Introduction
The moduli spaces M ‾ g , n \overline{\mathcal{M}}_{g,n} M g , n n n n g g g [8 , 20 ] are central objects of algebraic geometry, and their cohomology is the natural setting for intersection theory on moduli spaces, Gromov–Witten theory and quantum cohomology [22 ] . Much is known about this cohomology in special ranges. In genus zero, Keel described the cohomology ring and gave recursions for the Betti numbers [18 ] . In genus one, the even cohomology is tautological by a theorem of Petersen [23 ] , and the Hodge–Deligne polynomials can be computed explicitly [19 ] . In arbitrary genus, the cohomology in low degrees has been analyzed in recent work of Bergström, Faber and Payne [2 ] and of Canning, Larson and Payne [4 ] . We refer to [9 ] for a survey of tautological and non-tautological classes.
In this paper we take a complementary, global point of view: we fix the genus and let the number of markings grow. The total Betti number of M ‾ g , n \overline{\mathcal{M}}_{g,n} M g , n n ! ( e − 2 ) − n n!(e-2)^{-n} n ! ( e − 2 ) − n n n n shape of the cohomology:
how the Betti numbers are distributed over the degrees, and how large an individual Betti number is;
what proportion of the cohomology is tautological, and what proportion lies off the Hodge diagonal;
which log-concavity properties the Betti numbers satisfy near the middle degree.
Questions of this kind are classical for combinatorial sequences, where Gaussian limit laws are ubiquitous [10 ] . In earlier work we established Gaussian limit laws and central ultra-log-concavity for M ‾ 0 , n \overline{\mathcal{M}}_{0,n} M 0 , n P 1 [ n ] \mathbb{P}^{1}[n] P 1 [ n ] [5 ] , and for the Fulton–MacPherson compactifications of configuration spaces of smooth projective curves [6 ] . Here we treat all three questions in positive genus.
Throughout, g ≥ 1 g \ge1 g ≥ 1 g g g M ‾ g , n \overline{\mathcal{M}}_{g,n} M g , n
M ‾ g , 0 ∣ n = M ‾ g , ( 1 / n , … , 1 / n ) \overline{\mathcal{M}}_{g,0|n}=\overline{\mathcal{M}}_{g,(1/n,\ldots,1/n)} M g , 0∣ n = M g , ( 1/ n , … , 1/ n ) of [13 ] , whose markings have weight 1 / n 1/n 1/ n M ‾ g , n \overline{\mathcal{M}}_{g,n} M g , n heavy space and M ‾ g , 0 ∣ n \overline{\mathcal{M}}_{g,0|n} M g , 0∣ n light space. Both are smooth and proper Deligne–Mumford stacks of dimension
d n = n + 3 g − 3 , d_n=n+3g-3, d n = n + 3 g − 3 , so their Betti numbers are palindromic by Poincaré duality. We study the light spaces using their geometric stratification and apply the heavy/light substitution of Kannan, Serpente and Yun [16 ] to obtain results for M ‾ g , n \overline{\mathcal{M}}_{g,n} M g , n
Gaussian behavior of the Betti numbers
Let X g , n H X^H_{g,n} X g , n H k k k b 2 k ( M ‾ g , n ) b_{2k}(\overline{\mathcal{M}}_{g,n}) b 2 k ( M g , n ) X g , n L X^L_{g,n} X g , n L M ‾ g , 0 ∣ n \overline{\mathcal{M}}_{g,0|n} M g , 0∣ n X g , n H X^H_{g,n} X g , n H d n / 2 d_n/2 d n /2
ν = 3 − e 6 ( e − 2 ) = 0.0653685 … , \nu=\frac{3-e}{6(e-2)}=0.0653685\ldots, ν = 6 ( e − 2 ) 3 − e = 0.0653685 … , and let
Φ ( x ) = ∫ − ∞ x 1 2 π e − t 2 / 2 d t \Phi(x)=\int_{-\infty}^{x}\frac{1}{\sqrt{2\pi}}e^{-t^2/2}\,dt Φ ( x ) = ∫ − ∞ x 2 π 1 e − t 2 /2 d t denote the standard normal distribution function.
