflection at each vertex, ordered so that each vertex is a sink when it is reflected. For a path quiver Q Q Q S Q [ − 1 ] S_Q[-1] S Q [ − 1 ] [18 Definitions 5.8 and 5.13, Theorem 10.9] . Its inverse uses source reflections. For d ≥ 3 d\ge3 d ≥ 3 R R R R 2 ≃ S − 1 [ 1 ] R^2\simeq S^{-1}[1] R 2 ≃ S − 1 [ 1 ] Figure 4 illustrates these two rounds for $d=5.
Figure 4. R 2 ≃ S − 1 [ 1 ] R^2\simeq S^{-1}[1] R 2 ≃ S − 1 [ 1 ]
Proposition 3.3 also gives S d ≃ [ d − 2 ] S^d\simeq[d-2] S d ≃ [ d − 2 ]
R 2 d ≃ ( S − 1 [ 1 ] ) d ≃ S − d [ d ] ≃ [ 2 ] . R^{2d}\simeq\left(S^{-1}[1]\right)^d\simeq S^{-d}[d]\simeq[2]. R 2 d ≃ ( S − 1 [ 1 ] ) d ≃ S − d [ d ] ≃ [ 2 ] . Thus R d [ − 1 ] R^d[-1] R d [ − 1 ] Π \Pi Π d = 3 d=3 d = 3 A → B A\to B A → B C = Cone ( A → B ) C=\operatorname{Cone}(A\to B) C = Cone ( A → B )
( A → B ) ⟼ ( B → C ) ⟼ ( C → A [ 1 ] ) ⟼ ( A [ 1 ] → B [ 1 ] ) , (A\to B)\longmapsto(B\to C)\longmapsto(C\to A[1])\longmapsto(A[1]\to B[1]), ( A → B ) ⟼ ( B → C ) ⟼ ( C → A [ 1 ]) ⟼ ( A [ 1 ] → B [ 1 ]) , up to natural isomorphisms of diagrams. The final arrow lies in the opposite component, with the vertex indices reversed, as prescribed by Π \Pi Π [18 Theorem 4.14 and Proposition 5.4] . Their Corollary 4.11 identifies the shift [ 1 ] [1] [ 1 ] d − 1 d-1 d − 1 d d d x i x_i x i y i y_i y i X d − i [ 1 ] X_{d-i}[1] X d − i [ 1 ] Y d − i [ 1 ] Y_{d-i}[1] Y d − i [ 1 ] R d ≃ Π [ 1 ] R^d\simeq\Pi[1] R d ≃ Π [ 1 ] Figure 5 illustrates this identity for d = 5 d=5 d = 5 d = 2 d=2 d = 2 R ( X , Y ) = ( Y , X [ 1 ] ) R(X,Y)=(Y,X[1]) R ( X , Y ) = ( Y , X [ 1 ]) Π = id \Pi=\operatorname{id} Π = id
!Diagram showing E = ( X , Y ) E=(X,Y) E = ( X , Y ) R E = ( Y , C ) RE=(Y,C) R E = ( Y , C ) R 2 E = ( C , D ) R^2E=(C,D) R 2 E = ( C , D ) R 3 E = ( D , H ) R^3E=(D,H) R 3 E = ( D , H ) R 4 E = ( H , L ) R^4E=(H,L) R 4 E = ( H , L ) R 5 E ≃ Π E [ 1 ] R^5E\simeq\Pi E[1] R 5 E ≃ Π E [ 1 ] bbox=0.10,0.08,0.88,0.32
Figure 5. R d E ≃ Π E [ 1 ] R^dE\simeq\Pi E[1] R d E ≃ Π E [ 1 ]
These identities are realized by complexes of bimodules [18 Section 10] . Tensoring them with the identity bimodule of C \mathcal{C} C C \mathcal{C} C
Theorem 3.7. Let C \mathcal{C} C σ = ( Z , P ) \sigma=(Z,\mathcal{P}) σ = ( Z , P ) v : K 0 ( C ) → Λ v:K_0(\mathcal{C})\to\Lambda v : K 0 ( C ) → Λ
The category C ⊗ Q ^ d \mathcal{C}\otimes\widehat{\mathcal{Q}}_d C ⊗ Q d Σ = ( Z , P Σ ) \Sigma=(\mathcal{Z},\mathcal{P}_{\Sigma}) Σ = ( Z , P Σ ) Λ ⊕ 2 ( d − 1 ) \Lambda^{\oplus2(d-1)} Λ ⊕ 2 ( d − 1 ) ( Φ , 1 / d ) (\Phi,1/d) ( Φ , 1/ d ) Φ \Phi Φ
(i) the reflection functor R R R
(ii) the vertexwise action of every exact enhanced autoequivalence Ψ : C → C \Psi:\mathcal{C}\to\mathcal{C} Ψ : C → C σ \sigma σ ( Ψ , 1 / d ) (\Psi,1/d) ( Ψ , 1/ d )
Moreover, for every quiver diagram E E E
∑ w m σ ( E w ) ≤ B d m Σ ( E ) , B d = ( 1 + 2 cos ( π / d ) ) d sin ( π / ( 2 d ) ) − 4 , \sum_w m_{\sigma}(E_w)\le B_d m_{\Sigma}(E),\qquad B_d=(1+2\cos(\pi/d))^d\sin(\pi/(2d))^{-4}, w ∑ m σ ( E w ) ≤ B d m Σ ( E ) , B d = ( 1 + 2 cos ( π / d ) ) d sin ( π / ( 2 d ) ) − 4 , where the sum runs over all vertices. If σ \sigma σ
The proof occupies the rest of the section. Fix σ = ( Z , P ) \sigma=(Z,\mathcal{P}) σ = ( Z , P ) Z \mathcal{Z} Z C ⊗ Q ^ d \mathcal{C}\otimes\widehat{\mathcal{Q}}_d C ⊗ Q d U U U R U ⊂ U RU\subset U R U ⊂ U R j U R^jU R j U j ∈ Z j\in\mathbb{Z} j ∈ Z R j U ∩ ( R j + 1 U ) ⊥ R^jU\cap(R^{j+1}U)^\perp R j U ∩ ( R j + 1 U ) ⊥ π / d \pi/d π / d ( Ψ , 1 / d ) (\Psi,1/d) ( Ψ , 1/ d ) Ψ : C → C \Psi:\mathcal{C}\to\mathcal{C} Ψ : C → C σ \sigma σ ( Ψ , 1 / d ) (\Psi,1/d) ( Ψ , 1/ d )
The tilted heart
Let A = P ( ( 0 , 1 ] ) \mathcal{A}=\mathcal{P}((0,1]) A = P (( 0 , 1 ]) C \mathcal{C} C B = C ⊕ ( d − 1 ) \mathcal{B}=\mathcal{C}^{\oplus(d-1)} B = C ⊕ ( d − 1 ) P B \mathcal{P}_{\mathcal{B}} P B ϕ \phi ϕ P ( ϕ ) \mathcal{P}(\phi) P ( ϕ ) A B = A ⊕ ( d − 1 ) \mathcal{A}_{\mathcal{B}}=\mathcal{A}^{\oplus(d-1)} A B = A ⊕ ( d − 1 ) F F F B j B_j B j P ( ϕ ) \mathcal{P}(\phi) P ( ϕ ) ( F B ) j = B j − 1 ⊕ B j + 1 (FB)_j=B_{j-1}\oplus B_{j+1} ( F B ) j = B j − 1 ⊕ B j + 1 P ( ϕ ) \mathcal{P}(\phi) P ( ϕ ) B 0 = B d = 0 B_0=B_d=0 B 0 = B d = 0
r j = sin ( j π / d ) sin ( π / d ) , z ( B ) = ∑ j = 1 d − 1 r j Z ( B j ) . r_j=\frac{\sin(j\pi/d)}{\sin(\pi/d)},\qquad z(B)=\sum_{j=1}^{d-1}r_jZ(B_j). r j = sin ( π / d ) sin ( j π / d ) , z ( B ) = j = 1 ∑ d − 1 r j Z ( B j ) . The identities
r j − 1 + r j + 1 = λ r j , λ = 2 cos ( π / d ) , with r 0 = r d = 0 , r_{j-1}+r_{j+1}=\lambda r_j,\qquad\lambda=2\cos(\pi/d),\qquad\text{with }r_0=r_d=0, r j − 1 + r j + 1 = λ r j , λ = 2 cos ( π / d ) , with r 0 = r d = 0 , give z ( F B ) = λ z ( B ) z(FB)=\lambda z(B) z ( F B ) = λ z ( B ) r j = r d − j r_j=r_{d-j} r j = r d − j
By Lemma 3.2, H = Rep ( Q d □ Q d o p , A ) \mathcal{H}=\operatorname{Rep}(Q_d\mathbin{\Box} Q_d^{\mathrm{op}},\mathcal{A}) H = Rep ( Q d □ Q d op , A ) C ⊗ Q ^ d \mathcal{C}\otimes\widehat{\mathcal{Q}}_d C ⊗ Q d 2 ( d − 1 ) 2(d-1) 2 ( d − 1 )
t = 1 − 1 d , q = e − π i t = − e π i / d . t=1-\frac{1}{d},\qquad q=e^{-\pi i t}=-e^{\pi i/d}. t = 1 − d 1 , q = e − π i t = − e π i / d . Then q 2 + λ q + 1 = 0 q^2+\lambda q+1=0 q 2 + λ q + 1 = 0 Z \mathcal{Z} Z p p p
Z ( X , Y ) = z ( X ) + q z ( Y ) , p ( X , Y ) = Im z ( X ) + Im z ( Y ) . \mathcal{Z}(X,Y)=z(X)+qz(Y),\qquad p(X,Y)=\operatorname{Im}z(X)+\operatorname{Im}z(Y). Z ( X , Y ) = z ( X ) + q z ( Y ) , p ( X , Y ) = Im z ( X ) + Im z ( Y ) . The torsion pair will be determined by Im Z \operatorname{Im}\mathcal{Z} Im Z p p p q q q Y Y Y π t \pi t π t Z \mathcal{Z} Z H \mathcal{H} H [33 Section 3] , we call a nonzero object of H \mathcal{H} H p = 0 p=0 p = 0 Im Z > 0 \operatorname{Im}\mathcal{Z}>0 Im Z > 0 p > 0 p>0 p > 0 Im Z ≤ 0 \operatorname{Im}\mathcal{Z}\leq0 Im Z ≤ 0 p ≥ 0 p\geq0 p ≥ 0 H \mathcal{H} H p = 0 p=0 p = 0
z ( X ) = − a , z ( Y ) = − b , Z ( X , Y ) = − a + b e π i / d , a , b ≥ 0 , z(X)=-a,\qquad z(Y)=-b,\qquad\mathcal{Z}(X,Y)=-a+be^{\pi i/d},\qquad a,b\geq0, z ( X ) = − a , z ( Y ) = − b , Z ( X , Y ) = − a + b e π i / d , a , b ≥ 0 , with a , b a,b a , b Im Z ≥ 0 \operatorname{Im}\mathcal{Z}\geq0 Im Z ≥ 0 Figure 6 shows both cases separately.
Figure 6. Charges of high/low objects. Left: p > 0 p>0 p > 0 p = 0 p=0 p = 0
Proposition 3.8. There is a torsion pair ( T , F ) (\mathcal{T},\mathcal{F}) ( T , F ) H \mathcal{H} H
T = { E : every nonzero quotient of E in H is high } , \mathcal{T}=\{E:\text{ every nonzero quotient of }E\text{ in }\mathcal{H}\text{ is high}\}, T = { E : every nonzero quotient of E in H is high } , F = { E : every nonzero subobject of E in H is low } . \mathcal{F}=\{E:\text{ every nonzero subobject of }E\text{ in }\mathcal{H}\text{ is low}\}. F = { E : every nonzero subobject of E in H is low } . Neither rationality of the charge Z \mathcal{Z} Z A \mathcal{A} A
Proof. For a tuple Y ∈ A B Y \in\mathcal{A}_{B} Y ∈ A B Y > t ⊆ Y Y_{>t} \subseteq Y Y > t ⊆ Y t t t Y ≤ t = Y / Y > t Y_{\leq t}=Y/Y_{>t} Y ≤ t = Y / Y > t E = ( X , Y , m ) ∈ F E=(X,Y,m)\in\mathcal{F} E = ( X , Y , m ) ∈ F
Y ∈ P B ( ( 0 , t ] ) , m : X ↪ F Y . (16) Y\in\mathcal{P}_{B}((0,t]),\qquad m:X\hookrightarrow FY. \tag*{(16)} Y ∈ P B (( 0 , t ]) , m : X ↪ F Y . (16) Indeed, ( 0 , Y > t ) (0,Y_{>t}) ( 0 , Y > t ) ( ker m , 0 ) (\ker m,0) ( ker m , 0 ) E E E F F F P B ( ( 0 , t ] ) \mathcal{P}_{B}((0,t]) P B (( 0 , t ]) (16) puts all vertex phases in ( 0 , t ] (0,t] ( 0 , t ] (16) therefore has p > 0 p>0 p > 0
Given an object E = ( X , Y , m ) E=(X,Y,m) E = ( X , Y , m ) H \mathcal{H} H E ‾ \overline{E} E ( 0 , Y > t ) (0,Y_{>t}) ( 0 , Y > t ) ( ker ( X → F ( Y ≤ t ) ) , 0 ) (\ker(X\to F(Y_{\leq t})),0) ( ker ( X → F ( Y ≤ t )) , 0 ) E ‾ \overline{E} E (16) . The first subobject has Im Z > 0 \operatorname{Im} Z>0 Im Z > 0 q q q ( 0 , 1 / d ] (0,1/d] ( 0 , 1/ d ] Im Z = p ≥ 0 \operatorname{Im} Z=p\geq0 Im Z = p ≥ 0
Im Z ( E ‾ ) ≤ Im Z ( E ) , \operatorname{Im} Z(\overline{E})\leq\operatorname{Im} Z(E), Im Z ( E ) ≤ Im Z ( E ) , with equality only if both removed subobjects have p = 0 p=0 p = 0
Im Z ( E ‾ ) = Im Z ( E ) ⟹ p ( E ‾ ) = p ( E ) . \operatorname{Im} Z(\overline{E})=\operatorname{Im} Z(E)\quad\Longrightarrow\quad p(\overline{E})=p(E). Im Z ( E ) = Im Z ( E ) ⟹ p ( E ) = p ( E ) . *Finitely many possible charges. Fix 0 ≠ E ∈ H 0\neq E\in\mathcal{H} 0 = E ∈ H l > 0 l>0 l > 0 Q Q Q E E E (16) has vertex phases in [ l , t ] [l,t] [ l , t ] l > t l>t l > t
ϵ E = min ϕ ∈ [ l , t ] sin ( π ϕ ) > 0. \epsilon_E=\min_{\phi\in[l,t]}\sin(\pi\phi)>0. ϵ E = ϕ ∈ [ l , t ] min sin ( π ϕ ) > 0. For a support constant C σ C_{\sigma} C σ w w w
∥ v ( Q w ) ∥ ≤ C σ m σ ( Q w ) ≤ C σ ϵ E − 1 Im Z ( Q w ) ≤ C σ ϵ E − 1 Im Z ( E w ) . \lVert v(Q_w)\rVert\leq C_{\sigma}m_{\sigma}(Q_w)\leq C_{\sigma}\epsilon_E^{-1}\operatorname{Im}Z(Q_w)\leq C_{\sigma}\epsilon_E^{-1}\operatorname{Im}Z(E_w). ∥ v ( Q w )∥ ≤ C σ m σ ( Q w ) ≤ C σ ϵ E − 1 Im Z ( Q w ) ≤ C σ ϵ E − 1 Im Z ( E w ) . The last inequality follows from the nonnegative imaginary charge of the kernel of E w → Q w E_w\to Q_w E w → Q w Λ ⊕ 2 ( d − 1 ) \Lambda^{\oplus2(d-1)} Λ ⊕ 2 ( d − 1 ) Im Z ( Q ) \operatorname{Im}Z(Q) Im Z ( Q ) p ( Q ) p(Q) p ( Q )
*The torsion sequence. Among the quotients E ↠ Q E\twoheadrightarrow Q E ↠ Q (16) , including zero, choose one minimizing Im Z ( Q ) \operatorname{Im}Z(Q) Im Z ( Q ) p ( Q ) p(Q) p ( Q ) K = ker ( E → Q ) K=\ker(E\to Q) K = ker ( E → Q )
First, Q ∈ F Q\in\mathcal{F} Q ∈ F A ⊂ Q A\subset Q A ⊂ Q p ( A ) > 0 p(A)>0 p ( A ) > 0 Im Z ( A ) > 0 \operatorname{Im}Z(A)>0 Im Z ( A ) > 0 Q / A ‾ \overline{Q/A} Q / A E E E (16) and
Im Z ( Q / A ‾ ) ≤ Im Z ( Q / A ) = Im Z ( Q ) − Im Z ( A ) < Im Z ( Q ) , \operatorname{Im}Z(\overline{Q/A})\leq\operatorname{Im}Z(Q/A)=\operatorname{Im}Z(Q)-\operatorname{Im}Z(A)<\operatorname{Im}Z(Q), Im Z ( Q / A ) ≤ Im Z ( Q / A ) = Im Z ( Q ) − Im Z ( A ) < Im Z ( Q ) , contradicting minimality. Thus every such A A A
Next, suppose that K ∉ T K\notin\mathcal{T} K ∈ / T K ↠ L K\twoheadrightarrow L K ↠ L K ′ = ker ( K → L ) K'=\ker(K\to L) K ′ = ker ( K → L ) P = E / K ′ P=E/K' P = E / K ′
0 ⟶ L ⟶ P ⟶ Q ⟶ 0. 0\longrightarrow L\longrightarrow P\longrightarrow Q\longrightarrow0. 0 ⟶ L ⟶ P ⟶ Q ⟶ 0. The quotient P ‾ \overline{P} P E E E (16) , so minimality gives
Im Z ( Q ) ≤ Im Z ( P ‾ ) ≤ Im Z ( P ) = Im Z ( Q ) + Im Z ( L ) ≤ Im Z ( Q ) . \operatorname{Im}Z(Q)\leq\operatorname{Im}Z(\overline{P})\leq\operatorname{Im}Z(P)=\operatorname{Im}Z(Q)+\operatorname{Im}Z(L)\leq\operatorname{Im}Z(Q). Im Z ( Q ) ≤ Im Z ( P ) ≤ Im Z ( P ) = Im Z ( Q ) + Im Z ( L ) ≤ Im Z ( Q ) . Equality holds throughout. The construction of P ‾ \overline{P} P p p p
p ( P ‾ ) = p ( P ) = p ( Q ) + p ( L ) > p ( Q ) , p(\overline{P})=p(P)=p(Q)+p(L)>p(Q), p ( P ) = p ( P ) = p ( Q ) + p ( L ) > p ( Q ) , contrary to the choice of Q Q Q K ∈ T K\in\mathcal{T} K ∈ T
Finally, a nonzero image of a morphism from T \mathcal{T} T F \mathcal{F} F Hom ( T , F ) = 0 \operatorname{Hom}(\mathcal{T},\mathcal{F})=0 Hom ( T , F ) = 0 0 → K → E → Q → 0 0\to K\to E\to Q\to0 0 → K → E → Q → 0
We now tilt the heart H \mathcal{H} H ( T , F ) (\mathcal{T},\mathcal{F}) ( T , F ) § 2 . The tilted heart
A ′ = ⟨ F [ 1 ] , T ⟩ \mathcal{A}'=\langle\mathcal{F}[1],\mathcal{T}\rangle A ′ = ⟨ F [ 1 ] , T ⟩ consists precisely of objects E E E
F E [ 1 ] ⟶ E ⟶ T E ⟶ F E [ 2 ] , F E ∈ F , T E ∈ T . F_E[1]\longrightarrow E\longrightarrow T_E\longrightarrow F_E[2],\qquad F_E\in\mathcal{F},\quad T_E\in\mathcal{T}. F E [ 1 ] ⟶ E ⟶ T E ⟶ F E [ 2 ] , F E ∈ F , T E ∈ T . Thus the tilt keeps objects of T \mathcal{T} T F \mathcal{F} F − 1 -1 − 1 Z ( F E [ 1 ] ) = − Z ( F E ) Z(F_E[1])=-Z(F_E) Z ( F E [ 1 ]) = − Z ( F E ) V V V H \mathcal{H} H U = { E ∈ V : H H 0 ( E ) ∈ T } U=\{E\in V:H_{\mathcal{H}}^0(E)\in\mathcal{T}\} U = { E ∈ V : H H 0 ( E ) ∈ T } V [ 1 ] ⊂ U ⊂ V V[1]\subset U\subset V V [ 1 ] ⊂ U ⊂ V
A mass estimate
The central charge Z ( E ) Z(E) Z ( E ) E ∈ F E \in\mathcal{F} E ∈ F E = ( X , Y , m ) E=(X,Y,m) E = ( X , Y , m )
M ( E ) = ∑ j = 1 d − 1 r j ( m σ ( X j ) + m σ ( Y j ) ) . M(E)=\sum_{j=1}^{d-1}r_j\bigl(m_\sigma(X_j)+m_\sigma(Y_j)\bigr). M ( E ) = j = 1 ∑ d − 1 r j ( m σ ( X j ) + m σ ( Y j ) ) . Evaluation at a vertex commutes with shifts and preserves distinguished triangles. Shifting an object of C \mathcal{C} C M ( E [ n ] ) = M ( E ) M(E[n])=M(E) M ( E [ n ]) = M ( E ) n n n
E 1 ⟶ E 2 ⟶ E 3 ⟶ E 1 [ 1 ] E_1\longrightarrow E_2\longrightarrow E_3\longrightarrow E_1[1] E 1 ⟶ E 2 ⟶ E 3 ⟶ E 1 [ 1 ] gives a distinguished triangle at each vertex. Applying the mass triangle inequality (5) at each vertex and summing over all source and target vertices with the positive weights r j r_j r j
M ( E 2 ) ≤ M ( E 1 ) + M ( E 3 ) . (17) M(E_2)\leq M(E_1)+M(E_3). \tag*{(17)} M ( E 2 ) ≤ M ( E 1 ) + M ( E 3 ) . (17) Lemma 3.9. Let c = cos ( π t / 2 ) = sin ( π / ( 2 d ) ) > 0 c=\cos(\pi t/2)=\sin(\pi/(2d))>0 c = cos ( π t /2 ) = sin ( π / ( 2 d )) > 0 E ∈ F E\in\mathcal{F} E ∈ F arg ∈ ( − π , π ] \arg\in(-\pi,\pi] arg ∈ ( − π , π ]
arg Z ( E ) / π ∈ ( − t , 0 ] , ∣ Z ( E ) ∣ ≥ c 3 M ( E ) . (18) \arg Z(E)/\pi\in(-t,0],\qquad|Z(E)|\geq c^3M(E). \tag*{(18)} arg Z ( E ) / π ∈ ( − t , 0 ] , ∣ Z ( E ) ∣ ≥ c 3 M ( E ) . (18) Proof. We apply the following elementary geometric observation three times. If vectors ζ ν ∈ C \zeta_\nu\in\mathbb{C} ζ ν ∈ C π t < π \pi t<\pi π t < π θ \theta θ ζ ν \zeta_\nu ζ ν c ∣ ζ ν ∣ c|\zeta_\nu| c ∣ ζ ν ∣ c = cos ( π t / 2 ) c=\cos(\pi t/2) c = cos ( π t /2 )
∣ ∑ ν ζ ν ∣ ≥ Re ( e − i θ ∑ ν ζ ν ) ≥ c ∑ ν ∣ ζ ν ∣ . (19) \left|\sum_\nu\zeta_\nu\right|\geq\operatorname{Re}\left(e^{-i\theta}\sum_\nu\zeta_\nu\right)\geq c\sum_\nu|\zeta_\nu|. \tag*{(19)} ν ∑ ζ ν ≥ Re ( e − i θ ν ∑ ζ ν ) ≥ c ν ∑ ∣ ζ ν ∣. (19) and the sum lies in the same sector. In particular, it is nonzero if at least one vector is nonzero. We organize the proof into three claims, illustrated in Figure 7 .