Theorem A (Central limit theorem). Fix g ≥ 1 g \ge1 g ≥ 1 n → ∞ n \to\infty n → ∞ X g , n H X^H_{g,n} X g , n H X g , n L X^L_{g,n} X g , n L
Var ( X g , n H ) = ν n + ( g − 1 ) ( 11 − 4 e ) 12 ( e − 2 ) + O ( n − 1 / 2 ) , Var ( X g , n L ) = n − g + 1 12 + O ( n − 1 ) , \operatorname{Var}(X^H_{g,n})=\nu n+\frac{(g-1)(11-4e)}{12(e-2)}+O(n^{-1/2}), \qquad\operatorname{Var}(X^L_{g,n})=\frac{n-g+1}{12}+O(n^{-1}), Var ( X g , n H ) = ν n + 12 ( e − 2 ) ( g − 1 ) ( 11 − 4 e ) + O ( n − 1/2 ) , Var ( X g , n L ) = 12 n − g + 1 + O ( n − 1 ) , and, for J ∈ { H , L } J \in\{H,L\} J ∈ { H , L }
sup x ∈ R ∣ P ( X g , n J − d n / 2 Var ( X g , n J ) ≤ x ) − Φ ( x ) ∣ = O ( n − 1 / 2 ) . \sup_{x\in\mathbb{R}}\left|\mathbb{P}\left(\frac{X^J_{g,n}-d_n/2}{\sqrt{\operatorname{Var}(X^J_{g,n})}}\leq x\right)-\Phi(x)\right|=O(n^{-1/2}). x ∈ R sup P Var ( X g , n J ) X g , n J − d n /2 ≤ x − Φ ( x ) = O ( n − 1/2 ) . The same conclusions hold for the full Betti numbers b j b_j b j j j j d n d_n d n
The leading variance coefficients are independent of the genus. The coefficient ν \nu ν M ‾ 0 , n \overline{\mathcal{M}}_{0,n} M 0 , n [5 ] and of the Fulton–MacPherson spaces [6 ] , while 1 / 12 1/12 1/12 A ( t ) B ( t ) n A(t)B(t)^n A ( t ) B ( t ) n A ( t ) A(t) A ( t ) B ( t ) B(t) B ( t ) n n n
The Gaussian law of Theorem A concerns cumulative probabilities. We also prove a local limit theorem for the even Betti numbers (Theorem 5.3 ), giving a Gaussian approximation that is uniform over all degrees and includes higher-order corrections. Together with the asymptotics of the even Betti totals, these local estimates yield asymptotic formulas for individual even Betti numbers. They involve the weighted count
c g = ∑ Γ 1 ∣ Aut Γ ∣ . c_g=\sum_{\Gamma}\frac{1}{|\operatorname{Aut}\Gamma|}. c g = Γ ∑ ∣ Aut Γ∣ 1 . of connected trivalent graphs Γ \Gamma Γ g g g
∑ g ≥ 2 c g x g − 1 = log ∑ m ≥ 0 ( 6 m ) ! 2 3 m ( 3 m ) ! ( 3 ! ) 2 m ( 2 m ) ! x m , \sum_{g\geq2} c_g x^{g-1}=\log\sum_{m\geq0}\frac{(6m)!}{2^{3m}(3m)!(3!)^{2m}(2m)!}x^m, g ≥ 2 ∑ c g x g − 1 = log m ≥ 0 ∑ 2 3 m ( 3 m )! ( 3 ! ) 2 m ( 2 m )! ( 6 m )! x m , so that c 2 = 5 / 24 c_2=5/24 c 2 = 5/24 c 3 = 5 / 16 c_3=5/16 c 3 = 5/16
Theorem B (Individual Betti numbers). Fix g ≥ 1 g\geq1 g ≥ 1 x n , k = k − d n / 2 x_{n,k}=k-d_n/2 x n , k = k − d n /2 k k k
b 2 k ( M ‾ g , n ) = C g ( 1 ) 2 π ν n ! ( e − 2 ) − n n 3 g − 6 2 [ exp ( − x n , k 2 2 ν n ) + O ( n − 1 / 2 ) ] , b_{2k}(\overline{\mathcal M}_{g,n})=\frac{C_g(1)}{\sqrt{2\pi\nu}}\,n!(e-2)^{-n}n^{\frac{3g-6}{2}}\left[\exp\left(-\frac{x_{n,k}^2}{2\nu n}\right)+O(n^{-1/2})\right], b 2 k ( M g , n ) = 2 π ν C g ( 1 ) n ! ( e − 2 ) − n n 2 3 g − 6 [ exp ( − 2 ν n x n , k 2 ) + O ( n − 1/2 ) ] , b 2 k ( M ‾ g , 0 ∣ n ) = 6 π A g ( 1 ) n ! n 6 g − 9 2 [ exp ( − 6 x n , k 2 n ) + O ( n − 1 ) ] , b_{2k}(\overline{\mathcal M}_{g,0|n})=\sqrt{\frac{6}{\pi}}\,A_g(1)\,n!n^{\frac{6g-9}{2}}\left[\exp\left(-\frac{6x_{n,k}^2}{n}\right)+O(n^{-1})\right], b 2 k ( M g , 0∣ n ) = π 6 A g ( 1 ) n ! n 2 6 g − 9 [ exp ( − n 6 x n , k 2 ) + O ( n − 1 ) ] , where A 1 ( 1 ) = 1 / 2 A_1(1)=1/2 A 1 ( 1 ) = 1/2 C 1 ( 1 ) = 1 / 4 C_1(1)=1/4 C 1 ( 1 ) = 1/4 g ≥ 2 g\geq2 g ≥ 2
A g ( 1 ) = c g e − 2 ( g − 1 ) Γ ( 3 g − 3 ) , C g ( 1 ) = c g e − ( g − 1 ) / 2 Γ ( 3 g − 3 2 ) [ 2 ( e − 2 ) ] 3 g − 3 2 . A_g(1)=\frac{c_g e^{-2(g-1)}}{\Gamma(3g-3)},\qquad C_g(1)=\frac{c_g e^{-(g-1)/2}}{\Gamma\left(\frac{3g-3}{2}\right)[2(e-2)]^{\frac{3g-3}{2}}}. A g ( 1 ) = Γ ( 3 g − 3 ) c g e − 2 ( g − 1 ) , C g ( 1 ) = Γ ( 2 3 g − 3 ) [ 2 ( e − 2 ) ] 2 3 g − 3 c g e − ( g − 1 ) /2 . The asymptotics of the even Betti totals are established in Theorem 4.1 . The odd cohomology is negligible in comparison: it is a proportion O ( n − 1 / 2 ) O(n^{-1/2}) O ( n − 1/2 ) M ‾ g , n \overline{\mathcal M}_{g,n} M g , n O ( n − 1 ) O(n^{-1}) O ( n − 1 ) M ‾ g , 0 ∣ n \overline{\mathcal M}_{g,0|n} M g , 0∣ n [16 Question 5.3] .