Figure 7. Three applications of the projection estimate.
Claim 1. There is a filtration in H \mathcal{H} H
0 = E ( 0 ) ⊂ E ( 1 ) ⊂ ⋯ ⊂ E ( s ) = E (20) 0=E^{(0)}\subset E^{(1)}\subset\cdots\subset E^{(s)}=E \tag*{(20)} 0 = E ( 0 ) ⊂ E ( 1 ) ⊂ ⋯ ⊂ E ( s ) = E (20) with nonzero factors E k = E ( k ) / E ( k − 1 ) E_k=E^{(k)}/E^{(k-1)} E k = E ( k ) / E ( k − 1 ) E k E_k E k ϕ k \phi_k ϕ k t ≥ ϕ 1 > ⋯ > ϕ s > 0 t\geq\phi_1>\cdots>\phi_s>0 t ≥ ϕ 1 > ⋯ > ϕ s > 0
M ( E ) = ∑ k M ( E k ) , (21) M(E)=\sum_k M(E_k), \tag*{(21)} M ( E ) = k ∑ M ( E k ) , (21) arg Z ( E k ) / π ∈ [ ϕ k − t , ϕ k ] ⊂ ( − t , 1 ) , ∣ Z ( E k ) ∣ ≥ c M ( E k ) . (22) \arg Z(E_k)/\pi\in[\phi_k-t,\phi_k]\subset(-t,1),\qquad|Z(E_k)|\geq cM(E_k). \tag*{(22)} arg Z ( E k ) / π ∈ [ ϕ k − t , ϕ k ] ⊂ ( − t , 1 ) , ∣ Z ( E k ) ∣ ≥ c M ( E k ) . (22) Proof of Claim 1. Since E ∈ F E\in\mathcal{F} E ∈ F Y j Y_j Y j ( 0 , t ] (0,t] ( 0 , t ] m : X → F Y m:X\to FY m : X → F Y X j X_j X j Y j − 1 ⊕ Y j + 1 Y_{j-1}\oplus Y_{j+1} Y j − 1 ⊕ Y j + 1 ( 0 , t ] (0,t] ( 0 , t ] P ( ( 0 , t ] ) \mathcal{P}((0,t]) P (( 0 , t ]) A \mathcal{A} A E E E ( 0 , t ] (0,t] ( 0 , t ] ϕ 1 > ⋯ > ϕ s \phi_1 > \cdots> \phi_s ϕ 1 > ⋯ > ϕ s E E E 0 < ϕ k ≤ t 0 < \phi_k \le t 0 < ϕ k ≤ t k k k k k k ϕ k \phi_k ϕ k [8 Section 3] , so every arrow map restricts to a map between the chosen subobjects. They therefore form a quiver subdiagram E ( k ) ⊂ E E^{(k)} \subset E E ( k ) ⊂ E (20) .
At each vertex, the quotient E k = E ( k ) / E ( k − 1 ) E_k = E^{(k)}/E^{(k-1)} E k = E ( k ) / E ( k − 1 ) ϕ k \phi_k ϕ k ϕ k \phi_k ϕ k r j r_j r j (21) .
Write Z ( E k ) = a k + b k \mathcal{Z}(E_k) = a_k + b_k Z ( E k ) = a k + b k a k a_k a k b k b_k b k r j r_j r j a k a_k a k ϕ k \phi_k ϕ k b k b_k b k ϕ k − t \phi_k - t ϕ k − t ∣ a k ∣ + ∣ b k ∣ = M ( E k ) |a_k| + |b_k| = M(E_k) ∣ a k ∣ + ∣ b k ∣ = M ( E k ) π t \pi t π t Z ( E k ) ≠ 0 \mathcal{Z}(E_k) \ne0 Z ( E k ) = 0 (22) .
Claim 2. The charges Z ( E k ) \mathcal{Z}(E_k) Z ( E k ) ( − π t , 0 ] (-\pi t, 0] ( − π t , 0 ] Z ( E ) \mathcal{Z}(E) Z ( E )
c ∑ k ∣ Z ( E k ) ∣ . c\sum_k |\mathcal{Z}(E_k)|. c k ∑ ∣ Z ( E k ) ∣. Proof of Claim 2. An individual factor E k E_k E k F \mathcal{F} F Z ( E k ) \mathcal{Z}(E_k) Z ( E k ) E ( j ) E^{(j)} E ( j ) (20) is a subobject of E ∈ F E \in\mathcal{F} E ∈ F
∑ k ≤ j Im Z ( E k ) = Im Z ( E ( j ) ) ≤ 0 ( 1 ≤ j ≤ s ) . (23) \sum_{k\le j} \operatorname{Im}\mathcal{Z}(E_k) = \operatorname{Im}\mathcal{Z}(E^{(j)}) \le0 \qquad(1 \le j \le s). \tag*{(23)} k ≤ j ∑ Im Z ( E k ) = Im Z ( E ( j ) ) ≤ 0 ( 1 ≤ j ≤ s ) . (23) Start with the ordered list Z ( E 1 ) , … , Z ( E s ) \mathcal{Z}(E_1),\ldots,\mathcal{Z}(E_s) Z ( E 1 ) , … , Z ( E s ) (23) . If their imaginary parts cancel, remove the two terms from the ordered list and record them as a pair. Otherwise, split the term whose imaginary part has larger magnitude into two positive real multiples, one with imaginary part opposite to that of the other chosen term. Remove this piece and the other chosen term from the ordered list as a pair, keeping the second piece in its original position. The partial-sum inequalities (23) remain valid. Each operation removes at least one entry. Repeating therefore gives finitely many pairs and leaves only unpaired vectors with nonpositive imaginary part.
Each recorded pair consists of positive real multiples u u u v v v Z ( E k ) \mathcal{Z}(E_k) Z ( E k ) Z ( E l ) \mathcal{Z}(E_l) Z ( E l ) k < l k < l k < l Im u = − Im v < 0 \operatorname{Im}u = -\operatorname{Im}v < 0 Im u = − Im v < 0
0 < arg Z ( E l ) / π − arg Z ( E k ) / π ≤ ϕ l − ( ϕ k − t ) < t . 0 < \arg\mathcal{Z}(E_l)/\pi- \arg\mathcal{Z}(E_k)/\pi\le\phi_l - (\phi_k - t) < t. 0 < arg Z ( E l ) / π − arg Z ( E k ) / π ≤ ϕ l − ( ϕ k − t ) < t . The last inequality uses ϕ l < ϕ k \phi_l < \phi_k ϕ l < ϕ k π t \pi t π t
u + v > 0 , ∣ u + v ∣ ≥ c ( ∣ u ∣ + ∣ v ∣ ) . u+v > 0, \qquad|u+v| \ge c(|u|+|v|). u + v > 0 , ∣ u + v ∣ ≥ c ( ∣ u ∣ + ∣ v ∣ ) . Replace each pair by its positive real sum. By Claim 1, the unpaired vectors have phases in ( − t , 0 ] (-t,0] ( − t , 0 ] c c c c ∑ k ∣ Z ( E k ) ∣ c\sum_k |\mathcal{Z}(E_k)| c ∑ k ∣ Z ( E k ) ∣ Z ( E ) \mathcal{Z}(E) Z ( E )
Claim 3. The total charge satisfies arg Z ( E ) / π ∈ ( − t , 0 ] \arg\mathcal{Z}(E)/\pi\in(-t,0] arg Z ( E ) / π ∈ ( − t , 0 ]
∣ Z ( E ) ∣ ≥ c 2 ∑ k ∣ Z ( E k ) ∣ . (24) |\mathcal{Z}(E)| \ge c^2\sum_k |\mathcal{Z}(E_k)|. \tag*{(24)} ∣ Z ( E ) ∣ ≥ c 2 k ∑ ∣ Z ( E k ) ∣. (24) Proof of Claim 3. Apply the projection estimate to the vectors supplied by Claim 2. They lie in a sector of angle π t \pi t π t c c c (24) and, in particular, Z ( E ) ≠ 0 \mathcal{Z}(E) \ne0 Z ( E ) = 0 − t -t − t
Combining Claims 1 and 3 now gives
∣ Z ( E ) ∣ ≥ c 2 ∑ k ∣ Z ( E k ) ∣ ≥ c 3 ∑ k M ( E k ) = c 3 M ( E ) . \lvert\mathcal{Z}(E)\rvert\ge c^{2}\sum_{k}\lvert\mathcal{Z}(E_{k})\rvert\ge c^{3}\sum_{k}M(E_{k})=c^{3}M(E). ∣ Z ( E )∣ ≥ c 2 k ∑ ∣ Z ( E k )∣ ≥ c 3 k ∑ M ( E k ) = c 3 M ( E ) . Together with the phase bound in Claim 3, this proves the lemma.
Lemma 3.10. Every nonzero object E E E A ′ \mathcal{A}' A ′
Z ( E ) ∈ { z ∈ C : Im z > 0 } ∪ R < 0 . \mathcal{Z}(E)\in\{z\in\mathbb{C}:\operatorname{Im}z>0\}\cup\mathbb{R}_{<0}. Z ( E ) ∈ { z ∈ C : Im z > 0 } ∪ R < 0 . Proof. We first check the two kinds of objects used to form the tilted heart. Every nonzero T ∈ T T\in\mathcal{T} T ∈ T Im Z ( T ) > 0 \operatorname{Im}\mathcal{Z}(T)>0 Im Z ( T ) > 0 p ( T ) = 0 p(T)=0 p ( T ) = 0 T = ( X , Y , m ) T=(X,Y,m) T = ( X , Y , m ) z ( X ) = − a z(X)=-a z ( X ) = − a z ( Y ) = − b z(Y)=-b z ( Y ) = − b Z ( T ) = − a + b e π i / d \mathcal{Z}(T)=-a+b e^{\pi i/d} Z ( T ) = − a + b e π i / d a , b ≥ 0 a,b\ge0 a , b ≥ 0 a + b > 0 a+b>0 a + b > 0 sin ( π / d ) > 0 \sin(\pi/d)>0 sin ( π / d ) > 0 b > 0 b>0 b > 0 b = 0 b=0 b = 0
For a nonzero F ∈ F F\in\mathcal{F} F ∈ F Z ( F ) ≠ 0 \mathcal{Z}(F)\ne0 Z ( F ) = 0 arg Z ( F ) / π ∈ ( − t , 0 ] \arg\mathcal{Z}(F)/\pi\in(-t,0] arg Z ( F ) / π ∈ ( − t , 0 ]
Z ( F [ 1 ] ) = − Z ( F ) , arg Z ( F [ 1 ] ) / π ∈ ( 1 − t , 1 ] . \mathcal{Z}(F[1])=-\mathcal{Z}(F),\qquad\arg\mathcal{Z}(F[1])/\pi\in(1-t,1]. Z ( F [ 1 ]) = − Z ( F ) , arg Z ( F [ 1 ]) / π ∈ ( 1 − t , 1 ] . Thus Z ( F [ 1 ] ) \mathcal{Z}(F[1]) Z ( F [ 1 ])
Finally, every nonzero E ∈ A ′ E\in\mathcal{A}' E ∈ A ′ A ′ \mathcal{A}' A ′
0 ⟶ F E [ 1 ] ⟶ E ⟶ T E ⟶ 0 , F E ∈ F , T E ∈ T . 0\longrightarrow F_{E}[1]\longrightarrow E\longrightarrow T_{E}\longrightarrow0,\qquad F_{E}\in\mathcal{F},\quad T_{E}\in\mathcal{T}. 0 ⟶ F E [ 1 ] ⟶ E ⟶ T E ⟶ 0 , F E ∈ F , T E ∈ T . Both factors have nonnegative imaginary charge. If Im Z ( E ) = 0 \operatorname{Im}\mathcal{Z}(E)=0 Im Z ( E ) = 0
Action of the reflection functor
Path reversal functor Π \Pi Π H \mathcal{H} H V \mathcal{V} V r j = r d − j r_{j}=r_{d-j} r j = r d − j z ( X ) z(X) z ( X ) z ( Y ) z(Y) z ( Y ) Z \mathcal{Z} Z p p p ( T , F ) (\mathcal{T},\mathcal{F}) ( T , F ) U \mathcal{U} U R d ≃ Π [ 1 ] R^{d}\simeq\Pi[1] R d ≃ Π [ 1 ]
R d U = Π U [ 1 ] = U [ 1 ] . (25) R^{d}\mathcal{U}=\Pi\mathcal{U}[1]=\mathcal{U}[1]. \tag*{(25)} R d U = Π U [ 1 ] = U [ 1 ] . (25) For E = ( X , Y , m ) E=(X,Y,m) E = ( X , Y , m ) R E = ( Y , Cone ( m ) ) RE=(Y,\operatorname{Cone}(m)) R E = ( Y , Cone ( m )) z z z z ( F Y ) = λ z ( Y ) z(FY)=\lambda z(Y) z ( F Y ) = λ z ( Y )
z ( Cone ( m ) ) = z ( F Y ) − z ( X ) = λ z ( Y ) − z ( X ) . z(\operatorname{Cone}(m))=z(FY)-z(X)=\lambda z(Y)-z(X). z ( Cone ( m )) = z ( F Y ) − z ( X ) = λ z ( Y ) − z ( X ) . Since 1 + λ q = − q 2 1+\lambda q=-q^{2} 1 + λ q = − q 2 − q = e π i / d -q=e^{\pi i/d} − q = e π i / d
Z ( R E ) = z ( Y ) + q ( λ z ( Y ) − z ( X ) ) = − q ( z ( X ) + q z ( Y ) ) = e π i / d Z ( E ) . (26) \mathcal{Z}(RE)=z(Y)+q\bigl(\lambda z(Y)-z(X)\bigr)=-q\bigl(z(X)+qz(Y)\bigr)=e^{\pi i/d}\mathcal{Z}(E). \tag*{(26)} Z ( R E ) = z ( Y ) + q ( λ z ( Y ) − z ( X ) ) = − q ( z ( X ) + q z ( Y ) ) = e π i / d Z ( E ) . (26) Lemma 3.11. The reflection functor sends the tilted aisle into itself: R U ⊂ U R\mathcal{U}\subset\mathcal{U} R U ⊂ U
Proof. First, R R R H \mathcal{H} H V \mathcal{V} V A \mathcal{A} A A \mathcal{A} A − 1 -1 − 1 R R R E ∈ H E\in\mathcal{H} E ∈ H R E RE R E A \mathcal{A} A
Every object of V \mathcal{V} V H [ n ] \mathcal{H}[n] H [ n ] n ≥ 0 n\ge0 n ≥ 0 R R R R ( H ) [ n ] ⊂ V [ n ] ⊂ V R(\mathcal{H})[n]\subset\mathcal{V}[n]\subset\mathcal{V} R ( H ) [ n ] ⊂ V [ n ] ⊂ V R R R V \mathcal{V} V R V ⊂ V R\mathcal{V}\subset\mathcal{V} R V ⊂ V R R R
R ( V [ 1 ] ) = ( R V ) [ 1 ] ⊂ V [ 1 ] ⊂ U . R(\mathcal{V}[1])=(R\mathcal{V})[1]\subset\mathcal{V}[1]\subset\mathcal{U}. R ( V [ 1 ]) = ( R V ) [ 1 ] ⊂ V [ 1 ] ⊂ U . Every object of U \mathcal{U} U T \mathcal{T} T V [ 1 ] \mathcal{V}[1] V [ 1 ] U \mathcal{U} U R E ∈ U RE\in\mathcal{U} R E ∈ U E ∈ T E\in\mathcal{T} E ∈ T R E ∈ V RE\in\mathcal{V} R E ∈ V U \mathcal{U} U H H 0 ( R E ) ∈ T H^{0}_{\mathcal{H}}(RE)\in\mathcal{T} H H 0 ( R E ) ∈ T
Write E = ( X , Y , m ) E=(X,Y,m) E = ( X , Y , m ) x j x_{j} x j R E RE R E Y j Y_{j} Y j y k y_{k} y k m k : X k → Y k − 1 ⊕ Y k + 1 m_{k}:X_{k}\to Y_{k-1}\oplus Y_{k+1} m k : X k → Y k − 1 ⊕ Y k + 1 H \mathcal{H} H X k X_k X k Y j Y_j Y j A \mathcal{A} A Y j Y_j Y j C k : = coker ( m k ) C_k := \operatorname{coker}(m_k) C k := coker ( m k )
A = H H 0 ( R E ) = ( Y , C ) , C = coker ( m : X → F Y ) . A = H^0_{\mathcal{H}}(R\mathcal{E}) = (Y,C), \qquad C = \operatorname{coker}(m:X \to F Y). A = H H 0 ( R E ) = ( Y , C ) , C = coker ( m : X → F Y ) . The structure map Y → F C Y \to F C Y → F C F Y ↠ C F Y \twoheadrightarrow C F Y ↠ C F ⊣ F F \dashv F F ⊣ F
Y j ↪ Y k − 1 ⊕ Y k + 1 ↠ C k , ∣ j − k ∣ = 1. Y_j \hookrightarrow Y_{k-1} \oplus Y_{k+1} \twoheadrightarrow C_k,\qquad|j-k|=1. Y j ↪ Y k − 1 ⊕ Y k + 1 ↠ C k , ∣ j − k ∣ = 1. For fixed j j j Y j → ( F C ) j Y_j \to(F C)_j Y j → ( F C ) j k k k ( F Y ) k ↠ C k (F Y)_k \twoheadrightarrow C_k ( F Y ) k ↠ C k A ∉ T A \notin\mathcal{T} A ∈ / T A ↠ Q = ( Y ′ , C ′ ) ∈ F A \twoheadrightarrow Q=(Y',C')\in\mathcal{F} A ↠ Q = ( Y ′ , C ′ ) ∈ F
F Y → C ↓ ↓ F Y ′ → C ′ . \begin{CD}
F Y @>>> C \\
@VVV @VVV \\
F Y' @>>> C'.