Corollary C (Euler characteristics). For every fixed g ≥ 2 g\geq2 g ≥ 2
χ ( M ‾ g , 0 ∣ n ) χ ( M ‾ g , n ) = ( 2 ( e − 2 ) e ) 3 g − 3 2 Γ ( 3 g − 3 2 ) Γ ( 3 g − 3 ) n 3 g − 3 2 ( e − 2 ) n ( 1 + O ( n − 1 / 2 ) ) . \frac{\chi(\overline{\mathcal M}_{g,0|n})}{\chi(\overline{\mathcal M}_{g,n})}=\left(\frac{2(e-2)}{e}\right)^{\frac{3g-3}{2}}\frac{\Gamma\left(\frac{3g-3}{2}\right)}{\Gamma(3g-3)}n^{\frac{3g-3}{2}}(e-2)^n(1+O(n^{-1/2})). χ ( M g , n ) χ ( M g , 0∣ n ) = ( e 2 ( e − 2 ) ) 2 3 g − 3 Γ ( 3 g − 3 ) Γ ( 2 3 g − 3 ) n 2 3 g − 3 ( e − 2 ) n ( 1 + O ( n − 1/2 )) . In particular, the ratio tends to zero.
For g = 1 g=1 g = 1 2 ( e − 2 ) n 2(e-2)^n 2 ( e − 2 ) n [16 Corollary 5.2] . Our methods recover this.
Tautological classes
Let R H 2 k ( M ‾ g , n ) ⊂ H 2 k ( M ‾ g , n ) RH^{2k}(\overline{\mathcal M}_{g,n})\subset H^{2k}(\overline{\mathcal M}_{g,n}) R H 2 k ( M g , n ) ⊂ H 2 k ( M g , n ) [18 ] , and in genus one every even degree class is tautological [23 ] . For g ≥ 2 g\geq2 g ≥ 2 M ‾ 2 , 20 \overline{\mathcal M}_{2,20} M 2 , 20 [12 ] , and many further constructions are surveyed in [9 ] . Our next result shows that, nevertheless, non-tautological classes form a vanishing proportion of the even cohomology, of exact order n − 1 n^{-1} n − 1
Theorem D (Tautological classes). Fix g ≥ 2 g\geq2 g ≥ 2 0 < c g ′ ≤ C g ′ 0<c'_g\leq C'_g 0 < c g ′ ≤ C g ′ n n n
c g ′ n ≤ 1 − ∑ k dim R H 2 k ( M ‾ g , n ) ∑ k b 2 k ( M ‾ g , n ) ≤ C g ′ n . \frac{c'_g}{n}\leq1-\frac{\sum_k\dim RH^{2k}(\overline{\mathcal M}_{g,n})}{\sum_k b_{2k}(\overline{\mathcal M}_{g,n})}\leq\frac{C'_g}{n}. n c g ′ ≤ 1 − ∑ k b 2 k ( M g , n ) ∑ k dim R H 2 k ( M g , n ) ≤ n C g ′ . Consequently, the distribution of the graded dimensions of R H ∗ ( M ‾ g , n ) RH^*(\overline{\mathcal M}_{g,n}) R H ∗ ( M g , n ) O ( n − 1 ) O(n^{-1}) O ( n − 1 ) d n / 2 d_n/2 d n /2 ν n + O ( 1 ) \nu n+O(1) ν n + O ( 1 )
The two bounds have different origins. For the upper bound, we show that every even class restricting to zero on the complement of the preimage of the boundary strata of dimension at most one in M ‾ g \overline{\mathcal M}_g M g O ( n − 1 ) O(n^{-1}) O ( n − 1 ) ( p , q ) (p,q) ( p , q ) p − q = 22 p-q=22 p − q = 22 elliptic leaves , each carrying the weight-twelve cusp form Δ \Delta Δ ( k , k ) (k,k) ( k , k ) n − 1 n^{-1} n − 1 g ≥ 2 g \ge2 g ≥ 2 n n n M ‾ g , n \overline{\mathcal M}_{g,n} M g , n ( k , k ) (k,k) ( k , k )
Hodge types off the diagonal
For Y = M ‾ g , n Y=\overline{\mathcal M}_{g,n} Y = M g , n Y = M ‾ g , 0 ∣ n Y=\overline{\mathcal M}_{g,0|n} Y = M g , 0∣ n ( p , q ) (p,q) ( p , q ) h p , q ( Y ) / ∑ j b j ( Y ) h^{p,q}(Y)/\sum_j b_j(Y) h p , q ( Y ) / ∑ j b j ( Y ) D g , n H D^H_{g,n} D g , n H D g , n L D^L_{g,n} D g , n L p − q p-q p − q k ≥ 4 k \ge4 k ≥ 4 β k = dim S k ( S L 2 ( Z ) ) \beta_k=\dim S_k(\mathrm{SL}_2(\mathbb{Z})) β k = dim S k ( SL 2 ( Z )) k k k I m I_m I m
Z Q = ∑ m ≥ 3 m o d d β m + 1 m I m ( 2 ) , q m = q − m = β m + 1 m I m ( 2 ) 2 Z Q ( m ≥ 3 o d d ) , (1) Z_Q=\sum_{\substack{m\ge3\\m\ \mathrm{odd}}}\beta_{m+1}mI_m(2),\qquad q_m=q_{-m}=\frac{\beta_{m+1}mI_m(2)}{2Z_Q}\quad(m\ge3\ \mathrm{odd}), \tag*{(1)} Z Q = m ≥ 3 m odd ∑ β m + 1 m I m ( 2 ) , q m = q − m = 2 Z Q β m + 1 m I m ( 2 ) ( m ≥ 3 odd ) , (1) and q d = 0 q_d=0 q d = 0 d d d Q Q Q Z \mathbb{Z} Z
Theorem E. (Hodge types off the diagonal). Fix g ≥ 2 g\ge2 g ≥ 2 γ g H , γ g L > 0 \gamma^H_g,\gamma^L_g>0 γ g H , γ g L > 0
P ( D g , n H ≠ 0 ) = γ g H n − 1 / 2 + O ( n − 1 ) , P ( D g , n L ≠ 0 ) = γ g L n − 1 + O ( n − 2 ) . \mathbb{P}(D^H_{g,n}\ne0)=\gamma^H_g n^{-1/2}+O(n^{-1}),\qquad\mathbb{P}(D^L_{g,n}\ne0)=\gamma^L_g n^{-1}+O(n^{-2}). P ( D g , n H = 0 ) = γ g H n − 1/2 + O ( n − 1 ) , P ( D g , n L = 0 ) = γ g L n − 1 + O ( n − 2 ) . The same formulas hold for the probability that p + q p+q p + q O ( n − 1 ) O(n^{-1}) O ( n − 1 ) O ( n − 2 ) O(n^{-2}) O ( n − 2 ) J ∈ { H , L } J\in\{H,L\} J ∈ { H , L } D g , n J D^J_{g,n} D g , n J D g , n J ≠ 0 D^J_{g,n}\ne0 D g , n J = 0 Q Q Q x ∈ R x\in\mathbb{R} x ∈ R d ∈ Z d\in\mathbb{Z} d ∈ Z