\end{CD} F Y ↓ ⏐ F Y ′ C ↓ ⏐ C ′ . The top and right maps are epimorphisms, so the bottom map is an epimorphism as well. At each target vertex y k y_k y k Y k − 1 ′ ⊕ Y k + 1 ′ ↠ C k ′ Y'_{k-1}\oplus Y'_{k+1}\twoheadrightarrow C'_k Y k − 1 ′ ⊕ Y k + 1 ′ ↠ C k ′ (14) replaces C k ′ C'_k C k ′
Cone ( Y k − 1 ′ ⊕ Y k + 1 ′ → C k ′ ) [ − 1 ] ≃ ker ( Y k − 1 ′ ⊕ Y k + 1 ′ → C k ′ ) . \operatorname{Cone}(Y'_{k-1}\oplus Y'_{k+1}\to C'_k)[-1]\simeq\ker(Y'_{k-1}\oplus Y'_{k+1}\to C'_k). Cone ( Y k − 1 ′ ⊕ Y k + 1 ′ → C k ′ ) [ − 1 ] ≃ ker ( Y k − 1 ′ ⊕ Y k + 1 ′ → C k ′ ) . Indeed, because the map is an epimorphism, the shifted cone has cohomology only in degree zero, where it is the kernel. After relabeling, these kernels are the new source objects, and the Y j ′ Y'_j Y j ′ Y j ′ Y'_j Y j ′ A \mathcal{A} A
R − 1 Q ≃ ( ker ( F Y ′ → C ′ ) , Y ′ ) ∈ H . R^{-1}Q \simeq(\ker(FY'\to C'),Y')\in\mathcal{H}. R − 1 Q ≃ ( ker ( F Y ′ → C ′ ) , Y ′ ) ∈ H . Apply R − 1 R^{-1} R − 1 R E → A ↠ Q R E\to A\twoheadrightarrow Q R E → A ↠ Q E → R − 1 Q E\to R^{-1}Q E → R − 1 Q Y ↠ Y ′ Y\twoheadrightarrow Y' Y ↠ Y ′ J ∈ A B J\in\mathcal{A}_{\mathcal{B}} J ∈ A B H \mathcal{H} H
E ⟶ R − 1 Q ⟶ ( J , 0 ) ⟶ 0. E\longrightarrow R^{-1}Q\longrightarrow(J,0)\longrightarrow0. E ⟶ R − 1 Q ⟶ ( J , 0 ) ⟶ 0. Every diagram supported only at the sources belongs to T \mathcal{T} T Im Z = p ≥ 0 \operatorname{Im}\mathcal{Z}=p\geq0 Im Z = p ≥ 0 E → R − 1 Q E\to R^{-1}Q E → R − 1 Q T \mathcal{T} T E E E T \mathcal{T} T R − 1 Q ∈ T R^{-1}Q\in\mathcal{T} R − 1 Q ∈ T
The charge rotation (26) gives
Z ( R − 1 Q ) = e − π i / d Z ( Q ) . \mathcal{Z}(R^{-1}Q)=e^{-\pi i/d}\mathcal{Z}(Q). Z ( R − 1 Q ) = e − π i / d Z ( Q ) . Since 0 ≠ Q ∈ F 0\ne Q\in\mathcal{F} 0 = Q ∈ F (18) places this charge in the sector of phases ( − 1 , − 1 / d ] (-1,-1/d] ( − 1 , − 1/ d ] R − 1 Q ∈ T ⊂ A ′ R^{-1}Q\in\mathcal{T}\subset\mathcal{A}' R − 1 Q ∈ T ⊂ A ′ A ∈ T A\in\mathcal{T} A ∈ T R E ∈ U R E\in\mathcal{U} R E ∈ U
Lemma 3.12. For every quiver diagram E E E
M ( R E ) ≤ ( 1 + λ ) M ( E ) , λ = 2 cos ( π / d ) . (27) M(R E)\leq(1+\lambda)M(E),\qquad\lambda=2\cos(\pi/d). \tag*{(27)} M ( R E ) ≤ ( 1 + λ ) M ( E ) , λ = 2 cos ( π / d ) . (27) For d = 2 d=2 d = 2
Proof. Write E = ( X , Y , m ) E=(X,Y,m) E = ( X , Y , m ) R E = ( Y , Cone ( m ) ) R E=(Y,\operatorname{Cone}(m)) R E = ( Y , Cone ( m )) j j j
( F Y ) j ⟶ Cone ( m j ) ⟶ X j [ 1 ] ⟶ ( F Y ) j [ 1 ] . (FY)_j\longrightarrow\operatorname{Cone}(m_j)\longrightarrow X_j[1]\longrightarrow(FY)_j[1]. ( F Y ) j ⟶ Cone ( m j ) ⟶ X j [ 1 ] ⟶ ( F Y ) j [ 1 ] . The mass triangle inequality (5) , together with invariance under shifts and additivity under direct sums, gives
m σ ( Cone ( m j ) ) ≤ m σ ( ( F Y ) j ) + m σ ( X j [ 1 ] ) = m σ ( Y j − 1 ) + m σ ( Y j + 1 ) + m σ ( X j ) . m_\sigma(\operatorname{Cone}(m_j))\leq m_\sigma((FY)_j)+m_\sigma(X_j[1])=m_\sigma(Y_{j-1})+m_\sigma(Y_{j+1})+m_\sigma(X_j). m σ ( Cone ( m j )) ≤ m σ (( F Y ) j ) + m σ ( X j [ 1 ]) = m σ ( Y j − 1 ) + m σ ( Y j + 1 ) + m σ ( X j ) . Multiplying by r j r_j r j r j − 1 + r j + 1 = λ r j r_{j-1}+r_{j+1}=\lambda r_j r j − 1 + r j + 1 = λ r j Y j Y_j Y j R E R E R E
M ( R E ) ≤ ∑ j r j m σ ( X j ) + ( 1 + λ ) ∑ j r j m σ ( Y j ) ≤ ( 1 + λ ) M ( E ) . M(RE) \le\sum_j r_jm_\sigma(X_j)+(1+\lambda)\sum_j r_jm_\sigma(Y_j)\le(1+\lambda)M(E). M ( R E ) ≤ j ∑ r j m σ ( X j ) + ( 1 + λ ) j ∑ r j m σ ( Y j ) ≤ ( 1 + λ ) M ( E ) . For d = 2 d=2 d = 2 ( X , Y ) ↦ ( Y , X [ 1 ] ) (X,Y)\mapsto(Y,X[1]) ( X , Y ) ↦ ( Y , X [ 1 ]) M M M
Reflection blocks
We now use the nested aisles to divide the category into narrow phase sectors. For j ∈ Z j\in\mathbb{Z} j ∈ Z U j = R j U U_j=R^jU U j = R j U Lemma 3.11 and (25) give
U j + 1 ⊂ U j , U j + d = U j [ 1 ] . U_{j+1}\subset U_j,\qquad U_{j+d}=U_j[1]. U j + 1 ⊂ U j , U j + d = U j [ 1 ] . The exact autoequivalence R j R^j R j U U U A ′ \mathcal{A}' A ′ U j U_j U j R j A ′ R^j\mathcal{A}' R j A ′ [ 1 ] [1] [ 1 ]
H j = U j ∩ ( U j [ 1 ] ) ⊥ = U j ∩ U j + d ⊥ = R j A ′ . \mathcal{H}_j=U_j\cap(U_j[1])^\perp=U_j\cap U_{j+d}^{\perp}=R^j\mathcal{A}'. H j = U j ∩ ( U j [ 1 ] ) ⊥ = U j ∩ U j + d ⊥ = R j A ′ . We also define the block
S j = U j ∩ U j + 1 ⊥ . \mathcal{S}_j=U_j\cap U_{j+1}^{\perp}. S j = U j ∩ U j + 1 ⊥ . An object of S j \mathcal{S}_j S j H j \mathcal{H}_j H j H j + 1 [ − 1 ] \mathcal{H}_{j+1}[-1] H j + 1 [ − 1 ] Lemma 3.10 and (26) , the allowed half-planes for their rotated stability functions intersect in the sector
Z ( E ) ∈ R > 0 e i π θ , j / d < θ ≤ ( j + 1 ) / d ( 0 ≠ E ∈ S j ) . (28) Z(E)\in\mathbb{R}_{>0}e^{i\pi\theta},\qquad j/d<\theta\le(j+1)/d\qquad(0\ne E\in\mathcal{S}_j). \tag*{(28)} Z ( E ) ∈ R > 0 e iπ θ , j / d < θ ≤ ( j + 1 ) / d ( 0 = E ∈ S j ) . (28) The lower endpoint is excluded because a stability function on H j \mathcal{H}_j H j e − π i j / d e^{-\pi i j/d} e − π ij / d Figure 8 .
Figure 8. Charges in the reflection blocks for d = 5 d=5 d = 5
Lemma 3.13. Every object E ∈ S 0 E\in\mathcal{S}_0 E ∈ S 0 T ⊂ H \mathcal{T}\subset\mathcal{H} T ⊂ H E = ( X , Y , m ) E=(X,Y,m) E = ( X , Y , m ) C = coker ( F X → Y ) C=\operatorname{coker}(FX\to Y) C = coker ( F X → Y ) F X → Y FX\to Y F X → Y m : X → F Y m:X\to FY m : X → F Y
K ⟶ R − 1 E ⟶ ( C , 0 ) [ − 1 ] ⟶ K [ 1 ] , (29) K\longrightarrow R^{-1}E\longrightarrow(C,0)[-1]\longrightarrow K[1], \tag*{(29)} K ⟶ R − 1 E ⟶ ( C , 0 ) [ − 1 ] ⟶ K [ 1 ] , (29) where K = H H 0 ( R − 1 E ) ∈ F K=H^0_{\mathcal{H}}(R^{-1}E)\in\mathcal{F} K = H H 0 ( R − 1 E ) ∈ F C j C_j C j ( t , 1 ] (t,1] ( t , 1 ] t = 1 − 1 d t=1-\frac{1}{d} t = 1 − d 1
Proof. Let D = ( X , Y , m ) ∈ H D=(X,Y,m)\in\mathcal{H} D = ( X , Y , m ) ∈ H F X → Y FX\to Y F X → Y m : X → F Y m:X\to FY m : X → F Y (14) , followed by the shift [ 1 ] [1] [ 1 ]
R − 1 D [ 1 ] ≃ ( Cone ( F X → Y ) , X [ 1 ] ) . R^{-1}D[1]\simeq(\operatorname{Cone}(FX\to Y),X[1]). R − 1 D [ 1 ] ≃ ( Cone ( F X → Y ) , X [ 1 ]) . At each source, the cone has cohomology equal to the kernel in degree − 1 -1 − 1 0 0 0 X j [ 1 ] X_j[1] X j [ 1 ] − 1 -1 − 1 R − 1 D [ 1 ] ∈ V R^{-1}D[1]\in\mathcal{V} R − 1 D [ 1 ] ∈ V
H H 0 ( R − 1 D [ 1 ] ) = ( coker ( F X → Y ) , 0 ) . H^0_{\mathcal{H}}(R^{-1}D[1])=(\operatorname{coker}(FX\to Y),0). H H 0 ( R − 1 D [ 1 ]) = ( coker ( F X → Y ) , 0 ) . This zeroth cohomology diagram is supported only at the sources. It belongs to T \mathcal{T} T Im Z = p \operatorname{Im}Z=p Im Z = p U U U R − 1 D [ 1 ] ∈ U R^{-1}D[1]\in U R − 1 D [ 1 ] ∈ U H [ 1 ] ⊂ R U \mathcal{H}[1]\subset RU H [ 1 ] ⊂ R U
Now take E ∈ S 0 E\in\mathcal{S}_0 E ∈ S 0 U [ 1 ] ⊂ R U U[1]\subset RU U [ 1 ] ⊂ R U E ∈ U ∩ ( U [ 1 ] ) ⊥ = A ′ E\in U\cap(U[1])^\perp=\mathcal{A}' E ∈ U ∩ ( U [ 1 ] ) ⊥ = A ′
0 ⟶ F E [ 1 ] ⟶ E ⟶ T E ⟶ 0. 0\longrightarrow FE[1]\longrightarrow E\longrightarrow T_E\longrightarrow0. 0 ⟶ F E [ 1 ] ⟶ E ⟶ T E ⟶ 0. But F E [ 1 ] ∈ H [ 1 ] ⊂ R U F_E[1] \in\mathcal{H}[1] \subset\mathcal{R}U F E [ 1 ] ∈ H [ 1 ] ⊂ R U E ∈ ( R U ) ⊥ E \in(\mathcal{R}U)^\perp E ∈ ( R U ) ⊥ F E = 0 F_E=0 F E = 0 E ∈ T ⊂ H E \in\mathcal{T} \subset\mathcal{H} E ∈ T ⊂ H (14) gives
R − 1 E ≃ ( Cone ( F X → Y ) [ − 1 ] , X ) . R^{-1}E \simeq(\operatorname{Cone}(FX \to Y)[-1],X). R − 1 E ≃ ( Cone ( F X → Y ) [ − 1 ] , X ) . The shifted cone has the kernel in degree 0 and the cokernel in degree 1, so
H H 0 ( R − 1 E ) = ( ker ( F X → Y ) , X ) = K , H H 1 ( R − 1 E ) = ( C , 0 ) , H^0_{\mathcal{H}}(R^{-1}E)=(\ker(FX \to Y),X)=K,\qquad H^1_{\mathcal{H}}(R^{-1}E)=(C,0), H H 0 ( R − 1 E ) = ( ker ( F X → Y ) , X ) = K , H H 1 ( R − 1 E ) = ( C , 0 ) , with all other cohomology zero. The cohomology truncations therefore give the triangle (29) . For T ∈ T T \in\mathcal{T} T ∈ T Hom ( T , − ) \operatorname{Hom}(T,-) Hom ( T , − )