P ( p + q − d n 4 ν n ≤ x , D g , n H = d ∣ D g , n H ≠ 0 ) ⟶ Φ ( x ) q d , \mathbb{P}\left(\frac{p+q-dn}{\sqrt{4\nu n}}\le x,\ D^H_{g,n}=d\mathrel{\big|}D^H_{g,n}\ne0\right)\longrightarrow\Phi(x)q_d, P ( 4 ν n p + q − d n ≤ x , D g , n H = d D g , n H = 0 ) ⟶ Φ ( x ) q d , where Φ \Phi Φ 4 ν n \sqrt{4\nu n} 4 ν n n / 3 \sqrt{n/3} n /3 D g , n H D^H_{g,n} D g , n H D g , n L D^L_{g,n} D g , n L
The off-diagonal classes therefore form a vanishing proportion of the cohomology, but their conditional Hodge difference has a universal limiting distribution, independent of the genus and common to the heavy and light spaces. Among the off-diagonal classes, the proportion of odd-degree classes tends to 1. Since β 12 = 1 \beta_{12}=1 β 12 = 1 ± 11 \pm11 ± 11
q 11 + q − 11 = 11 I 11 ( 2 ) Z Q > 0.9999. q_{11}+q_{-11}=\frac{11I_{11}(2)}{Z_Q}>0.9999. q 11 + q − 11 = Z Q 11 I 11 ( 2 ) > 0.9999. Geometrically, the leading off-diagonal classes come from boundary strata whose stable core is trivalent and rational except for a single elliptic leaf. For odd m ≥ 3 m\ge3 m ≥ 3 Sym m − 1 V \operatorname{Sym}^{m-1}V Sym m − 1 V M ‾ 1 , 1 \overline{\mathcal M}_{1,1} M 1 , 1 m + 1 m+1 m + 1 ± m \pm m ± m V V V p p p q q q M ‾ 1 , 1 \overline{\mathcal M}_{1,1} M 1 , 1 H 11 ( M ‾ g , n ) ≠ 0 H^{11}(\overline{\mathcal M}_{g,n})\ne0 H 11 ( M g , n ) = 0 g = 1 g=1 g = 1 n ≥ 11 n\ge11 n ≥ 11 M ‾ 1 , 11 \overline{\mathcal M}_{1,11} M 1 , 11 [4 ] . Remark 6.6 interprets the Bessel weights in q m q_m q m
Ultra-log-concavity
A positive sequence ( a k ) k = 0 d (a_k)_{k=0}^d ( a k ) k = 0 d log-concave at an index 1 ≤ k ≤ d − 1 1 \le k \le d-1 1 ≤ k ≤ d − 1 a k 2 ≥ a k − 1 a k + 1 a_k^2 \ge a_{k-1}a_{k+1} a k 2 ≥ a k − 1 a k + 1 r ≥ 0 r \ge0 r ≥ 0 r r r ultra-log-concave at k k k ( a k / ( d k ) r ) k \left(a_k/\binom{d}{k}^r\right)_k ( a k / ( k d ) r ) k k k k r = 0 r=0 r = 0 r = 1 r=1 r = 1 [3 ] . Larger values of r r r M ‾ g , n \overline{\mathcal{M}}_{g,n} M g , n M ‾ g , 0 ∣ n \overline{\mathcal{M}}_{g,0|n} M g , 0∣ n
Theorem F (Central ultra-log-concavity). Fix g ≥ 1 g \ge1 g ≥ 1 0 < c 0 < 2 0 < c_0 < \sqrt{2} 0 < c 0 < 2
r H = 1 4 v = 3 ( e − 2 ) 2 ( 3 − e ) = 3.82447 … , r L = 3. r_H = \frac{1}{4v} = \frac{3(e-2)}{2(3-e)} = 3.82447\ldots,\qquad r_L = 3. r H = 4 v 1 = 2 ( 3 − e ) 3 ( e − 2 ) = 3.82447 … , r L = 3. For J ∈ { H , L } J \in\{H,L\} J ∈ { H , L } ( a k ) (a_k) ( a k ) M ‾ g , n \overline{\mathcal{M}}_{g,n} M g , n J = H J=H J = H M ‾ g , 0 ∣ n \overline{\mathcal{M}}_{g,0|n} M g , 0∣ n J = L J=L J = L k k k ∣ k + j − d n / 2 ∣ ≤ c 0 Var ( X g , n J ) log n \lvert k+j-d_n/2\rvert\le c_0\sqrt{\operatorname{Var}(X^J_{g,n})\log n} ∣ k + j − d n /2 ∣ ≤ c 0 Var ( X g , n J ) log n j = − 1 , 0 , 1 j=-1,0,1 j = − 1 , 0 , 1 n n n ( a k ) (a_k) ( a k ) r r r k k k r ≤ r J r \le r_J r ≤ r J r r r k k k r > r J r > r_J r > r J
The central range has width of order n log n \sqrt{n\log n} n log n n n n v n vn v n ( n k ) \binom{n}{k} ( k n ) n / 4 n/4 n /4 1 / ( 4 v ) 1/(4v) 1/ ( 4 v ) r = r J r=r_J r = r J n − 1 n^{-1} n − 1 M ‾ 0 , 2 ∣ n \overline{\mathcal{M}}_{0,2|n} M 0 , 2∣ n
Method
We work with the exponential generating functions L g ( w ; u , v ) \mathcal{L}_g(w;u,v) L g ( w ; u , v ) H g ( z ; u , v ) \mathcal{H}_g(z;u,v) H g ( z ; u , v ) [16 ] , L g ( w ) = H g ( R ( w , u v ) ) \mathcal{L}_g(w)=\mathcal{H}_g(R(w,uv)) L g ( w ) = H g ( R ( w , uv )) R R R t = u s t=us t = u s T ( z , t ) T(z,t) T ( z , t ) R ( w , t ) R(w,t) R ( w , t )
ρ ( t ) = t 1 t − 1 − 1 − t − 1 , ρ ( 1 ) = e − 2. \rho(t)=t^{\frac{1}{t-1}}-1-t^{-1},\qquad\rho(1)=e-2. ρ ( t ) = t t − 1 1 − 1 − t − 1 , ρ ( 1 ) = e − 2. This explains the factor ( e − 2 ) − n (e-2)^{-n} ( e − 2 ) − n
The light series are computed geometrically. Forgetting the markings of a light curve and stabilizing produces a stable core in M ‾ g \overline{\mathcal{M}}_g M g ∣ E ∣ \lvert E\rvert ∣ E ∣ ∣ E ∣ \lvert E\rvert ∣ E ∣ 3 g − 3 3g-3 3 g − 3 c g c_g c g
Singularity analysis then gives uniform quasi-powers estimates, and Hwang’s theorem [15 ] yields Theorem A. For the local limit theorem we refine Hwang’s Fourier method [14 ] . Poincaré duality makes the distributions symmetric about their means, so all odd cumulants of the centered variables vanish. The resulting local expansion is precise enough to compare neighboring Betti numbers at the critical exponent.
For Hodge types we substitute ( u , v ) = ( − s ξ , − s ξ − 1 ) (u,v)=(-s\xi,-s\xi^{-1}) ( u , v ) = ( − s ξ , − s ξ − 1 ) ξ \xi ξ p − q p-q p − q m + 1 m+1 m + 1 [19 ] , and the Bessel functions arise from summing over the markings on the leaf. The same analysis, applied to two elliptic leaves, gives the lower bound in Theorem D.
Layout
Section 2 collects the Hodge-theoretic and analytic tools. Section 3 computes the singular structure of the Hodge generating series. Section 4 proves the coefficient asymptotics, Theorem A and Corollary C. Section 5 proves the local limit theorem, Theorem B and Theorem F . Section 6 develops the Hodge-refined series and the analysis of strata by dimension, and proves Theorem E. Section 7 proves Theorem D.
The main theorems correspond to results in the body as follows: Theorem A to Theorem 4.3 and Corollary 4.4; Theorem B to Corollary 5.4; Corollary C to Corollary 4.6 ; Theorem D to Theorem 7.1 ; Theorem E to Theorem 6.5, where γ g H \gamma_g^H γ g H γ g L \gamma_g^L γ g L (78) ; and Theorem F to Theorems 5.6 and 5.7 .
Preliminaries
This section collects the Hodge-theoretic identities behind our generating series and the analytic tools used to extract asymptotics from them.
Hodge–Deligne polynomials and finite quotients
For a complex algebraic variety, or a Deligne–Mumford stack, Y Y Y E ( Y ; u , v ) E(Y;u,v) E ( Y ; u , v ) α \alpha α
E α ( Y ; u , v ) = ∑ i , p , q ( − 1 ) i Tr ( α ∣ Gr F p Gr p + q W H c i ( Y ; C ) ) u p v q . E_{\alpha}(Y;u,v)=\sum_{i,p,q}(-1)^i\operatorname{Tr}\left(\alpha\mid\operatorname{Gr}_{F}^{p}\operatorname{Gr}_{p+q}^{W}H_c^i(Y;\mathbb{C})\right)u^pv^q. E α ( Y ; u , v ) = i , p , q ∑ ( − 1 ) i Tr ( α ∣ Gr F p Gr p + q W H c i ( Y ; C ) ) u p v q . If a finite group G G G Y Y Y
E ( Y / G ; u , v ) = 1 ∣ G ∣ ∑ α ∈ G E α ( Y ; u , v ) . (2) E(Y/G;u,v)=\frac{1}{|G|}\sum_{\alpha\in G}E_{\alpha}(Y;u,v). \tag*{(2)} E ( Y / G ; u , v ) = ∣ G ∣ 1 α ∈ G ∑ E α ( Y ; u , v ) . (2) The same argument applies to stacks, because the higher cohomology of a finite group with rational coefficients vanishes. The polynomials E α E_{\alpha} E α α \alpha α α \alpha α
For either Y = M ‾ g , n Y=\overline{\mathcal{M}}_{g,n} Y = M g , n Y = M ‾ g , 0 ∣ n Y=\overline{\mathcal{M}}_{g,0|n} Y = M g , 0∣ n Y Y Y H m ( Y ; Q ) H^m(Y;\mathbb{Q}) H m ( Y ; Q ) m m m u p v q u^pv^q u p v q E ( Y ; u , v ) E(Y;u,v) E ( Y ; u , v ) ( − 1 ) p + q h p , q ( Y ) (-1)^{p+q}h^{p,q}(Y) ( − 1 ) p + q h p , q ( Y )
P Y e v ( t ) = E ( Y ; t , t ) + E ( Y ; − t , − t ) 2 , P Y ( q ) = E ( Y ; − q , − q ) , (3) P_Y^{\mathrm{ev}}(t)=\frac{E(Y;\sqrt{t},\sqrt{t})+E(Y;-\sqrt{t},-\sqrt{t})}{2},\qquad P_Y(q)=E(Y;-q,-q), \tag*{(3)} P Y ev ( t ) = 2 E ( Y ; t , t ) + E ( Y ; − t , − t ) , P Y ( q ) = E ( Y ; − q , − q ) , (3) where P Y ( q ) = ∑ j b j ( Y ) q j P_Y(q)=\sum_j b_j(Y)q^j P Y ( q ) = ∑ j b j ( Y ) q j P Y e v ( t ) = ∑ k b 2 k ( Y ) t k P_Y^{\mathrm{ev}}(t)=\sum_k b_{2k}(Y)t^k P Y ev ( t ) = ∑ k b 2 k ( Y ) t k (3) retains all Hodge types with p + q p+q p + q
Singularity analysis
We write [ z n ] f ( z ) [z^n]f(z) [ z n ] f ( z ) z n z^n z n f f f
A ( z , t ) = ∑ n ≥ 0 A n ( t ) z n n ! , A n ( t ) = n ! [ z n ] A ( z , t ) , \mathcal{A}(z,t)=\sum_{n\geq0}A_n(t)\frac{z^n}{n!},\qquad A_n(t)=n![z^n]\mathcal{A}(z,t), A ( z , t ) = n ≥ 0 ∑ A n ( t ) n ! z n , A n ( t ) = n ! [ z n ] A ( z , t ) , so that a product of two such series distributes the labels between the factors, with the binomial coefficients supplied automatically.