Hom ( T , K ) ≃ Hom ( T , R − 1 E ) = 0. \operatorname{Hom}(T,K)\simeq\operatorname{Hom}(T,R^{-1}E)=0. Hom ( T , K ) ≃ Hom ( T , R − 1 E ) = 0. The isomorphism uses Hom ( T , ( C , 0 ) [ − k ] ) = 0 \operatorname{Hom}(T,(C,0)[-k])=0 Hom ( T , ( C , 0 ) [ − k ]) = 0 k = 1 , 2 k=1,2 k = 1 , 2 T T T ( C , 0 ) (C,0) ( C , 0 ) H \mathcal{H} H T ∈ U T \in\mathcal{U} T ∈ U R − 1 E ∈ U ⊥ R^{-1}E \in\mathcal{U}^{\perp} R − 1 E ∈ U ⊥ K ∈ F K \in\mathcal{F} K ∈ F
At each target y j y_j y j Y j ↠ C j Y_j \twoheadrightarrow C_j Y j ↠ C j X j − 1 X_{j-1} X j − 1 X j + 1 X_{j+1} X j + 1 E ↠ ( 0 , C ) E \twoheadrightarrow(0,C) E ↠ ( 0 , C ) H \mathcal{H} H E ∈ T E \in\mathcal{T} E ∈ T T \mathcal{T} T ( 0 , C ) ∈ T (0,C) \in\mathcal{T} ( 0 , C ) ∈ T
Suppose some C j C_j C j t t t C j ↠ B C_j \twoheadrightarrow B C j ↠ B 0 < ϕ ≤ t 0<\phi\leq t 0 < ϕ ≤ t ( 0 , C ) (0,C) ( 0 , C ) B B B y j y_j y j p = r j Im Z ( B ) > 0 p=r_j\operatorname{Im}Z(B)>0 p = r j Im Z ( B ) > 0 0 < ϕ < 1 0<\phi<1 0 < ϕ < 1 r j q Z ( B ) r_jqZ(B) r j q Z ( B ) ϕ − t ∈ ( − t , 0 ] \phi-t\in(-t,0] ϕ − t ∈ ( − t , 0 ] q = e − π i t q=e^{-\pi i t} q = e − π i t T \mathcal{T} T C C C ( t , 1 ] (t,1] ( t , 1 ]
Lemma 3.14. Recall that c = cos ( π t / 2 ) = sin ( π / ( 2 d ) ) c=\cos(\pi t/2)=\sin(\pi/(2d)) c = cos ( π t /2 ) = sin ( π / ( 2 d )) E ∈ S 0 E\in\mathcal{S}_0 E ∈ S 0
M ( E ) ≤ ( 1 + λ ) c − 4 ∣ Z ( E ) ∣ . (30) M(E)\leq(1+\lambda)c^{-4}|Z(E)|. \tag*{(30)} M ( E ) ≤ ( 1 + λ ) c − 4 ∣ Z ( E ) ∣. (30) Proof. Let K K K C C C Z ( C j ) Z(C_j) Z ( C j ) z ( C ) = ∑ j r j Z ( C j ) z(C)=\sum_j r_jZ(C_j) z ( C ) = ∑ j r j Z ( C j ) ( t , 1 ] (t,1] ( t , 1 ] ∑ j r j m σ ( C j ) = M ( ( C , 0 ) ) \sum_j r_jm_\sigma(C_j)=M((C,0)) ∑ j r j m σ ( C j ) = M (( C , 0 )) π / d ≤ π t \pi/d\leq\pi t π / d ≤ π t c c c
M ( ( C , 0 ) ) ≤ c − 1 ∣ z ( C ) ∣ . (31) M((C,0))\leq c^{-1}|z(C)|. \tag*{(31)} M (( C , 0 )) ≤ c − 1 ∣ z ( C ) ∣. (31) We can now estimate the two cohomology objects in the triangle (29) . Additivity of the charge and (26) give
e − π i / d Z ( E ) = Z ( K ) − z ( C ) . e^{-\pi i/d}Z(E)=Z(K)-z(C). e − π i / d Z ( E ) = Z ( K ) − z ( C ) . Since K ∈ F K\in\mathcal{F} K ∈ F (18) puts Z ( K ) Z(K) Z ( K ) ( − t , 0 ] (-t,0] ( − t , 0 ] C C C − z ( C ) -z(C) − z ( C ) ( − 1 / d , 0 ] ⊆ ( − t , 0 ] (-1/d,0]\subseteq(-t,0] ( − 1/ d , 0 ] ⊆ ( − t , 0 ]
∣ Z ( K ) ∣ + ∣ z ( C ) ∣ ≤ c − 1 ∣ Z ( E ) ∣ . (32) |Z(K)|+|z(C)|\leq c^{-1}|Z(E)|. \tag*{(32)} ∣ Z ( K ) ∣ + ∣ z ( C ) ∣ ≤ c − 1 ∣ Z ( E ) ∣. (32) Combining these estimates, we obtain
M ( R − 1 E ) ≤ M ( K ) + M ( ( C , 0 ) ) ≤ c − 3 ( ∣ Z ( K ) ∣ + ∣ z ( C ) ∣ ) ≤ c − 4 ∣ Z ( E ) ∣ . M(R^{-1}E)\leq M(K)+M((C,0))\leq c^{-3}(|Z(K)|+|z(C)|)\leq c^{-4}|Z(E)|. M ( R − 1 E ) ≤ M ( K ) + M (( C , 0 )) ≤ c − 3 ( ∣ Z ( K ) ∣ + ∣ z ( C ) ∣ ) ≤ c − 4 ∣ Z ( E ) ∣. The first inequality applies the mass triangle inequality (5) at each vertex of (29) , using invariance under shifts. The second uses the bound for K K K (18) , the bound for C C C (31) , and c ≤ 1 c\leq1 c ≤ 1 (32) . Applying (27) to R − 1 E R^{-1}E R − 1 E M ( E ) M(E) M ( E )
Lemma 3.15. Put B d = ( 1 + λ ) d c − 4 B_d=(1+\lambda)^d c^{-4} B d = ( 1 + λ ) d c − 4 j ∈ Z j\in\mathbb{Z} j ∈ Z E ∈ S j E\in\mathcal{S}_j E ∈ S j
M ( E ) ≤ B d ∣ Z ( E ) ∣ . (33) M(E)\leq B_d|Z(E)|. \tag*{(33)} M ( E ) ≤ B d ∣ Z ( E ) ∣. (33) In particular, every nonzero block object has nonzero charge. Fix a norm on Λ R \Lambda_{\mathbb{R}} Λ R C σ C_\sigma C σ σ \sigma σ Λ ⊕ 2 ( d − 1 ) \Lambda^{\oplus2(d-1)} Λ ⊕ 2 ( d − 1 )
∥ v ( E ) ∥ ≤ C σ M ( E ) ≤ C σ B d ∣ Z ( E ) ∣ ( E ∈ S j ) . (34) \|v(E)\|\leq C_\sigma M(E)\leq C_\sigma B_d|Z(E)|\qquad(E\in\mathcal{S}_j). \tag*{(34)} ∥ v ( E ) ∥ ≤ C σ M ( E ) ≤ C σ B d ∣ Z ( E ) ∣ ( E ∈ S j ) . (34) Proof. First suppose 0 ≤ j < d 0 \le j < d 0 ≤ j < d R − j E ∈ S 0 R^{-j}E \in\mathcal{S}_0 R − j E ∈ S 0 (27) j j j (30) , we obtain
M ( E ) ≤ ( 1 + λ ) j M ( R − j E ) ≤ ( 1 + λ ) j + 1 c − 4 ∣ Z ( E ) ∣ ≤ B d ∣ Z ( E ) ∣ . M(E) \le(1+\lambda)^j M(R^{-j}E) \le(1+\lambda)^{j+1}c^{-4}|Z(E)| \le B_d|Z(E)|. M ( E ) ≤ ( 1 + λ ) j M ( R − j E ) ≤ ( 1 + λ ) j + 1 c − 4 ∣ Z ( E ) ∣ ≤ B d ∣ Z ( E ) ∣. Here (26) gives ∣ Z ( R − j E ) ∣ = ∣ Z ( E ) ∣ |Z(R^{-j}E)|=|Z(E)| ∣ Z ( R − j E ) ∣ = ∣ Z ( E ) ∣ j + 1 ≤ d j+1\le d j + 1 ≤ d 1 + λ ≥ 1 1+\lambda\ge1 1 + λ ≥ 1 S j + d = S j [ 1 ] \mathcal{S}_{j+d}=\mathcal{S}_j[1] S j + d = S j [ 1 ] M M M j ∈ Z j\in\mathbb{Z} j ∈ Z E ≠ 0 E\ne0 E = 0 M ( E ) > 0 M(E)>0 M ( E ) > 0 Z ( E ) ≠ 0 Z(E)\ne0 Z ( E ) = 0
For any object A ∈ C A\in\mathcal{C} A ∈ C (4) to its HN factors with respect to σ \sigma σ ∥ v ( A ) ∥ ≤ C σ m σ ( A ) \|v(A)\|\le C_\sigma m_\sigma(A) ∥ v ( A ) ∥ ≤ C σ m σ ( A ) r k ≥ 1 r_k\ge1 r k ≥ 1 ∥ v ( E ) ∥ ≤ C σ M ( E ) \|v(E)\|\le C_\sigma M(E) ∥ v ( E ) ∥ ≤ C σ M ( E ) (33) , this proves (34) . □
The stronger estimate (33) will also provide the mass–Hom bound after constructing HN filtrations.
Harder–Narasimhan filtrations
Successive truncations along U ⊃ R U ⊃ ⋯ ⊃ R d U = U [ 1 ] U\supset RU\supset\cdots\supset R^dU=U[1] U ⊃ R U ⊃ ⋯ ⊃ R d U = U [ 1 ] (34) . A similar use of nested aisles appears in [33 §4] .
Lemma 3.16. Let a < b a<b a < b E ∈ U a ∩ U b ⊥ E\in U_a\cap U_b^\perp E ∈ U a ∩ U b ⊥ S b − 1 , … , S a \mathcal{S}_{b-1},\ldots,\mathcal{S}_a S b − 1 , … , S a a < b a<b a < b j ∈ Z j\in\mathbb{Z} j ∈ Z ( T j , S j ) (\mathcal{T}_j,\mathcal{S}_j) ( T j , S j ) H j \mathcal{H}_j H j
T j = U j + 1 ∩ H j = ⟨ S j + d − 1 , … , S j + 1 ⟩ . \mathcal{T}_j=U_{j+1}\cap\mathcal{H}_j=\langle\mathcal{S}_{j+d-1},\ldots,\mathcal{S}_{j+1}\rangle. T j = U j + 1 ∩ H j = ⟨ S j + d − 1 , … , S j + 1 ⟩ . Moreover, Hom ( S l , S j ) = 0 \operatorname{Hom}(\mathcal{S}_l,\mathcal{S}_j)=0 Hom ( S l , S j ) = 0 l > j l>j l > j
Proof. Suppose E ∈ U a ∩ U b ⊥ E\in U_a\cap U_b^\perp E ∈ U a ∩ U b ⊥ a < b a<b a < b U a + 1 U_{a+1} U a + 1
E ′ ⟶ E ⟶ E ′ ′ ⟶ E ′ [ 1 ] , E ′ ∈ U a + 1 , E ′ ′ ∈ U a + 1 ⊥ . E'\longrightarrow E\longrightarrow E''\longrightarrow E'[1],\qquad E'\in U_{a+1},\quad E''\in U_{a+1}^\perp. E ′ ⟶ E ⟶ E ′′ ⟶ E ′ [ 1 ] , E ′ ∈ U a + 1 , E ′′ ∈ U a + 1 ⊥ . The rotated triangle E → E ′ ′ → E ′ [ 1 ] E\to E''\to E'[1] E → E ′′ → E ′ [ 1 ] E ′ ′ ∈ U a E''\in U_a E ′′ ∈ U a E ′ ′ ∈ U a ∩ U a + 1 ⊥ = S a E''\in U_a\cap U_{a+1}^\perp=\mathcal{S}_a E ′′ ∈ U a ∩ U a + 1 ⊥ = S a E ′ ∈ U b ⊥ E'\in U_b^\perp E ′ ∈ U b ⊥ A ∈ U b A\in U_b A ∈ U b A [ 1 ] ∈ U a + 1 A[1]\in U_{a+1} A [ 1 ] ∈ U a + 1 Hom ( A , E ′ ′ [ − 1 ] ) = Hom ( A [ 1 ] , E ′ ′ ) = 0 \operatorname{Hom}(A,E''[-1])=\operatorname{Hom}(A[1],E'')=0 Hom ( A , E ′′ [ − 1 ]) = Hom ( A [ 1 ] , E ′′ ) = 0 Hom ( A , − ) \operatorname{Hom}(A,-) Hom ( A , − ) E ′ ′ [ − 1 ] → E ′ → E E''[-1]\to E'\to E E ′′ [ − 1 ] → E ′ → E Hom ( A , E ′ ) = 0 \operatorname{Hom}(A,E')=0 Hom ( A , E ′ ) = 0
We can therefore repeat the construction with E ′ ∈ U a + 1 ∩ U b ⊥ E'\in U_{a+1}\cap U_b^\perp E ′ ∈ U a + 1 ∩ U b ⊥ U b ∩ U b ⊥ = 0 U_b\cap U_b^\perp=0 U b ∩ U b ⊥ = 0 S b − 1 , … , S a \mathcal{S}_{b-1},\ldots,\mathcal{S}_a S b − 1 , … , S a U j + d = U j [ 1 ] U_{j+d}=U_j[1] U j + d = U j [ 1 ] U a ∩ U b ⊥ U_a\cap U_b^\perp U a ∩ U b ⊥ a < b a<b a < b
For E ∈ H j E\in\mathcal{H}_j E ∈ H j a = j a=j a = j b = j + d b=j+d b = j + d H j \mathcal{H}_j H j
( T j , S j ) , T j = U j + 1 ∩ H j = ⟨ S j + d − 1 , … , S j + 1 ⟩ , (\mathcal{T}_j,\mathcal{S}_j),\qquad\mathcal{T}_j=U_{j+1}\cap\mathcal{H}_j=\langle\mathcal{S}_{j+d-1},\ldots,\mathcal{S}_{j+1}\rangle, ( T j , S j ) , T j = U j + 1 ∩ H j = ⟨ S j + d − 1 , … , S j + 1 ⟩ , where orthogonality follows from S j ⊂ U j + 1 ⊥ \mathcal{S}_j\subset U_{j+1}^\perp S j ⊂ U j + 1 ⊥ S j \mathcal{S}_j S j H j \mathcal{H}_j H j l > j l>j l > j S l ⊂ U j + 1 \mathcal{S}_l\subset U_{j+1} S l ⊂ U j + 1 Hom ( S l , S j ) = 0 \operatorname{Hom}(\mathcal{S}_l,\mathcal{S}_j)=0 Hom ( S l , S j ) = 0
By Lemma 3.10 and (26) , every nonzero E ∈ H j E\in\mathcal{H}_j E ∈ H j φ ∈ ( j / d , j / d + 1 ] \varphi\in(j/d,j/d+1] φ ∈ ( j / d , j / d + 1 ]
Z ( E ) ∈ R > 0 e π i φ . Z(E)\in\mathbb{R}_{>0}e^{\pi i\varphi}. Z ( E ) ∈ R > 0 e π i φ . We call E E E H j \mathcal{H}_j H j [10 Proposition 3.1 and Lemma 3.2] .