A dominant singularity of a power series is a singularity on its circle of convergence. For the logarithmic and algebraic singularities considered here, a unique dominant singularity determines the coefficient asymptotics: its location gives the exponential growth rate, and the local form of the function gives the polynomial factor. The basic transfer models are
[ z n ] log 1 1 − z / ρ = ρ − n n , [ z n ] ( 1 − z / ρ ) − α ∼ ρ − n n α − 1 Γ ( α ) , (4) [z^n]\log\frac{1}{1-z/\rho}=\frac{\rho^{-n}}{n},\qquad[z^n](1-z/\rho)^{-\alpha}\sim\frac{\rho^{-n}n^{\alpha-1}}{\Gamma(\alpha)}, \tag*{(4)} [ z n ] log 1 − z / ρ 1 = n ρ − n , [ z n ] ( 1 − z / ρ ) − α ∼ Γ ( α ) ρ − n n α − 1 , (4) valid for n ≥ 1 n\geq1 n ≥ 1 α ∉ { 0 , − 1 , − 2 , … } \alpha\notin\{0,-1,-2,\ldots\} α ∈ / { 0 , − 1 , − 2 , … } m m m n m − 1 n^{m-1} n m − 1 n − 3 / 2 n^{-3/2} n − 3/2
To transfer a local expansion together with its error term, one needs analytic continuation slightly beyond the circle of convergence. A Δ \Delta Δ z = ρ z=\rho z = ρ
∣ z / ρ ∣ < 1 + η , ∣ arg ( z / ρ − 1 ) ∣ > θ , η > 0 , 0 < θ < π / 2. |z/\rho|<1+\eta,\qquad|\arg(z/\rho-1)|>\theta,\qquad\eta>0,\quad0<\theta<\pi/2. ∣ z / ρ ∣ < 1 + η , ∣ arg ( z / ρ − 1 ) ∣ > θ , η > 0 , 0 < θ < π /2. The transfer theorem [10 Theorem VI.4] converts a local expansion with remainder on such a domain into a coefficient expansion with remainder. We use it with a complex parameter t t t t t t Δ \Delta Δ ρ ( t ) \rho(t) ρ ( t ) t t t t t t [10 Lemma IX.2] .
Probability generating functions and quasi-powers
Let A ( t ) = ∑ k = 0 d a k t k A(t)=\sum_{k=0}^{d}a_k t^k A ( t ) = ∑ k = 0 d a k t k A ( 1 ) > 0 A(1)>0 A ( 1 ) > 0 Y Y Y P ( Y = k ) = a k / A ( 1 ) \mathbb{P}(Y=k)=a_k/A(1) P ( Y = k ) = a k / A ( 1 ) probability generating function (PGF) is
G Y ( t ) = E t Y = A ( t ) A ( 1 ) . (5) G_Y(t)=\mathbb{E}t^Y=\frac{A(t)}{A(1)}. \tag*{(5)} G Y ( t ) = E t Y = A ( 1 ) A ( t ) . (5) Applied to an even Betti polynomial, this gives the distribution of half the cohomological degree. For a function f f f f ( 1 ) ≠ 0 f(1)\ne0 f ( 1 ) = 0
m ( f ) = f ′ ( 1 ) f ( 1 ) , v ( f ) = f ′ ′ ( 1 ) f ( 1 ) + f ′ ( 1 ) f ( 1 ) − ( f ′ ( 1 ) f ( 1 ) ) 2 . (6) \mathfrak{m}(f)=\frac{f'(1)}{f(1)},\qquad\mathfrak{v}(f)=\frac{f''(1)}{f(1)}+\frac{f'(1)}{f(1)}-\left(\frac{f'(1)}{f(1)}\right)^2. \tag*{(6)} m ( f ) = f ( 1 ) f ′ ( 1 ) , v ( f ) = f ( 1 ) f ′′ ( 1 ) + f ( 1 ) f ′ ( 1 ) − ( f ( 1 ) f ′ ( 1 ) ) 2 . (6) For a local holomorphic logarithm of f f f
d d s log f ( e s ) ∣ s = 0 = m ( f ) , d 2 d s 2 log f ( e s ) ∣ s = 0 = v ( f ) , (7) \left.\frac{d}{ds}\log f(e^s)\right|_{s=0}=\mathfrak{m}(f),\qquad\left.\frac{d^2}{ds^2}\log f(e^s)\right|_{s=0}=\mathfrak{v}(f), \tag*{(7)} d s d log f ( e s ) s = 0 = m ( f ) , d s 2 d 2 log f ( e s ) s = 0 = v ( f ) , (7) so E Y = m ( G Y ) \mathbb{E}Y=\mathfrak{m}(G_Y) E Y = m ( G Y ) Var ( Y ) = v ( G Y ) \operatorname{Var}(Y)=\mathfrak{v}(G_Y) Var ( Y ) = v ( G Y ) [10 p. 645] . The expressions m ( f ) \mathfrak{m}(f) m ( f ) v ( f ) \mathfrak{v}(f) v ( f ) f f f A A A A ( t ) = t d A ( t − 1 ) A(t)=t^dA(t^{-1}) A ( t ) = t d A ( t − 1 ) Y Y Y d − Y d-Y d − Y E Y = d / 2 \mathbb{E}Y=d/2 E Y = d /2 M ‾ g , n \overline{\mathcal{M}}_{g,n} M g , n M ‾ g , 0 ∣ n \overline{\mathcal{M}}_{g,0|n} M g , 0∣ n d n / 2 d_n/2 d n /2
Our coefficient estimates, divided by their values at t = 1 t=1 t = 1 [15 ] as stated in [10 Theorem IX.8] .