Lemma 3.17. For every j ∈ Z j\in\mathbb{Z} j ∈ Z E ∈ S j E\in\mathcal{S}_j E ∈ S j H j \mathcal{H}_j H j S j \mathcal{S}_j S j
Proof. Fix 0 ≠ E ∈ S j 0\ne E\in\mathcal{S}_j 0 = E ∈ S j ( T j , S j ) (\mathcal{T}_j,\mathcal{S}_j) ( T j , S j ) A ⊂ E A\subset E A ⊂ E H j \mathcal{H}_j H j E / A ∈ S j E/A\in\mathcal{S}_j E / A ∈ S j
It suffices to test semistability on saturated subobjects. Indeed, given 0 ≠ G ⊂ E 0\ne G\subset E 0 = G ⊂ E G ‾ \overline{G} G E E E T j \mathcal{T}_j T j E / G E/G E / G E / G ‾ ∈ S j E/\overline{G}\in\mathcal{S}_j E / G ∈ S j G ‾ \overline{G} G G ‾ / G ∈ T j \overline{G}/G\in\mathcal{T}_j G / G ∈ T j ( ( j + 1 ) / d , j / d + 1 ] ((j+1)/d,j/d+1] (( j + 1 ) / d , j / d + 1 ] G ∈ S j G\in\mathcal{S}_j G ∈ S j G G G π \pi π G G G
There are only finitely many possible charges of saturated subobjects of E E E
b j ( w ) = Re ( e − π i ( j + 1 / 2 ) / d w ) . b_j(w)=\operatorname{Re}(e^{-\pi i(j+1/2)/d}w). b j ( w ) = Re ( e − π i ( j + 1/2 ) / d w ) . By (28) , the sector has angle π / d ≤ π t \pi/d\le\pi t π / d ≤ π t (19) therefore gives b j ( Z ( A ) ) ≥ c ∣ Z ( A ) ∣ > 0 b_j(\mathcal{Z}(A))\ge c|\mathcal{Z}(A)|>0 b j ( Z ( A )) ≥ c ∣ Z ( A ) ∣ > 0 A ∈ S j A\in\mathcal{S}_j A ∈ S j A ⊂ E A\subset E A ⊂ E S j \mathcal{S}_j S j b j ( Z ( A ) ) ≤ b j ( Z ( E ) ) b_j(\mathcal{Z}(A))\le b_j(\mathcal{Z}(E)) b j ( Z ( A )) ≤ b j ( Z ( E )) (34) now gives
∥ v ( A ) ∥ ≤ C σ B d ∣ Z ( A ) ∣ ≤ C σ B d c − 1 b j ( Z ( E ) ) . \lVert v(A)\rVert\le C_\sigma B_d|\mathcal{Z}(A)|\le C_\sigma B_dc^{-1}b_j(\mathcal{Z}(E)). ∥ v ( A )∥ ≤ C σ B d ∣ Z ( A ) ∣ ≤ C σ B d c − 1 b j ( Z ( E )) . A bounded subset of the vertex lattice is finite, and Z ( A ) \mathcal{Z}(A) Z ( A ) v ( A ) v(A) v ( A )
Choose a nonzero saturated subobject A 1 ⊂ E A_1\subset E A 1 ⊂ E E E E A 1 A_1 A 1 E E E E E E S j \mathcal{S}_j S j A 1 A_1 A 1 A 1 A_1 A 1 B ⊂ E / A 1 B\subset E/A_1 B ⊂ E / A 1 A 1 A_1 A 1 E E E A 1 A_1 A 1
Repeat this construction on E / A 1 E/A_1 E / A 1 0 = E 0 ⊂ E 1 ⊂ ⋯ ⊂ E 0=E_0\subset E_1\subset\cdots\subset E 0 = E 0 ⊂ E 1 ⊂ ⋯ ⊂ E E E E S j \mathcal{S}_j S j b j ( Z ( E r ) ) b_j(\mathcal{Z}(E_r)) b j ( Z ( E r )) E E E
Lemma 3.18. The charge Z \mathcal{Z} Z Σ = ( Z , P Σ ) \Sigma=(\mathcal{Z},\mathcal{P}_\Sigma) Σ = ( Z , P Σ ) A ′ \mathcal{A}' A ′ Λ ⊕ 2 ( d − 1 ) \Lambda^{\oplus2(d-1)} Λ ⊕ 2 ( d − 1 ) Σ \Sigma Σ ( R , 1 / d ) (R,1/d) ( R , 1/ d ) E E E
M ( E ) ≤ B d m Σ ( E ) , B d = ( 1 + 2 cos ( π / d ) ) d sin ( π / ( 2 d ) ) − 4 . (35) M(E)\le B_dm_\Sigma(E),\qquad B_d=(1+2\cos(\pi/d))^d\sin(\pi/(2d))^{-4}. \tag*{(35)} M ( E ) ≤ B d m Σ ( E ) , B d = ( 1 + 2 cos ( π / d ) ) d sin ( π / ( 2 d ) ) − 4 . (35) Proof. For each ϕ ∈ R \phi\in\mathbb{R} ϕ ∈ R j j j j / d < ϕ ≤ ( j + 1 ) / d j/d<\phi\le(j+1)/d j / d < ϕ ≤ ( j + 1 ) / d P Σ ( ϕ ) \mathcal{P}_\Sigma(\phi) P Σ ( ϕ ) S j \mathcal{S}_j S j ϕ \phi ϕ H j \mathcal{H}_j H j [8 Definition 2.2 and the following paragraph] imply that extensions and finite direct sums of semistable objects of phase ϕ \phi ϕ H j \mathcal{H}_j H j ϕ \phi ϕ S j \mathcal{S}_j S j S j \mathcal{S}_j S j P Σ ( ϕ ) \mathcal{P}_\Sigma(\phi) P Σ ( ϕ )
We next check the vanishing of morphisms in decreasing phase. Let A ∈ P Σ ( ϕ ) A\in\mathcal{P}_\Sigma(\phi) A ∈ P Σ ( ϕ ) B ∈ P Σ ( ψ ) B\in\mathcal{P}_\Sigma(\psi) B ∈ P Σ ( ψ ) ϕ > ψ \phi>\psi ϕ > ψ A A A B B B S j \mathcal{S}_j S j f : A → B f:A\to B f : A → B I I I H j \mathcal{H}_j H j f ≠ 0 f\ne0 f = 0 I ≠ 0 I\ne0 I = 0 A A A B B B ϕ ( I ) ≥ ϕ \phi(I)\ge\phi ϕ ( I ) ≥ ϕ ϕ ( I ) ≤ ψ \phi(I)\le\psi ϕ ( I ) ≤ ψ f = 0 f=0 f = 0 A A A B B B A A A
Every object has the block filtration of Lemma 3.16. Refine each block factor by its HN filtration from Lemma 3.17. The phases decrease within each block. They also decrease when passing to a block of smaller index, since every phase in S j \mathcal{S}_j S j j / d j/d j / d j / d j/d j / d P Σ \mathcal{P}_\Sigma P Σ
The shift [ 1 ] [1] [ 1 ] H j \mathcal{H}_j H j H j + d \mathcal{H}_{j+d} H j + d S j \mathcal{S}_j S j S j + d \mathcal{S}_{j+d} S j + d Z ( E [ 1 ] ) = − Z ( E ) \mathcal{Z}(E[1])=-\mathcal{Z}(E) Z ( E [ 1 ]) = − Z ( E ) 1 1 1 P Σ ( ϕ + 1 ) = P Σ ( ϕ ) [ 1 ] \mathcal{P}_\Sigma(\phi+1)=\mathcal{P}_\Sigma(\phi)[1] P Σ ( ϕ + 1 ) = P Σ ( ϕ ) [ 1 ] P Σ ( ϕ ) \mathcal{P}_\Sigma(\phi) P Σ ( ϕ ) Σ = ( Z , P Σ ) \Sigma=(\mathcal{Z},\mathcal{P}_\Sigma) Σ = ( Z , P Σ ) H 0 = A ′ \mathcal{H}_0=\mathcal{A}' H 0 = A ′ S 0 , … , S d − 1 \mathcal{S}_0,\ldots,\mathcal{S}_{d-1} S 0 , … , S d − 1 A ′ \mathcal{A}' A ′ P Σ ( ϕ ) \mathcal{P}_\Sigma(\phi) P Σ ( ϕ ) 0 < ϕ ≤ 1 0<\phi\le1 0 < ϕ ≤ 1 P Σ ( ( 0 , 1 ] ) = A ′ \mathcal{P}_\Sigma((0,1])=\mathcal{A}' P Σ (( 0 , 1 ]) = A ′ R : H j → ∼ H j + 1 R:\mathcal{H}_j \xrightarrow{\sim} \mathcal{H}_{j+1} R : H j ∼ H j + 1 S j S_j S j S j + 1 S_{j+1} S j + 1 R E RE R E H j + 1 \mathcal{H}_{j+1} H j + 1 R R R E E E H j \mathcal{H}_j H j E E E 1 / d 1/d 1/ d E E E R E RE R E
R P Σ ( ϕ ) = P Σ ( ϕ + 1 / d ) . RP_{\Sigma}(\phi)=P_{\Sigma}(\phi+1/d). R P Σ ( ϕ ) = P Σ ( ϕ + 1/ d ) . Together with (26), this is the Gepner property of Definition 1.1.
Every Σ \Sigma Σ E 1 , … , E s E_1,\ldots,E_s E 1 , … , E s E E E M M M
M ( E ) ≤ ∑ k = 1 s M ( E k ) ≤ B d ∑ k = 1 s ∣ Z ( E k ) ∣ = B d m Σ ( E ) , (36) M(E)\leq\sum_{k=1}^{s}M(E_k)\leq B_d\sum_{k=1}^{s}\lvert Z(E_k)\rvert=B_dm_{\Sigma}(E), \tag*{(36)} M ( E ) ≤ k = 1 ∑ s M ( E k ) ≤ B d k = 1 ∑ s ∣ Z ( E k )∣ = B d m Σ ( E ) , (36) which proves (35). □
Lemma 3.19. The vertex class map
K 0 ( C ⊗ Q ^ d ) ⟶ Λ ⊕ 2 ( d − 1 ) , [ E ] ⟼ ( v ( E j ) ) j , K_0(\mathcal{C}\otimes\widehat{\mathbb{Q}}_d)\longrightarrow\Lambda^{\oplus2(d-1)},\qquad[E]\longmapsto\bigl(v(E_j)\bigr)_j, K 0 ( C ⊗ Q d ) ⟶ Λ ⊕ 2 ( d − 1 ) , [ E ] ⟼ ( v ( E j ) ) j , is numerical: it factors through K n u m ( C ⊗ Q ^ d ) K_{\mathrm{num}}(\mathcal{C}\otimes\widehat{\mathbb{Q}}_d) K num ( C ⊗ Q d ) j j j v ( E j ) v(E_j) v ( E j ) E j ∈ C E_j\in\mathcal{C} E j ∈ C
Proof. Fix a vertex j j j P j ⊗ C B P_j\otimes_{\mathcal{C}}B P j ⊗ C B
Hom ( P j ⊗ C B , E [ n ] ) ≅ Hom C ( B , E j [ n ] ) ( B ∈ C , n ∈ Z ) . \operatorname{Hom}(P_j\otimes_{\mathcal{C}}B,E[n])\cong\operatorname{Hom}_{\mathcal{C}}(B,E_j[n])\qquad(B\in\mathcal{C},\ n\in\mathbb{Z}). Hom ( P j ⊗ C B , E [ n ]) ≅ Hom C ( B , E j [ n ]) ( B ∈ C , n ∈ Z ) . Taking alternating sums of dimensions gives
χ C ( B , E j ) = χ C ⊗ Q ^ d ( P j ⊗ C B , E ) . (37) \chi_{\mathcal{C}}(B,E_j)=\chi_{\mathcal{C}\otimes\widehat{\mathbb{Q}}_d}(P_j\otimes_{\mathcal{C}}B,E). \tag*{(37)} χ C ( B , E j ) = χ C ⊗ Q d ( P j ⊗ C B , E ) . (37) Both categories have Serre functors, so their left and right Euler radicals agree, as recalled in Section 2. If a class ξ ∈ K 0 ( C ⊗ Q ^ d ) \xi\in K_0(\mathcal{C}\otimes\widehat{\mathbb{Q}}_d) ξ ∈ K 0 ( C ⊗ Q d ) ξ \xi ξ B ∈ C B\in\mathcal{C} B ∈ C j j j C \mathcal{C} C v v v
Corollary 3.20. If σ \sigma σ Σ \Sigma Σ G G G C \mathcal{C} C
∑ n ∈ Z dim C Hom ( G , E [ n ] ) ≤ κ m σ ( E ) for every E ∈ C , \sum_{n\in\mathbb{Z}}\dim_{\mathcal{C}}\operatorname{Hom}(G,E[n])\leq\kappa m_{\sigma}(E)\qquad\text{for every }E\in\mathcal{C}, n ∈ Z ∑ dim C Hom ( G , E [ n ]) ≤ κ m σ ( E ) for every E ∈ C , then the diagram category has a split generator P P P
∑ n ∈ Z dim C Hom ( P , E [ n ] ) ≤ κ B d m Σ ( E ) for every diagram E . \sum_{n\in\mathbb{Z}}\dim_{\mathcal{C}}\operatorname{Hom}(P,E[n])\leq\kappa B_dm_{\Sigma}(E)\qquad\text{for every diagram }E. n ∈ Z ∑ dim C Hom ( P , E [ n ]) ≤ κ B d m Σ ( E ) for every diagram E . Proof. Take P = ⨁ j P j ⊗ G P=\bigoplus_jP_j\otimes G P = ⨁ j P j ⊗ G P j P_j P j j j j P P P r j ≥ 1 r_j\geq1 r j ≥ 1
∑ n ∈ Z dim C Hom ( P , E [ n ] ) = ∑ j ∑ n ∈ Z dim C Hom ( G , E j [ n ] ) ≤ κ ∑ j m σ ( E j ) ≤ κ M ( E ) ≤ κ B d m Σ ( E ) . \sum_{n\in\mathbb{Z}}\dim_{\mathcal{C}}\operatorname{Hom}(P,E[n])
=\sum_j\sum_{n\in\mathbb{Z}}\dim_{\mathcal{C}}\operatorname{Hom}(G,E_j[n])
\leq\kappa\sum_jm_{\sigma}(E_j)\leq\kappa M(E)\leq\kappa B_dm_{\Sigma}(E). n ∈ Z ∑ dim C Hom ( P , E [ n ]) = j ∑ n ∈ Z ∑ dim C Hom ( G , E j [ n ]) ≤ κ j ∑ m σ ( E j ) ≤ κ M ( E ) ≤ κ B d m Σ ( E ) . Since P P P A ∈ C ⊗ Q ^ d A\in\mathcal{C}\otimes\widehat{\mathbb{Q}}_d A ∈ C ⊗ Q d C A > 0 C_A>0 C A > 0 E E E E E E
∑ n ∈ Z dim C Hom ( A , E [ n ] ) ≤ C A m Σ ( E ) . \sum_{n\in\mathbb{Z}}\dim_{\mathcal{C}}\operatorname{Hom}(A,E[n])\leq C_Am_{\Sigma}(E). n ∈ Z ∑ dim C Hom ( A , E [ n ]) ≤ C A m Σ ( E ) . Compatibility with autoequivalences
Suppose that the stability condition σ \sigma σ ( Ψ , 1 / d ) (\Psi,1/d) ( Ψ , 1/ d ) Ψ \Psi Ψ C \mathcal{C} C Ψ \Psi Ψ Ψ \Psi Ψ R R R Σ \Sigma Σ ( Ψ , 1 / d ) (\Psi,1/d) ( Ψ , 1/ d ) Z ( X , Y ) = z ( X ) + q z ( Y ) \mathcal{Z}(X,Y)=z(X)+qz(Y) Z ( X , Y ) = z ( X ) + q z ( Y )
Z ( Ψ E ) = e π i / d Z ( E ) . (38) \mathcal{Z}(\Psi E)=e^{\pi i/d}\mathcal{Z}(E). \tag*{(38)} Z ( Ψ E ) = e π i / d Z ( E ) . (38) To prove the corresponding statement for semistability, we first show that Ψ \Psi Ψ U U U
Lemma 3.21. One has Ψ U ⊂ U \Psi U\subset U Ψ U ⊂ U
Proof. Recall that V V V H \mathcal{H} H U U U T \mathcal{T} T V [ 1 ] V[1] V [ 1 ] Ψ \Psi Ψ C \mathcal{C} C 1 / d 1/d 1/ d Ψ V ⊂ V \Psi V\subset V Ψ V ⊂ V Ψ ( V [ 1 ] ) ⊂ V [ 1 ] ⊂ U \Psi(V[1])\subset V[1]\subset U Ψ ( V [ 1 ]) ⊂ V [ 1 ] ⊂ U E ∈ T E\in\mathcal{T} E ∈ T Ψ E ∈ V \Psi E\in V Ψ E ∈ V H H 0 ( Ψ E ) ∈ T H^0_{\mathcal{H}}(\Psi E)\in\mathcal{T} H H 0 ( Ψ E ) ∈ T
At a vertex j j j E j E_j E j E j > t ⊂ E j E_j^{>t}\subset E_j E j > t ⊂ E j t t t E j > t = 0 E_j^{>t}=0 E j > t = 0 E j / E j > t E_j/E_j^{>t} E j / E j > t ( 0 , t ] (0,t] ( 0 , t ] E j → E k E_j\to E_k E j → E k E j > t E_j^{>t} E j > t E k > t E_k^{>t} E k > t E k → E k / E k > t E_k\to E_k/E_k^{>t} E k → E k / E k > t t t t t t t E j > t E_j^{>t} E j > t E E E E l o w E_{\mathrm{low}} E low E j / E j > t E_j/E_j^{>t} E j / E j > t T \mathcal{T} T E ∈ T E\in\mathcal{T} E ∈ T T \mathcal{T} T H \mathcal{H} H Ψ \Psi Ψ E j > t E_j^{>t} E j > t H [ 1 ] \mathcal{H}[1] H [ 1 ] E l o w E_{\mathrm{low}} E low H \mathcal{H} H
H H 0 ( Ψ E ) = Ψ E l o w ∈ H . H^0_{\mathcal{H}}(\Psi E)=\Psi E_{\mathrm{low}}\in\mathcal{H}. H H 0 ( Ψ E ) = Ψ E low ∈ H . Suppose that Ψ E l o w \Psi E_{\mathrm{low}} Ψ E low T \mathcal{T} T H \mathcal{H} H
0 ⟶ L ⟶ Ψ E l o w ⟶ Q ⟶ 0 , 0 ≠ Q ∈ F . 0\longrightarrow L\longrightarrow\Psi E_{\mathrm{low}}\longrightarrow Q\longrightarrow0,\qquad0\ne Q\in\mathcal{F}. 0 ⟶ L ⟶ Ψ E low ⟶ Q ⟶ 0 , 0 = Q ∈ F . The objects in C \mathcal{C} C ( 1 / d , 1 ] (1/d,1] ( 1/ d , 1 ] A \mathcal{A} A Q Q Q Ψ E l o w \Psi E_{\mathrm{low}} Ψ E low Ψ − 1 Q ∈ H \Psi^{-1}Q\in\mathcal{H} Ψ − 1 Q ∈ H
Applying Ψ − 1 \Psi^{-1} Ψ − 1
E l o w ⟶ Ψ − 1 Q ⟶ H H 1 ( Ψ − 1 L ) ⟶ 0. E_{\mathrm{low}}\longrightarrow\Psi^{-1}Q\longrightarrow H^1_{\mathcal{H}}(\Psi^{-1}L)\longrightarrow0. E low ⟶ Ψ − 1 Q ⟶ H H 1 ( Ψ − 1 L ) ⟶ 0. The image of the first map belongs to T \mathcal{T} T E l o w E_{\mathrm{low}} E low L L L ( 0 , 1 / d ] (0,1/d] ( 0 , 1/ d ] Ψ − 1 \Psi^{-1} Ψ − 1 [ 1 ] [1] [ 1 ] ( t , 1 ] (t,1] ( t , 1 ] q = e − π i / d q=e^{-\pi i/d} q = e − π i / d ( 0 , 1 / d ] (0,1/d] ( 0 , 1/ d ] Im Z ( Ψ − 1 Q ) ≥ 0 \operatorname{Im}\mathcal{Z}(\Psi^{-1}Q)\ge0 Im Z ( Ψ − 1 Q ) ≥ 0
Z ( Ψ − 1 Q ) = e − π i / d Z ( Q ) , \mathcal{Z}(\Psi^{-1}Q)=e^{-\pi i/d}\mathcal{Z}(Q), Z ( Ψ − 1 Q ) = e − π i / d Z ( Q ) , whose phase lies in ( − 1 , − 1 / d ] (-1,-1/d] ( − 1 , − 1/ d ] 0 ≠ Q ∈ F 0\ne Q\in\mathcal{F} 0 = Q ∈ F H H 0 ( Ψ E ) ∈ T H^0_{\mathcal{H}}(\Psi E)\in\mathcal{T} H H 0 ( Ψ E ) ∈ T □ \square □
Lemma 3.22. If σ \sigma σ ( Ψ , 1 / d ) (\Psi,1/d) ( Ψ , 1/ d ) Ψ \Psi Ψ C \mathcal{C} C Σ \Sigma Σ ( Ψ , 1 / d ) (\Psi,1/d) ( Ψ , 1/ d ) C ⋊ Q ^ d \mathcal{C}\rtimes\widehat{\mathbb{Q}}_d C ⋊ Q d
Proof. The functor Ψ d [ − 1 ] \Psi^d[-1] Ψ d [ − 1 ] P ( ϕ ) \mathcal{P}(\phi) P ( ϕ ) Z Z Z σ \sigma σ C \mathcal{C} C H \mathcal{H} H Z \mathcal{Z} Z P \mathcal{P} P U U U Ψ d U = U [ 1 ] \Psi^dU=U[1] Ψ d U = U [ 1 ]
U [ 1 ] = Ψ d U ⊂ Ψ U ⊂ U , U ⊂ Ψ − 1 U ⊂ U [ − 1 ] . U[1]=\Psi^dU\subset\Psi U\subset U,\qquad U\subset\Psi^{-1}U\subset U[-1]. U [ 1 ] = Ψ d U ⊂ Ψ U ⊂ U , U ⊂ Ψ − 1 U ⊂ U [ − 1 ] . The second chain shows that, for E ∈ A ′ E\in\mathcal{A}' E ∈ A ′ Ψ − 1 E \Psi^{-1}E Ψ − 1 E A ′ \mathcal{A}' A ′ A ′ [ − 1 ] \mathcal{A}'[-1] A ′ [ − 1 ] A ′ \mathcal{A}' A ′ Σ \Sigma Σ ( − 1 , 1 ] (-1,1] ( − 1 , 1 ]
To finish, transport the slicing by Ψ \Psi Ψ
P ′ ( ϕ ) = Ψ P Σ ( ϕ − 1 / d ) . \mathcal{P}'(\phi)=\Psi\mathcal{P}_{\Sigma}(\phi-1/d). P ′ ( ϕ ) = Ψ P Σ ( ϕ − 1/ d ) . By (37), this is a slicing with central charge Z Z Z Σ ′ = ( Z , P ′ ) \Sigma'=(Z,\mathcal{P}') Σ ′ = ( Z , P ′ ) Σ \Sigma Σ P ′ \mathcal{P}' P ′ E E E P Σ \mathcal{P}_{\Sigma} P Σ Ψ − 1 E \Psi^{-1}E Ψ − 1 E 1 / d 1/d 1/ d
A ′ ⊂ P ′ ( ( − 1 + 1 / d , 1 + 1 / d ] ) ⊂ P ′ ( ( − 1 , 2 ] ) . \mathcal{A}'\subset\mathcal{P}'((-1+1/d,1+1/d])\subset\mathcal{P}'((-1,2]). A ′ ⊂ P ′ (( − 1 + 1/ d , 1 + 1/ d ]) ⊂ P ′ (( − 1 , 2 ]) . Collins–Polishchuk’s uniqueness criterion [10 Proposition 4.2(a)] says that two stability conditions with the same charge coincide if the heart of the first lies in phases ( − 1 , 2 ] (-1,2] ( − 1 , 2 ] Σ ′ = Σ \Sigma'=\Sigma Σ ′ = Σ
Ψ P Σ ( ϕ ) = P Σ ( ϕ + 1 / d ) . \Psi\mathcal{P}_{\Sigma}(\phi)=\mathcal{P}_{\Sigma}(\phi+1/d). Ψ P Σ ( ϕ ) = P Σ ( ϕ + 1/ d ) . Together with (37), this is the Gepner property ( Ψ , 1 / d ) (\Psi,1/d) ( Ψ , 1/ d )
This proves Theorem 3.7.
Variations on the Thom–Sebastiani theorem
We now transfer the stability conditions of Section 3 to Fermat hypersurfaces.
Theorem 4.1. For every d ≥ 2 d\geq2 d ≥ 2 M ≥ 1 M\geq1 M ≥ 1 1 1 1
F d , M = x 1 d + ⋯ + x M d . F_{d,M}=x_{1}^{d}+\cdots+x_{M}^{d}. F d , M = x 1 d + ⋯ + x M d . The category MF g r ( F d , M ) \operatorname{MF}^{\mathrm{gr}}(F_{d,M}) MF gr ( F d , M ) ( τ , 2 / d ) (\tau,2/d) ( τ , 2/ d ) K n u m ( MF g r ( F d , M ) ) K_{\mathrm{num}}(\operatorname{MF}^{\mathrm{gr}}(F_{d,M})) K num ( MF gr ( F d , M )) τ \tau τ
We first relate matrix factorizations of a sum of polynomials to the tensor product of the factorization categories of its summands. We then construct compatible stability conditions on the tensor product and prove the theorem at the end of the section.
Let f 1 , … , f r f_{1},\ldots,f_{r} f 1 , … , f r d d d x d x^{d} x d W = f 1 + ⋯ + f r W=f_{1}+\cdots+f_{r} W = f 1 + ⋯ + f r
L = ⨁ i = 1 r Z e i ⟨ d e i − d e j : 1 ≤ i , j ≤ r ⟩ , π : L ⟶ Z , e i ⟼ 1. L=\frac{\bigoplus_{i=1}^{r}\mathbb{Z}e_{i}}{\langle de_{i}-de_{j}:1\leq i,j\leq r\rangle},\qquad\pi:L\longrightarrow\mathbb{Z},\quad e_{i}\longmapsto1. L = ⟨ d e i − d e j : 1 ≤ i , j ≤ r ⟩ ⨁ i = 1 r Z e i , π : L ⟶ Z , e i ⟼ 1. A variable of weight a a a i i i L L L a e i ae_{i} a e i W W W d e i de_{i} d e i
L = Z e r ⊕ H , H = ker π = ⨁ i = 1 r − 1 ( Z / d Z ) ( e i − e r ) . L=\mathbb{Z}e_{r}\oplus H,\qquad H=\ker\pi=\bigoplus_{i=1}^{r-1}(\mathbb{Z}/d\mathbb{Z})(e_{i}-e_{r}). L = Z e r ⊕ H , H = ker π = i = 1 ⨁ r − 1 ( Z / d Z ) ( e i − e r ) . Proposition 4.2. Set T = ⨂ i MF Z ( f i ) \mathcal{T}=\bigotimes_{i}\operatorname{MF}^{\mathbb{Z}}(f_{i}) T = ⨂ i MF Z ( f i ) D = MF Z ( W ) \mathcal{D}=\operatorname{MF}^{\mathbb{Z}}(W) D = MF Z ( W ) τ i \tau_{i} τ i i i i T \mathcal{T} T H H H T \mathcal{T} T e i − e r e_{i}-e_{r} e i − e r τ i τ r − 1 \tau_{i}\tau_{r}^{-1} τ i τ r − 1
D ≃ T H . \mathcal{D}\simeq\mathcal{T}^{H}. D ≃ T H . Under this equivalence, the ordinary grading shift on D \mathcal{D} D τ r \tau_{r} τ r
Proof. There are two steps. The Thom–Sebastiani theorem identifies T \mathcal{T} T W W W L L L H H H
The block grading and the Thom–Sebastiani theorem. We recall the part of [3 Theorem 1.1] that we use. Suppose that two polynomials u u u v v v χ : G 1 → C ∗ \chi:G_{1}\to\mathbb{C}^{*} χ : G 1 → C ∗ ψ : G 2 → C ∗ \psi:G_{2}\to\mathbb{C}^{*} ψ : G 2 → C ∗ u + v u+v u + v
G 1 × C ∗ G 2 = { ( g 1 , g 2 ) : χ ( g 1 ) = ψ ( g 2 ) } . G_{1}\times_{\mathbb{C}^{*}}G_{2}=\{(g_{1},g_{2}):\chi(g_{1})=\psi(g_{2})\}. G 1 × C ∗ G 2 = {( g 1 , g 2 ) : χ ( g 1 ) = ψ ( g 2 )} . The equivalence is enhanced. The hypotheses are that the character ( g 1 , g 2 ) ↦ χ ( g 1 ) ψ ( g 2 ) − 1 (g_1,g_2)\mapsto\chi(g_1)\psi(g_2)^{-1} ( g 1 , g 2 ) ↦ χ ( g 1 ) ψ ( g 2 ) − 1 { u + v = 0 } \{u+v=0\} { u + v = 0 } { u = 0 } × { v = 0 } \{u=0\}\times\{v=0\} { u = 0 } × { v = 0 }
For r ≥ 2 r\ge2 r ≥ 2 f 1 f_1 f 1 f 2 f_2 f 2 f k f_k f k t 1 d = ⋯ = t k − 1 d t_1^d=\cdots=t_{k-1}^d t 1 d = ⋯ = t k − 1 d C ∗ \mathbb{C}^* C ∗ t 1 d t k − d t_1^d t_k^{-d} t 1 d t k − d t k t_k t k t k d = t 1 d t_k^d=t_1^d t k d = t 1 d
G = { ( t 1 , … , t r ) ∈ ( C ∗ ) r : t 1 d = ⋯ = t r d } = Hom ( L , C ∗ ) . G=\{(t_1,\ldots,t_r)\in(\mathbb{C}^*)^r:t_1^d=\cdots=t_r^d\}=\operatorname{Hom}(L,\mathbb{C}^*). G = {( t 1 , … , t r ) ∈ ( C ∗ ) r : t 1 d = ⋯ = t r d } = Hom ( L , C ∗ ) . Its character group is L L L G G G L L L a a a i i i a e i ae_i a e i W W W d e i = d e r de_i=de_r d e i = d e r T ≃ MF L ( W ) \mathcal{T}\simeq\operatorname{MF}^L(W) T ≃ MF L ( W ) τ i \tau_i τ i e i e_i e i e i − e r e_i-e_r e i − e r τ i τ r − 1 \tau_i\tau_r^{-1} τ i τ r − 1 L L L N N N N ( h ) ( h ′ ) = N ( h + h ′ ) N(h)(h')=N(h+h') N ( h ) ( h ′ ) = N ( h + h ′ ) h , h ′ ∈ H h,h'\in H h , h ′ ∈ H T \mathcal{T} T
Write S S S L = Z e r ⊕ H L=\mathbb{Z}e_r\oplus H L = Z e r ⊕ H M M M M n M_n M n n n n J ( M ) n e r + h = M n J(M)_{ne_r+h}=M_n J ( M ) n e r + h = M n n ∈ Z n\in\mathbb{Z} n ∈ Z h ∈ H h\in H h ∈ H s ∈ S s\in S s ∈ S L L L a e r + b ae_r+b a e r + b s s s M n M_n M n h h h M n + a M_{n+a} M n + a h + b h+b h + b M M M W W W L L L d e r de_r d e r ρ h : J ( M ) ( h ) → J ( M ) \rho_h:J(M)(h)\to J(M) ρ h : J ( M ) ( h ) → J ( M ) S ( n ) S(n) S ( n ) ⨁ h ∈ H S ( n e r + h ) \bigoplus_{h\in H}S(ne_r+h) ⨁ h ∈ H S ( n e r + h )
Conversely, let ( N , ρ ) (N,\rho) ( N , ρ ) L L L ρ h : N ( h ) → N \rho_h:N(h)\to N ρ h : N ( h ) → N ρ 0 = id \rho_0=\operatorname{id} ρ 0 = id ρ h + h ′ = ρ h ∘ ρ h ′ ( h ) \rho_{h+h'}=\rho_h\circ\rho_{h'}(h) ρ h + h ′ = ρ h ∘ ρ h ′ ( h ) M n = N n e r M_n=N_{ne_r} M n = N n e r s ∈ S a e r + b s\in S_{ae_r+b} s ∈ S a e r + b M M M M n = N n e r → s N ( n + a ) e r + b → ρ b N ( n + a ) e r = M n + a M_n=N_{ne_r}\xrightarrow{s}N_{(n+a)e_r+b}\xrightarrow{\rho_b}N_{(n+a)e_r}=M_{n+a} M n = N n e r s N ( n + a ) e r + b ρ b N ( n + a ) e r = M n + a M M M H H H W W W H H H M M M J ( M ) J(M) J ( M ) ρ h : N n e r + h → N n e r \rho_h:N_{ne_r+h}\to N_{ne_r} ρ h : N n e r + h → N n e r N N N J ( M ) J(M) J ( M )
The same constructions apply to morphisms and commute with their differentials, so they identify the enhanced categories after perfect completion. By [15 Theorem 8.7] , this completion of the strict equivariant model computes the equivariant objects in the homotopy category. We obtain
D ≃ MF L ( W ) H ≃ T H . \mathcal{D}\simeq\operatorname{MF}^L(W)^H\simeq\mathcal{T}^H. D ≃ MF L ( W ) H ≃ T H . Finally, increasing the integer grading by one increases the L L L e r e_r e r J ( M ( 1 ) ) = J ( M ) ( e r ) J(M(1))=J(M)(e_r) J ( M ( 1 )) = J ( M ) ( e r ) τ r \tau_r τ r
Lemma 4.3. Let a finite group H H H T \mathcal{T} T σ = ( Z , P ) \sigma=(Z,\mathcal{P}) σ = ( Z , P ) v : K 0 ( T ) → Λ v:K_0(\mathcal{T})\to\Lambda v : K 0 ( T ) → Λ T H \mathcal{T}^H T H U : T H → T U:\mathcal{T}^H\to\mathcal{T} U : T H → T
Z H = Z ∘ U ∗ , P H ( ϕ ) = { E : U ( E ) ∈ P ( ϕ ) } Z^H=Z\circ U_*,\qquad\mathcal{P}^H(\phi)=\{E:U(E)\in\mathcal{P}(\phi)\} Z H = Z ∘ U ∗ , P H ( ϕ ) = { E : U ( E ) ∈ P ( ϕ )} define a stability condition supported on the finite-rank lattice Λ H = im ( v ∘ U ∗ ) ⊆ Λ \Lambda^H=\operatorname{im}(v\circ U_*)\subseteq\Lambda Λ H = im ( v ∘ U ∗ ) ⊆ Λ v v v v ∘ U ∗ v\circ U_* v ∘ U ∗ m σ H ( E ) = m σ ( U ( E ) ) m_{\sigma^H}(E)=m_\sigma(U(E)) m σ H ( E ) = m σ ( U ( E )) σ \sigma σ σ H \sigma^H σ H
Proof. The forgetful functor is exact, faithful, and conservative. The shift and orthogonality axioms follow from those in T \mathcal{T} T U ( E ) U(E) U ( E ) T H \mathcal{T}^H T H
Choose a norm on Λ R \Lambda_{\mathbb{R}} Λ R C C C σ \sigma σ E E E U ( E ) U(E) U ( E )
∥ v ( U ( E ) ) ∥ ≤ C ∣ Z ( U ( E ) ) ∣ = C ∣ Z H ( E ) ∣ . \|v(U(E))\|\le C|Z(U(E))|=C|Z^H(E)|. ∥ v ( U ( E )) ∥ ≤ C ∣ Z ( U ( E )) ∣ = C ∣ Z H ( E ) ∣. Restricting the norm to Λ R H \Lambda^H_{\mathbb{R}} Λ R H
The HN construction also proves m σ H ( E ) = m σ ( U ( E ) ) m_{\sigma^H}(E)=m_{\sigma}(U(E)) m σ H ( E ) = m σ ( U ( E )) I ( A ) = ⨁ h ∈ H F h ( A ) I(A)=\bigoplus_{h\in H}F_h(A) I ( A ) = ⨁ h ∈ H F h ( A ) U U U [15 Lemma 3.8] . Hence χ T ( U ∗ x , A ) = χ T H ( x , I ( A ) ) \chi_{\mathcal T}(U_*x,A)=\chi_{\mathcal T^H}(x,I(A)) χ T ( U ∗ x , A ) = χ T H ( x , I ( A )) x x x A ∈ T A\in\mathcal T A ∈ T U ∗ x U_*x U ∗ x v ∘ U ∗ v\circ U_* v ∘ U ∗ T H \mathcal T^H T H
Finally, faithfulness of U U U A ∈ T H A\in\mathcal T^H A ∈ T H
∑ n ∈ Z dim C Hom ( A , E [ n ] ) ≤ ∑ n ∈ Z dim C Hom ( U ( A ) , U ( E ) [ n ] ) ≤ C U ( A ) m σ ( U ( E ) ) = C U ( A ) m σ H ( E ) . \sum_{n\in\mathbb Z}\dim_{\mathbb C}\operatorname{Hom}(A,E[n])\leq\sum_{n\in\mathbb Z}\dim_{\mathbb C}\operatorname{Hom}(U(A),U(E)[n])
\leq C_{U(A)}m_{\sigma}(U(E))=C_{U(A)}m_{\sigma^H}(E). n ∈ Z ∑ dim C Hom ( A , E [ n ]) ≤ n ∈ Z ∑ dim C Hom ( U ( A ) , U ( E ) [ n ]) ≤ C U ( A ) m σ ( U ( E )) = C U ( A ) m σ H ( E ) . This proves the total mass–Hom bound. □
Corollary 4.4. With the notation of Proposition 4.2 , suppose the category T \mathcal T T σ T = ( Z T , P T ) \sigma_{\mathcal T}=(Z_{\mathcal T},\mathcal P_{\mathcal T}) σ T = ( Z T , P T ) v : K 0 ( T ) → Λ v:K_0(\mathcal T)\to\Lambda v : K 0 ( T ) → Λ σ T \sigma_{\mathcal T} σ T ( τ i , 2 / d ) (\tau_i,2/d) ( τ i , 2/ d ) τ i \tau_i τ i U : D ≃ T H → T U:\mathcal D\simeq\mathcal T^H\to\mathcal T U : D ≃ T H → T
Z D = Z T ∘ U ∗ , P D ( ϕ ) = { E : U ( E ) ∈ P T ( ϕ ) } Z_{\mathcal D}=Z_{\mathcal T}\circ U_*,\qquad\mathcal P_{\mathcal D}(\phi)=\{E:U(E)\in\mathcal P_{\mathcal T}(\phi)\} Z D = Z T ∘ U ∗ , P D ( ϕ ) = { E : U ( E ) ∈ P T ( ϕ )} define a numerical stability condition on D = MF Z ( W ) \mathcal D=\operatorname{MF}^{\mathbb Z}(W) D = MF Z ( W ) Λ D = im ( v ∘ U ∗ ) ⊆ Λ \Lambda_{\mathcal D}=\operatorname{im}(v\circ U_*)\subseteq\Lambda Λ D = im ( v ∘ U ∗ ) ⊆ Λ ( τ , 2 / d ) (\tau,2/d) ( τ , 2/ d ) τ \tau τ σ T \sigma_{\mathcal T} σ T
Proof. The shifts τ i \tau_i τ i τ i τ r − 1 \tau_i\tau_r^{-1} τ i τ r − 1 H H H σ T \sigma_{\mathcal T} σ T Lemma 4.3 gives the stated condition, including its support lattice, numericality, and the mass–Hom bound. By Proposition 4.2 , the ordinary grading shift is induced by τ r \tau_r τ r U τ ≃ τ r U U\tau\simeq\tau_rU U τ ≃ τ r U ( τ , 2 / d ) (\tau,2/d) ( τ , 2/ d )
To apply Corollary 4.4 , we need a stability condition on the tensor product that is of Gepner type ( τ i , 2 / d ) (\tau_i,2/d) ( τ i , 2/ d ) Theorem 3.7 lets us add one quiver tensor factor at a time while preserving these Gepner properties. Fix d ≥ 2 d\geq2 d ≥ 2 Q d Q_d Q d Q d Q_d Q d
r j = sin ( j π / d ) sin ( π / d ) , q = − e π i / d , a j = { r j , j odd , q r j , j even . (39) r_j=\frac{\sin(j\pi/d)}{\sin(\pi/d)},\qquad q=-e^{\pi i/d},\qquad a_j=\begin{cases}r_j,&j\ \text{odd},\\qr_j,&j\ \text{even}.\end{cases} \tag*{(39)} r j = sin ( π / d ) sin ( j π / d ) , q = − e π i / d , a j = { r j , q r j , j odd , j even . (39) Proposition 4.5. For every d ≥ 2 d\geq2 d ≥ 2 M ≥ 1 M\geq1 M ≥ 1 D d , M = Q d ⊗ M \mathcal D_{d,M}=\mathcal Q_d^{\otimes M} D d , M = Q d ⊗ M Σ d , M = ( Z d , M , P d , M ) \Sigma_{d,M}=(Z_{d,M},\mathcal P_{d,M}) Σ d , M = ( Z d , M , P d , M ) K n u m ( D d , M ) ≃ Z ( d − 1 ) M K_{\mathrm{num}}(\mathcal D_{d,M})\simeq\mathbb Z^{(d-1)M} K num ( D d , M ) ≃ Z ( d − 1 ) M τ i \tau_i τ i S Q d − 1 [ 1 ] S_{\mathcal Q_d}^{-1}[1] S Q d − 1 [ 1 ] i i i Σ d , M \Sigma_{d,M} Σ d , M ( τ i , 2 / d ) (\tau_i,2/d) ( τ i , 2/ d ) i i i
Z d , M ( E ) = − ∑ α ∈ { 1 , … , d − 1 } M ( ∏ i = 1 M a α i ) n α ( E ) , (40) Z_{d,M}(E)=-\sum_{\alpha\in\{1,\ldots,d-1\}^M}\left(\prod_{i=1}^{M}a_{\alpha_i}\right)n_\alpha(E), \tag*{(40)} Z d , M ( E ) = − α ∈ { 1 , … , d − 1 } M ∑ ( i = 1 ∏ M a α i ) n α ( E ) , (40) where α \alpha α n α ( E ) = ∑ k ( − 1 ) k dim H k ( E α ) n_\alpha(E)=\sum_k(-1)^k\dim H^k(E_\alpha) n α ( E ) = ∑ k ( − 1 ) k dim H k ( E α ) d = 2 d=2 d = 2 Q 2 = Perf ( C ) \mathcal Q_2=\operatorname{Perf}(\mathbb C) Q 2 = Perf ( C ) a 1 = 1 a_1=1 a 1 = 1
Proof. For d = 2 d=2 d = 2 Perf ( C ) \operatorname{Perf}(\mathbb C) Perf ( C ) τ i \tau_i τ i [ 1 ] [1] [ 1 ] − dim -\dim − dim
∑ n ∈ Z dim C Hom ( C , E [ n ] ) = ∑ n ∈ Z dim C H n ( E ) = m ( E ) , \sum_{n\in\mathbb Z}\dim_{\mathbb C}\operatorname{Hom}(\mathbb C,E[n])=\sum_{n\in\mathbb Z}\dim_{\mathbb C}H^n(E)=m(E), n ∈ Z ∑ dim C Hom ( C , E [ n ]) = n ∈ Z ∑ dim C H n ( E ) = m ( E ) , giving the mass–Hom bound.