Theorem 2.1 (Quasi-powers theorem). Let Y n Y_n Y n p n ( t ) p_n(t) p n ( t ) t = 1 t=1 t = 1
p n ( t ) = A ( t ) B ( t ) β n ( 1 + O ( ε n ) ) , β n → ∞ , ε n → 0 , (8) p_n(t)=A(t)B(t)^{\beta_n}(1+O(\varepsilon_n)),\qquad\beta_n\to\infty,\qquad\varepsilon_n\to0, \tag*{(8)} p n ( t ) = A ( t ) B ( t ) β n ( 1 + O ( ε n )) , β n → ∞ , ε n → 0 , (8) where A A A B B B n n n A ( 1 ) = B ( 1 ) = 1 A(1)=B(1)=1 A ( 1 ) = B ( 1 ) = 1 v ( B ) > 0 \mathfrak{v}(B)>0 v ( B ) > 0 log B \log B log B μ n \mu_n μ n σ n 2 \sigma_n^2 σ n 2 Y n Y_n Y n
μ n = β n m ( B ) + m ( A ) + O ( ε n ) , σ n 2 = β n v ( B ) + v ( A ) + O ( ε n ) , \mu_n=\beta_n\mathfrak{m}(B)+\mathfrak{m}(A)+O(\varepsilon_n),\qquad\sigma_n^2=\beta_n\mathfrak{v}(B)+\mathfrak{v}(A)+O(\varepsilon_n), μ n = β n m ( B ) + m ( A ) + O ( ε n ) , σ n 2 = β n v ( B ) + v ( A ) + O ( ε n ) , and
sup x ∈ R ∣ P ( Y n − μ n σ n ≤ x ) − Φ ( x ) ∣ = O ( β n − 1 / 2 + ε n ) , \sup_{x\in\mathbb{R}}\left|\mathbb{P}\left(\frac{Y_n-\mu_n}{\sigma_n}\leq x\right)-\Phi(x)\right|=O(\beta_n^{-1/2}+\varepsilon_n), x ∈ R sup P ( σ n Y n − μ n ≤ x ) − Φ ( x ) = O ( β n − 1/2 + ε n ) , where Φ \Phi Φ
The factors A A A B B B ν ( B ) \nu(B) ν ( B ) B B B
Cumulants and Fourier inversion
To go beyond the mean and the variance we use cumulants, in the logarithmic coordinate t = e s t=e^s t = e s Y Y Y moment generating function M Y ( s ) = E e s Y = G Y ( e s ) M_Y(s)=\mathbb{E}e^{sY}=G_Y(e^s) M Y ( s ) = E e s Y = G Y ( e s ) M Y ( 0 ) = 1 M_Y(0)=1 M Y ( 0 ) = 1
Λ Y ( s ) = log M Y ( s ) = ∑ j ≥ 1 κ j ( Y ) s j j ! , Λ Y ( 0 ) = 0 , \Lambda_Y(s)=\log M_Y(s)=\sum_{j\geq1}\kappa_j(Y)\frac{s^j}{j!},\qquad\Lambda_Y(0)=0, Λ Y ( s ) = log M Y ( s ) = j ≥ 1 ∑ κ j ( Y ) j ! s j , Λ Y ( 0 ) = 0 , is the cumulant generating function , and κ j ( Y ) \kappa_j(Y) κ j ( Y ) j j j μ = E Y \mu=\mathbb{E}Y μ = E Y σ 2 = Var ( Y ) \sigma^2=\operatorname{Var}(Y) σ 2 = Var ( Y )
κ 1 ( Y ) = μ , κ 2 ( Y ) = σ 2 , κ 3 ( Y ) = E ( Y − μ ) 3 , κ 4 ( Y ) = E ( Y − μ ) 4 − 3 σ 4 . (9) \kappa_1(Y)=\mu,\qquad\kappa_2(Y)=\sigma^2,\qquad\kappa_3(Y)=\mathbb{E}(Y-\mu)^3,\qquad\kappa_4(Y)=\mathbb{E}(Y-\mu)^4-3\sigma^4. \tag*{(9)} κ 1 ( Y ) = μ , κ 2 ( Y ) = σ 2 , κ 3 ( Y ) = E ( Y − μ ) 3 , κ 4 ( Y ) = E ( Y − μ ) 4 − 3 σ 4 . (9) Cumulants are additive for independent sums, and the cumulants of order at least three vanish for a Gaussian variable. They therefore measure the departure from a Gaussian law. If G Y G_Y G Y d d d M Y ( s ) = e d s M Y ( − s ) M_Y(s)=e^{ds}M_Y(-s) M Y ( s ) = e d s M Y ( − s ) Λ Y ( s ) − d s / 2 \Lambda_Y(s)-ds/2 Λ Y ( s ) − d s /2 Y − d / 2 Y-d/2 Y − d /2 § 5 .