We now assume d ≥ 3 d\geq3 d ≥ 3 Perf ( C ) \operatorname{Perf}(\mathbb C) Perf ( C ) Theorem 3.7 M M M Q ^ d \widehat{\mathcal Q}_d Q d Q ^ d ⊗ M \widehat{\mathcal Q}_d^{\otimes M} Q d ⊗ M ( R i , 1 / d ) (R_i,1/d) ( R i , 1/ d )
We next choose the orientation Q d Q_d Q d Q ^ d \widehat{Q}_d Q d Q d Q_d Q d Q d o p Q_d^{\mathrm{op}} Q d op M M M 2 M 2^M 2 M D d , M \mathcal{D}_{d,M} D d , M Q d Q_d Q d R i 2 R_i^2 R i 2 S Q d − 1 [ 1 ] S_{Q_d}^{-1}[1] S Q d − 1 [ 1 ] i i i τ i = R i 2 \tau_i=R_i^2 τ i = R i 2 ( τ i , 2 / d ) (\tau_i,2/d) ( τ i , 2/ d ) i i i
It remains to identify the charge and the lattice. Adding a factor multiplies the coefficient at its j j j a j a_j a j − dim -\dim − dim K 0 ( D d , M ) K_0(\mathcal{D}_{d,M}) K 0 ( D d , M ) Z ( d − 1 ) M \mathbb{Z}^{(d-1)M} Z ( d − 1 ) M Q d Q_d Q d 1 1 1
Proof of Theorem 4.1. By Proposition 3.3, the enhanced equivalence MF g r ( x i d ) ≃ Q d \operatorname{MF}^{\mathrm{gr}}(x_i^d)\simeq\mathcal{Q}_d MF gr ( x i d ) ≃ Q d S Q d − 1 [ 1 ] S_{Q_d}^{-1}[1] S Q d − 1 [ 1 ] ( τ i , 2 / d ) (\tau_i,2/d) ( τ i , 2/ d ) i i i f i = x i d f_i=x_i^d f i = x i d MF g r ( F d , M ) \operatorname{MF}^{\mathrm{gr}}(F_{d,M}) MF gr ( F d , M ) M = 1 M=1 M = 1 M ≥ 2 M\geq2 M ≥ 2
Restriction to weighted hyperplanes
We now pass from Fermat hypersurfaces to their smooth linear sections. Derived restriction and extension of scalars give adjoint functors between the corresponding singularity categories.
Lemma 5.1. Let C C C D D D
F : C ⟶ D , R : D ⟶ C F:C\longrightarrow D,\qquad R:D\longrightarrow C F : C ⟶ D , R : D ⟶ C be exact dg functors with F ⊣ R F\dashv R F ⊣ R F F F D D D σ = ( Z , P ) \sigma=(Z,\mathcal{P}) σ = ( Z , P ) C C C v : K 0 ( C ) ⟶ Λ v:K_0(C)\longrightarrow\Lambda v : K 0 ( C ) ⟶ Λ Λ \Lambda Λ
( R ∘ F ) ( P ( ϕ ) ) ⊆ P [ ϕ , ∞ ) (41) (R\circ F)(\mathcal{P}(\phi))\subseteq\mathcal{P}[\phi,\infty) \tag*{(41)} ( R ∘ F ) ( P ( ϕ )) ⊆ P [ ϕ , ∞ ) (41) for every ϕ ∈ R \phi\in\mathbb{R} ϕ ∈ R D D D
P D ( ϕ ) = { E ∈ D : R ( E ) ∈ P ( ϕ ) } , (42) \mathcal{P}_D(\phi)=\{E\in D:R(E)\in\mathcal{P}(\phi)\}, \tag*{(42)} P D ( ϕ ) = { E ∈ D : R ( E ) ∈ P ( ϕ )} , (42) Z D ( E ) = Z ( R ( E ) ) . (43) Z_D(E)=Z(R(E)). \tag*{(43)} Z D ( E ) = Z ( R ( E )) . (43) It satisfies the support property relative to the class map
v D = v ∘ R ∗ : K 0 ( D ) ⟶ Λ D : = im ( v ∘ R ∗ ) ⊆ Λ . v_D=v\circ R_*:K_0(D)\longrightarrow\Lambda_D:=\operatorname{im}(v\circ R_*)\subseteq\Lambda. v D = v ∘ R ∗ : K 0 ( D ) ⟶ Λ D := im ( v ∘ R ∗ ) ⊆ Λ. Moreover, R R R I ⊆ R I\subseteq\mathbb{R} I ⊆ R E ∈ P D ( I ) E\in\mathcal{P}_D(I) E ∈ P D ( I ) R ( E ) ∈ P ( I ) R(E)\in\mathcal{P}(I) R ( E ) ∈ P ( I ) σ D = ( Z D , P D ) \sigma_D=(Z_D,\mathcal{P}_D) σ D = ( Z D , P D ) m σ D ( E ) = m σ ( R ( E ) ) m_{\sigma_D}(E)=m_\sigma(R(E)) m σ D ( E ) = m σ ( R ( E )) σ \sigma σ σ D \sigma_D σ D
Proof. We adapt Polishchuk’s induction argument [36 Theorem 2.1.2] ; compare [29 Theorem A.17 and Corollary A.23] . First extend the adjunction to
F : Ind ( C ) ⟶ Ind ( D ) , R : Ind ( D ) ⟶ Ind ( C ) . F:\operatorname{Ind}(C)\longrightarrow\operatorname{Ind}(D),\qquad R:\operatorname{Ind}(D)\longrightarrow\operatorname{Ind}(C). F : Ind ( C ) ⟶ Ind ( D ) , R : Ind ( D ) ⟶ Ind ( C ) . Here Ind ( C ) \operatorname{Ind}(\mathcal{C}) Ind ( C ) Ind ( D ) \operatorname{Ind}(\mathcal{D}) Ind ( D ) C \mathcal{C} C D \mathcal{D} D § 2.2 . The extended right adjoint is particularly concrete: it restricts a dg module along F F F
( R M ) ( A ) = M ( F ( A ) ) . (RM)(A)=M(F(A)). ( R M ) ( A ) = M ( F ( A )) . Restriction preserves quasi-isomorphisms and arbitrary coproducts because both are computed at each object. Its left adjoint is derived extension of scalars, which sends the representable module h A h_A h A h F ( A ) h_{F(A)} h F ( A ) F F F [25 Section 3.8] . For B ∈ D B \in\mathcal{D} B ∈ D
( R h B ) ( A ) = Hom D ∙ ( F ( A ) , B ) ≃ Hom C ∙ ( A , R ( B ) ) . (Rh_B)(A)=\operatorname{Hom}^{\bullet}_{\mathcal{D}}(F(A),B)\simeq\operatorname{Hom}^{\bullet}_{\mathcal{C}}(A,R(B)). ( R h B ) ( A ) = Hom D ∙ ( F ( A ) , B ) ≃ Hom C ∙ ( A , R ( B )) . Thus the restricted module R h B Rh_B R h B R ( B ) R(B) R ( B ) R R R R ( E ) = 0 R(E)=0 R ( E ) = 0
Hom ( F ( A ) , E [ k ] ) = Hom ( A , R ( E ) [ k ] ) = 0 ( A ∈ C , k ∈ Z ) . \operatorname{Hom}(F(A),E[k])=\operatorname{Hom}(A,R(E)[k])=0 \qquad(A\in\mathcal{C},\ k\in\mathbb{Z}). Hom ( F ( A ) , E [ k ]) = Hom ( A , R ( E ) [ k ]) = 0 ( A ∈ C , k ∈ Z ) . Since F ( C ) F(\mathcal{C}) F ( C ) D \mathcal{D} D D \mathcal{D} D F ( A ) F(A) F ( A ) E E E E = 0 E=0 E = 0 E E E R ( E ) R(E) R ( E ) A ∈ C A\in\mathcal{C} A ∈ C
⨁ k Hom ( F ( A ) , E [ k ] ) ≃ ⨁ k Hom ( A , R ( E ) [ k ] ) . \bigoplus_k\operatorname{Hom}(F(A),E[k])\simeq\bigoplus_k\operatorname{Hom}(A,R(E)[k]). k ⨁ Hom ( F ( A ) , E [ k ]) ≃ k ⨁ Hom ( A , R ( E ) [ k ]) . The right-hand side is finite dimensional by properness of C \mathcal{C} C D \mathcal{D} D F ( A ) F(A) F ( A ) [44 Lemma 2.8(3)] . This criterion makes E E E
Fix a ∈ R a\in\mathbb{R} a ∈ R a a a C \mathcal{C} C P ( ( a , ∞ ) ) \mathcal{P}((a,\infty)) P (( a , ∞ )) P ( ( − ∞ , a ] ) \mathcal{P}((-\infty,a]) P (( − ∞ , a ]) [1 ] is an aisle [1 Theorem A.1] . Choose representatives of the isomorphism classes in P ( ( a , ∞ ) ) \mathcal{P}((a,\infty)) P (( a , ∞ )) Ind ( C ) \operatorname{Ind}(\mathcal{C}) Ind ( C ) F F F Ind ( D ) \operatorname{Ind}(\mathcal{D}) Ind ( D )
U a ⊂ Ind ( C ) , generated by P ( ( a , ∞ ) ) , \mathcal{U}_a\subset\operatorname{Ind}(\mathcal{C}),\qquad\text{generated by }\mathcal{P}((a,\infty)), U a ⊂ Ind ( C ) , generated by P (( a , ∞ )) , V a ⊂ Ind ( D ) , generated by F ( P ( ( a , ∞ ) ) ) . \mathcal{V}_a\subset\operatorname{Ind}(\mathcal{D}),\qquad\text{generated by }F\bigl(\mathcal{P}((a,\infty))\bigr). V a ⊂ Ind ( D ) , generated by F ( P (( a , ∞ )) ) . In both cases, “generated” means closure under the three operations just listed. The t-structure with aisle U a \mathcal{U}_a U a C \mathcal{C} C U a \mathcal{U}_a U a U a ⊥ \mathcal{U}_a^\perp U a ⊥ Ind ( C ) \operatorname{Ind}(\mathcal{C}) Ind ( C ) [36 Lemma 2.1.1] .
Adjunction identifies the other right orthogonal explicitly:
V a ⊥ = { E : Hom ( F ( A ) , E ) = 0 for all A ∈ P ( ( a , ∞ ) ) } = R − 1 ( U a ⊥ ) . \mathcal{V}_a^\perp=\{E:\operatorname{Hom}(F(A),E)=0\text{ for all }A\in\mathcal{P}((a,\infty))\}=R^{-1}(\mathcal{U}_a^\perp). V a ⊥ = { E : Hom ( F ( A ) , E ) = 0 for all A ∈ P (( a , ∞ ))} = R − 1 ( U a ⊥ ) . Every object of P ( ( a , ∞ ) ) \mathcal{P}((a,\infty)) P (( a , ∞ )) a a a (41) gives R F ( P ( ( a , ∞ ) ) ) ⊂ P ( ( a , ∞ ) ) RF(\mathcal{P}((a,\infty)))\subset\mathcal{P}((a,\infty)) R F ( P (( a , ∞ ))) ⊂ P (( a , ∞ )) R R R R ( V a ) ⊂ U a R(\mathcal{V}_a)\subset\mathcal{U}_a R ( V a ) ⊂ U a R R R V a \mathcal{V}_a V a U a \mathcal{U}_a U a U a ⊥ \mathcal{U}_a^\perp U a ⊥ R R R R R R
For E ∈ D E\in\mathcal{D} E ∈ D R ( E ) R(E) R ( E ) C \mathcal{C} C U a \mathcal{U}_a U a C \mathcal{C} C R R R E E E V a \mathcal{V}_a V a R ( M ) R(M) R ( M ) M M M Ind ( D ) \operatorname{Ind}(\mathcal{D}) Ind ( D ) D \mathcal{D} D E E E D \mathcal{D} D Ind ( D ) \operatorname{Ind}(\mathcal{D}) Ind ( D ) D \mathcal{D} D
V a ∩ D = { E ∈ D : R ( E ) ∈ P ( ( a , ∞ ) ) } , \mathcal{V}_a\cap\mathcal{D}=\{E\in\mathcal{D}:R(E)\in\mathcal{P}((a,\infty))\}, V a ∩ D = { E ∈ D : R ( E ) ∈ P (( a , ∞ ))} , V a ⊥ ∩ D = { E ∈ D : R ( E ) ∈ P ( ( − ∞ , a ] ) } . \mathcal{V}_a^\perp\cap\mathcal{D}=\{E\in\mathcal{D}:R(E)\in\mathcal{P}((-\infty,a])\}. V a ⊥ ∩ D = { E ∈ D : R ( E ) ∈ P (( − ∞ , a ])} . For the first equality, if R ( E ) R(E) R ( E ) E E E R R R C \mathcal{C} C P ( [ a , ∞ ) ) \mathcal{P}([a,\infty)) P ([ a , ∞ )) P ( ( − ∞ , a ) ) \mathcal{P}((-\infty,a)) P (( − ∞ , a )) D \mathcal{D} D a a a R R R (42) follows because R R R E ∈ P D ( ϕ ) E \in\mathcal{P}_{\mathcal{D}}(\phi) E ∈ P D ( ϕ ) E ′ ∈ P D ( ψ ) E' \in\mathcal{P}_{\mathcal{D}}(\psi) E ′ ∈ P D ( ψ ) ϕ > ψ \phi> \psi ϕ > ψ a a a ϕ > a ≥ ψ \phi> a \geq\psi ϕ > a ≥ ψ E E E E ′ E' E ′ Hom ( E , E ′ ) = 0 \operatorname{Hom}(E,E')=0 Hom ( E , E ′ ) = 0
For 0 ≠ E ∈ D 0 \neq E \in\mathcal{D} 0 = E ∈ D ϕ 1 > ⋯ > ϕ s \phi_1 > \cdots> \phi_s ϕ 1 > ⋯ > ϕ s R ( E ) R(E) R ( E ) E E E [ ϕ 1 , ∞ ) [\phi_1,\infty) [ ϕ 1 , ∞ ) R R R R ( E ) R(E) R ( E ) P D ( ϕ 1 ) \mathcal{P}_{\mathcal{D}}(\phi_1) P D ( ϕ 1 ) [ ϕ 2 , ∞ ) [\phi_2,\infty) [ ϕ 2 , ∞ ) j j j P D ( ϕ j ) \mathcal{P}_{\mathcal{D}}(\phi_j) P D ( ϕ j ) R ( E ) R(E) R ( E ) s s s R R R E E E E E E R ( E ) R(E) R ( E )
If 0 ≠ E ∈ P D ( ϕ ) 0 \neq E \in\mathcal{P}_{\mathcal{D}}(\phi) 0 = E ∈ P D ( ϕ ) R ( E ) R(E) R ( E ) ϕ \phi ϕ Z D ( E ) = Z ( R ( E ) ) ∈ R > 0 e i π ϕ Z_{\mathcal{D}}(E)=Z(R(E)) \in\mathbb{R}_{>0}e^{i\pi\phi} Z D ( E ) = Z ( R ( E )) ∈ R > 0 e iπ ϕ C C C
∥ v D ( E ) ∥ = ∥ v ( R ( E ) ) ∥ ≤ C ∣ Z ( R ( E ) ) ∣ = C ∣ Z D ( E ) ∣ , \lVert v_{\mathcal{D}}(E)\rVert=\lVert v(R(E))\rVert\leq C\lvert Z(R(E))\rvert=C\lvert Z_{\mathcal{D}}(E)\rvert, ∥ v D ( E )∥ = ∥ v ( R ( E ))∥ ≤ C ∣ Z ( R ( E ))∣ = C ∣ Z D ( E )∣ , proving the support inequality on the finite-rank image lattice Λ D \Lambda_{\mathcal{D}} Λ D
Preservation of HN filtrations and equality of charges give m σ D ( E ) = m σ ( R ( E ) ) m_{\sigma_{\mathcal{D}}}(E)=m_{\sigma}(R(E)) m σ D ( E ) = m σ ( R ( E )) G G G C \mathcal{C} C
∑ n ∈ Z dim C Hom ( G , A [ n ] ) ≤ C G m σ ( A ) for every A ∈ C . \sum_{n\in\mathbb{Z}}\dim_{\mathbb{C}}\operatorname{Hom}(G,A[n])\leq C_Gm_{\sigma}(A)\qquad\text{for every }A\in\mathcal{C}. n ∈ Z ∑ dim C Hom ( G , A [ n ]) ≤ C G m σ ( A ) for every A ∈ C . Since F ( C ) F(\mathcal{C}) F ( C ) D \mathcal{D} D F ( G ) F(G) F ( G )
∑ n ∈ Z dim C Hom ( F ( G ) , E [ n ] ) = ∑ n ∈ Z dim C Hom ( G , R ( E ) [ n ] ) ≤ C G m σ ( R ( E ) ) = C G m σ D ( E ) , \begin{aligned}
\sum_{n\in\mathbb{Z}}\dim_{\mathbb{C}}\operatorname{Hom}(F(G),E[n])
&=\sum_{n\in\mathbb{Z}}\dim_{\mathbb{C}}\operatorname{Hom}(G,R(E)[n])\\
&\leq C_Gm_{\sigma}(R(E))=C_Gm_{\sigma_{\mathcal{D}}}(E),
\end{aligned} n ∈ Z ∑ dim C Hom ( F ( G ) , E [ n ]) = n ∈ Z ∑ dim C Hom ( G , R ( E ) [ n ]) ≤ C G m σ ( R ( E )) = C G m σ D ( E ) , and Lemma 2.1 gives the total mass–Hom bound. □
Let S = C [ x 1 , … , x N ] S=\mathbb{C}[x_1,\ldots,x_N] S = C [ x 1 , … , x N ] W ∈ S W\in S W ∈ S d d d z z z a a a
A = S / ( W ) , B = A / ( z ) , w = W ∣ z = 0 . A=S/(W),\qquad B=A/(z),\qquad w=W|_{z=0}. A = S / ( W ) , B = A / ( z ) , w = W ∣ z = 0 . Assume that w w w w ≠ 0 w\neq0 w = 0 z z z W W W z z z A A A
0 ⟶ A ( − a ) → z A ⟶ B ⟶ 0. (44) 0\longrightarrow A(-a)\xrightarrow{z}A\longrightarrow B\longrightarrow0. \tag*{(44)} 0 ⟶ A ( − a ) z A ⟶ B ⟶ 0. (44) Since the origin is a critical point of W W W W ∈ ( x 1 , … , x N ) 2 W\in(x_1,\ldots,x_N)^2 W ∈ ( x 1 , … , x N ) 2 w w w
C = D s g g r ( A ) ≃ MF g r ( W ) , D = D s g g r ( B ) ≃ MF g r ( w ) . \mathcal{C}=D_{\mathrm{sg}}^{\mathrm{gr}}(A)\simeq\operatorname{MF}^{\mathrm{gr}}(W),\qquad\mathcal{D}=D_{\mathrm{sg}}^{\mathrm{gr}}(B)\simeq\operatorname{MF}^{\mathrm{gr}}(w). C = D sg gr ( A ) ≃ MF gr ( W ) , D = D sg gr ( B ) ≃ MF gr ( w ) . The weighted form of Orlov’s theorem shows that these categories are smooth and proper: they are either admissible components of the derived categories of smooth proper hypersurface Deligne–Mumford stacks or are obtained from those derived categories by adjoining finitely many exceptional objects. See [34 Theorem 2.5 and Remark 3.12] and [2 Theorem 6.13] . Denote internal grading shift by τ \tau τ τ d ≃ [ 2 ] \tau^d\simeq[2] τ d ≃ [ 2 ]
Derived extension of scalars F = B ⊗ A L − F=B\otimes_A^{\mathbf{L}}- F = B ⊗ A L − R : D → C R:\mathcal{D}\to\mathcal{C} R : D → C (44) . This finite free resolution of B B B A A A F F F R R R B B B A A A B B B A A A F F F A A A B B B
F : C ⟶ D , R : D ⟶ C , F ⊣ R F : C \longrightarrow D,\qquad R : D \longrightarrow C,\qquad F \dashv R F : C ⟶ D , R : D ⟶ C , F ⊣ R by [34 Lemma 1.1] , and extends to their idempotent completions. Both functors commute with internal grading shifts.
Lemma 5.2. Let k A k_A k A k B k_B k B
F ( k A ) ≃ k B ⊕ k B ( − a ) [ 1 ] in D g r b ( B ) . F(k_A) \simeq k_B \oplus k_B(-a)[1]\qquad\text{in }D_{\mathrm{gr}}^b(B). F ( k A ) ≃ k B ⊕ k B ( − a ) [ 1 ] in D gr b ( B ) . Here D g r b ( B ) D_{\mathrm{gr}}^b(B) D gr b ( B ) B B B F ( C ) F(C) F ( C ) D D D
Proof. Resolution (43) computes the cohomology of F ( k A ) F(k_A) F ( k A ) k B k_B k B k B ( − a ) k_B(-a) k B ( − a ) − 1 -1 − 1
k B ( − a ) [ 1 ] ⟶ F ( k A ) ⟶ k B → e k B ( − a ) [ 2 ] . (45) k_B(-a)[1]\longrightarrow F(k_A)\longrightarrow k_B \xrightarrow{e} k_B(-a)[2]. \tag*{(45)} k B ( − a ) [ 1 ] ⟶ F ( k A ) ⟶ k B e k B ( − a ) [ 2 ] . (45) We show that e e e Hom ( − , k B ( − a ) [ 1 ] ) \operatorname{Hom}(-,k_B(-a)[1]) Hom ( − , k B ( − a ) [ 1 ])
0 ⟶ Ext g r , B 1 ( k B , k B ( − a ) ) ⟶ Ext g r , A 1 ( k A , k A ( − a ) ) ⟶ C → e Ext g r , B 2 ( k B , k B ( − a ) ) . (46) 0\longrightarrow\operatorname{Ext}^{1}_{\mathrm{gr},B}(k_B,k_B(-a))\longrightarrow\operatorname{Ext}^{1}_{\mathrm{gr},A}(k_A,k_A(-a))\longrightarrow\mathbb{C}\xrightarrow{e}\operatorname{Ext}^{2}_{\mathrm{gr},B}(k_B,k_B(-a)). \tag*{(46)} 0 ⟶ Ext gr , B 1 ( k B , k B ( − a )) ⟶ Ext gr , A 1 ( k A , k A ( − a )) ⟶ C e Ext gr , B 2 ( k B , k B ( − a )) . (46) The last map sends 1 1 1 e e e a a a a a a W W W w w w A A A B B B z z z a a a 1 ^1 1 C \mathbb{C} C e = 0 e=0 e = 0
The graded residue fields k B ( j ) k_B(j) k B ( j ) D D D dim B > 0 \dim B>0 dim B > 0 B B B [2 Corollary 4.14] , applied to the grading action of G m \mathbb{G}_m G m Spec B \operatorname{Spec} B Spec B 1 ( d ) = [ 2 ] (d)=[2] ( d ) = [ 2 ] k B ( j ) k_B(j) k B ( j ) 0 ≤ j < d 0\leq j<d 0 ≤ j < d F ( k A ( j ) ) F(k_A(j)) F ( k A ( j )) F F F D D D
Proposition 5.3. Suppose a stability condition ( Z , P ) (Z,\mathcal{P}) ( Z , P ) C C C v : K 0 ( C ) → Λ v:K_0(C)\to\Lambda v : K 0 ( C ) → Λ
τ − a [ 1 ] P ( ϕ ) ⊆ P ( [ ϕ , ∞ ) ) ( ϕ ∈ R ) . (47) \tau^{-a}[1]\mathcal{P}(\phi)\subseteq\mathcal{P}([\phi,\infty))\qquad(\phi\in\mathbb{R}). \tag*{(47)} τ − a [ 1 ] P ( ϕ ) ⊆ P ([ ϕ , ∞ )) ( ϕ ∈ R ) . (47) Then restriction of scalars induces a stability condition on D D D im ( v ∘ R ∗ ) ⊂ Λ \operatorname{im}(v\circ R_*)\subset\Lambda im ( v ∘ R ∗ ) ⊂ Λ ( Z , P ) (Z,\mathcal{P}) ( Z , P ) ( τ , 2 / d ) (\tau,2/d) ( τ , 2/ d ) a ≤ d / 2 a\leq d/2 a ≤ d /2
Proof. Tensoring (43) with an object gives the functorial triangle
τ − a E ⟶ E ⟶ R F ( E ) ⟶ τ − a E [ 1 ] . (48) \tau^{-a}E \longrightarrow E \longrightarrow RF(E)\longrightarrow\tau^{-a}E[1]. \tag*{(48)} τ − a E ⟶ E ⟶ R F ( E ) ⟶ τ − a E [ 1 ] . (48) Rotating this triangle expresses R F ( E ) RF(E) R F ( E ) τ − a E [ 1 ] \tau^{-a}E[1] τ − a E [ 1 ] E E E E ∈ P ( ϕ ) E\in\mathcal{P}(\phi) E ∈ P ( ϕ ) P ( [ ϕ , ∞ ) ) \mathcal{P}([\phi,\infty)) P ([ ϕ , ∞ )) F ( C ) F(C) F ( C ) D D D ( Z , P ) (Z,\mathcal{P}) ( Z , P ) ( τ , 2 / d ) (\tau,2/d) ( τ , 2/ d ) τ − a E [ 1 ] \tau^{-a}E[1] τ − a E [ 1 ] ϕ + 1 − 2 a / d \phi+1-2a/d ϕ + 1 − 2 a / d ϕ \phi ϕ a ≤ d / 2 a\leq d/2 a ≤ d /2 a = d / 2 a=d/2 a = d /2 P ( ϕ ) \mathcal{P}(\phi) P ( ϕ ) R R R τ \tau τ ( τ , 2 / d ) (\tau,2/d) ( τ , 2/ d )
The containment (46) also holds for the induced slicing P D \mathcal{P}_D P D E ∈ P D ( ϕ ) E \in\mathcal{P}_D(\phi) E ∈ P D ( ϕ ) R ( E ) ∈ P ( ϕ ) R(E) \in\mathcal{P}(\phi) R ( E ) ∈ P ( ϕ )
R ( τ − a E [ 1 ] ) ≅ τ − a R ( E ) [ 1 ] ∈ P ( [ ϕ , ∞ ) ) . R(\tau^{-a}E[1]) \cong\tau^{-a}R(E)[1] \in\mathcal{P}([\phi,\infty)). R ( τ − a E [ 1 ]) ≅ τ − a R ( E ) [ 1 ] ∈ P ([ ϕ , ∞ )) . The characterization of phase intervals in Lemma 5.1 now gives τ − a E [ 1 ] ∈ P D ( [ ϕ , ∞ ) ) \tau^{-a}E[1] \in\mathcal{P}_D([\phi,\infty)) τ − a E [ 1 ] ∈ P D ([ ϕ , ∞ ))
Lemma 5.4. Let f ∈ Sym d ( V ∗ ) f \in\operatorname{Sym}^{d}(V^*) f ∈ Sym d ( V ∗ ) d ≥ 2 d \geq2 d ≥ 2 X ⊂ P ( V ) X \subset\mathbb{P}(V) X ⊂ P ( V ) M M M P ( V ) ↪ P M − 1 \mathbb{P}(V) \hookrightarrow\mathbb{P}^{M-1} P ( V ) ↪ P M − 1 X X X
Y M = { y 1 d + ⋯ + y M d = 0 } ⊂ P M − 1 . Y_M=\left\{y_1^d+\cdots+y_M^d=0\right\}\subset\mathbb{P}^{M-1}. Y M = { y 1 d + ⋯ + y M d = 0 } ⊂ P M − 1 . Moreover, there is a flag of projective subspaces from the embedded P ( V ) \mathbb{P}(V) P ( V ) P M − 1 \mathbb{P}^{M-1} P M − 1 Y M Y_M Y M
Proof. A Waring decomposition realizes a hypersurface as a linear section of a Fermat hypersurface; see [12 § 1] . We recall the construction and choose smooth intermediate sections. Writing N = dim V N=\dim V N = dim V X = Y M ∩ L N ⊂ Y M ∩ L N + 1 ⊂ ⋯ ⊂ Y M ∩ L M = Y M X=Y_M\cap L_N\subset Y_M\cap L_{N+1}\subset\cdots\subset Y_M\cap L_M=Y_M X = Y M ∩ L N ⊂ Y M ∩ L N + 1 ⊂ ⋯ ⊂ Y M ∩ L M = Y M L k ≅ P k − 1 L_k\cong\mathbb{P}^{k-1} L k ≅ P k − 1 L k L_k L k L k + 1 L_{k+1} L k + 1 Y M ∩ L k Y_M\cap L_k Y M ∩ L k
Powers of linear forms span Sym d ( V ∗ ) \operatorname{Sym}^{d}(V^*) Sym d ( V ∗ )
m 1 ⋯ m d = 1 d ! ∑ I ⊂ { 1 , … , d } ( − 1 ) d − ∣ I ∣ ( ∑ i ∈ I m i ) d . m_1\cdots m_d=\frac{1}{d!}\sum_{I\subset\{1,\ldots,d\}}(-1)^{d-|I|}\left(\sum_{i\in I}m_i\right)^d. m 1 ⋯ m d = d ! 1 I ⊂ { 1 , … , d } ∑ ( − 1 ) d − ∣ I ∣ ( i ∈ I ∑ m i ) d . The coefficients can be absorbed into d d d f = ∑ i = 1 M 0 ℓ i d f=\sum_{i=1}^{M_0}\ell_i^d f = ∑ i = 1 M 0 ℓ i d ℓ i \ell_i ℓ i V ∗ V^* V ∗ v v v f ( v ) = 0 f(v)=0 f ( v ) = 0 d f ( v ) = d ∑ i ℓ i ( v ) d − 1 ℓ i = 0 df(v)=d\sum_i\ell_i(v)^{d-1}\ell_i=0 df ( v ) = d ∑ i ℓ i ( v ) d − 1 ℓ i = 0 v ↦ ( ℓ i ( v ) ) v\mapsto(\ell_i(v)) v ↦ ( ℓ i ( v )) M ≥ M 0 M\geq M_0 M ≥ M 0 X X X
Write L = P ( V ) L=\mathbb{P}(V) L = P ( V ) L L L f f f X X X L L L X X X M = N M=N M = N L M = P M − 1 L_M=\mathbb{P}^{M-1} L M = P M − 1 Y M ∩ L k Y_M\cap L_k Y M ∩ L k k > N + 1 k>N+1 k > N + 1 L k L_k L k L L L X X X [20 III, Corollary 10.9 and Remark 10.9.2] , a general such hyperplane L k − 1 L_{k-1} L k − 1 X X X X X X L N + 1 L_{N+1} L N + 1 L N = L L_N=L L N = L X X X
Proposition 5.5. Fix d ≥ 2 d\geq2 d ≥ 2 M M M MF g r ( ∑ i = 1 M y i d ) \operatorname{MF}^{\mathrm{gr}}\left(\sum_{i=1}^{M}y_i^d\right) MF gr ( ∑ i = 1 M y i d ) a = 1 a=1 a = 1 MF g r ( f ) \operatorname{MF}^{\mathrm{gr}}(f) MF gr ( f ) d d d { f = 0 } \{f=0\} { f = 0 } ( τ , 2 / d ) (\tau,2/d) ( τ , 2/ d )
Proof. Choose M M M a = 1 a=1 a = 1 ( τ , 2 / d ) (\tau,2/d) ( τ , 2/ d ) 1 ≤ d / 2 1\leq d/2 1 ≤ d /2
The following theorem implies Theorem 1.2.
Theorem 5.6. Give each variable x i x_i x i W ∈ C [ x 1 , … , x N ] W\in\mathbb{C}[x_1,\ldots,x_N] W ∈ C [ x 1 , … , x N ] d ≥ 2 d\geq2 d ≥ 2 MF g r ( W ) \operatorname{MF}^{\mathrm{gr}}(W) MF gr ( W ) ( τ , 2 / d ) (\tau,2/d) ( τ , 2/ d ) K n u m ( MF g r ( W ) ) K_{\mathrm{num}}(\operatorname{MF}^{\mathrm{gr}}(W)) K num ( MF gr ( W )) τ \tau τ
0 ) ⊂ P N − 1 0) \subset\mathbb{P}^{N-1} 0 ) ⊂ P N − 1 N > d N > d N > d Ku ( X ) \operatorname{Ku}(X) Ku ( X ) N = d N = d N = d D b ( X ) \operatorname{D}^{b}(X) D b ( X )
Proof. If N = 1 N = 1 N = 1 W W W x d x^{d} x d Theorem 4.1 . For N ≥ 2 N \ge2 N ≥ 2 X = ( W = 0 ) ⊂ P N − 1 X = (W = 0) \subset\mathbb{P}^{N-1} X = ( W = 0 ) ⊂ P N − 1 Theorem 4.1 supplies a stability condition of Gepner type ( τ , 2 / d ) (\tau, 2/d) ( τ , 2/ d ) τ − 1 [ 1 ] P ( ϕ ) = P ( ϕ + 1 − 2 / d ) ⊆ P ( [ ϕ , ∞ ) ) \tau^{-1}[1]\mathcal{P}(\phi) = \mathcal{P}(\phi+ 1 - 2/d) \subseteq\mathcal{P}([\phi,\infty)) τ − 1 [ 1 ] P ( ϕ ) = P ( ϕ + 1 − 2/ d ) ⊆ P ([ ϕ , ∞ )) d ≥ 2 d \ge2 d ≥ 2 ( τ , 2 / d ) (\tau, 2/d) ( τ , 2/ d ) MF g r ( W ) \operatorname{MF}^{\mathrm{gr}}(W) MF gr ( W ) F : C → D F : \mathcal{C} \to\mathcal{D} F : C → D R : D → C R : \mathcal{D} \to\mathcal{C} R : D → C χ C ( A , R ( E ) ) = χ D ( F ( A ) , E ) \chi_{\mathcal{C}}(A, R(E)) = \chi_{\mathcal{D}}(F(A), E) χ C ( A , R ( E )) = χ D ( F ( A ) , E ) R ∗ R_{*} R ∗ C \mathcal{C} C D \mathcal{D} D N ≥ 2 N \ge2 N ≥ 2 MF g r ( W ) \operatorname{MF}^{\mathrm{gr}}(W) MF gr ( W ) D b ( X ) \operatorname{D}^{b}(X) D b ( X ) D b ( X ) \operatorname{D}^{b}(X) D b ( X ) Lemma 2.1 gives the full numerical support property. Finally, Orlov’s theorem [34 Theorem 3.11] identifies MF g r ( W ) \operatorname{MF}^{\mathrm{gr}}(W) MF gr ( W ) Ku ( X ) \operatorname{Ku}(X) Ku ( X ) N > d N > d N > d D b ( X ) \operatorname{D}^{b}(X) D b ( X ) N = d N = d N = d
Proper moduli spaces
Proof of Corollary 1.3. By Orlov’s equivalence, C \mathcal{C} C Theorem 5.6 gives the full numerical support property and a total mass–Hom bound. Since the class map is numerical, σ \sigma σ [21 Example 2.29] . Over C \mathbb{C} C [21 Theorem 2.35 and Remark 2.36] . Choose a a a ϕ ∈ ( a , a + 1 ] \phi\in(a, a + 1] ϕ ∈ ( a , a + 1 ] P ( ( a , a + 1 ] ) \mathcal{P}((a, a + 1]) P (( a , a + 1 ]) v v v ϕ \phi ϕ [21 Proposition 2.41] , their stack is algebraic of finite type with affine diagonal and admits a proper good moduli space. □
Recall that a morphism π : M → M \pi: \mathcal{M} \to M π : M → M π ∗ \pi_{*} π ∗ O M → π ∗ O M \mathcal{O}_{M} \to\pi_{*}\mathcal{O}_{\mathcal{M}} O M → π ∗ O M