The quasi-powers hypothesis controls all cumulants. Suppose (8) holds with β n = n \beta_n=n β n = n
λ ( s ) = log B ( e s ) , ψ ( s ) = log A ( e s ) , \lambda(s)=\log B(e^s),\qquad\psi(s)=\log A(e^s), λ ( s ) = log B ( e s ) , ψ ( s ) = log A ( e s ) , with both logarithms vanishing at s = 0 s=0 s = 0 (8) becomes
M n ( s ) = exp { n λ ( s ) + ψ ( s ) } ( 1 + O ( ϵ n ) ) (10) M_n(s)=\exp\{n\lambda(s)+\psi(s)\}(1+O(\epsilon_n)) \tag*{(10)} M n ( s ) = exp { nλ ( s ) + ψ ( s )} ( 1 + O ( ϵ n )) (10) uniformly on a fixed complex neighborhood of 0, where M n = M Y n M_n=M_{Y_n} M n = M Y n λ ′ ′ ( 0 ) = ν ( B ) > 0 \lambda''(0)=\nu(B)>0 λ ′′ ( 0 ) = ν ( B ) > 0 n n n M n ( s ) exp ( − n λ ( s ) − ψ ( s ) ) M_n(s)\exp(-n\lambda(s)-\psi(s)) M n ( s ) exp ( − nλ ( s ) − ψ ( s )) R n ( s ) = O ( ϵ n ) R_n(s)=O(\epsilon_n) R n ( s ) = O ( ϵ n ) R n ( 0 ) = 0 R_n(0)=0 R n ( 0 ) = 0 R n ( j ) ( s ) = O j ( ϵ n ) R_n^{(j)}(s)=O_j(\epsilon_n) R n ( j ) ( s ) = O j ( ϵ n ) j j j
κ j ( Y n ) = n λ ( j ) ( 0 ) + ψ ( j ) ( 0 ) + O j ( ϵ n ) . (11) \kappa_j(Y_n)=n\lambda^{(j)}(0)+\psi^{(j)}(0)+O_j(\epsilon_n). \tag*{(11)} κ j ( Y n ) = n λ ( j ) ( 0 ) + ψ ( j ) ( 0 ) + O j ( ϵ n ) . (11) Here O j ( ϵ n ) O_j(\epsilon_n) O j ( ϵ n ) C j ϵ n C_j\epsilon_n C j ϵ n C j C_j C j j j j n n n O ( n ) O(n) O ( n ) κ 4 ( Y n ) / Var ( Y n ) 2 \kappa_4(Y_n)/\operatorname{Var}(Y_n)^2 κ 4 ( Y n ) / Var ( Y n ) 2 O ( n − 1 ) O(n^{-1}) O ( n − 1 )
A central limit theorem controls cumulative probabilities, whereas a local limit theorem controls the individual probabilities P ( Y n = k ) \mathbb{P}(Y_n=k) P ( Y n = k ) O ( n − 1 / 2 ) O(n^{-1/2}) O ( n − 1/2 ) B N B_N B N N N N 1 / 2 1/2 1/2 2 B N 2B_N 2 B N
For an integer-valued variable Y Y Y
P ( Y = k ) = 1 2 π ∫ − π π e − i k θ M Y ( i θ ) d θ . (12) \mathbb{P}(Y=k)=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{-ik\theta}M_Y(i\theta)\,d\theta. \tag*{(12)} P ( Y = k ) = 2 π 1 ∫ − π π e − ik θ M Y ( i θ ) d θ . (12) In § 5 we evaluate this integral using the cumulant expansion near θ = 0 \theta=0 θ = 0 ∣ M Y ( i θ ) ∣ \lvert M_Y(i\theta)\rvert ∣ M Y ( i θ )∣ θ = 0 \theta=0 θ = 0
Hodge series and the heavy/light substitution
Weighted pointed curves and generating series
We recall Hassett’s moduli spaces of weighted pointed stable curves [13 ] . Let A = ( a 1 , … , a n ) A=(a_1,\ldots,a_n) A = ( a 1 , … , a n ) 0 < a i ≤ 1 0<a_i\leq1 0 < a i ≤ 1 2 g − 2 + ∑ i a i > 0 2g-2+\sum_i a_i>0 2 g − 2 + ∑ i a i > 0 C C C g g g p 1 , … , p n p_1,\ldots,p_n p 1 , … , p n A A A
(1) whenever the markings indexed by I ⊂ { 1 , … , n } I\subset\{1,\ldots,n\} I ⊂ { 1 , … , n } ∑ i ∈ I a i ≤ 1 \sum_{i\in I}a_i\leq1 ∑ i ∈ I a i ≤ 1
(2) the Q \mathbb{Q} Q K C + ∑ i a i p i K_C+\sum_i a_i p_i K C + ∑ i a i p i
The second condition means that
2 g v − 2 + d v + ∑ p i ∈ C v a i > 0 2g_v-2+d_v+\sum_{p_i\in C_v}a_i>0 2 g v − 2 + d v + p i ∈ C v ∑ a i > 0 for every irreducible component C v C_v C v g v g_v g v d v d_v d v C v C_v C v d v d_v d v M ‾ g , A \overline{\mathcal{M}}_{g,A} M g , A g ≥ 1 g\geq1 g ≥ 1 n ≥ 1 n\geq1 n ≥ 1
M ‾ g , n = M ‾ g , ( 1 , … , 1 ) , M ‾ g , 0 ∣ n = M ‾ g , ( 1 / n , … , 1 / n ) . \overline{\mathcal{M}}_{g,n}=\overline{\mathcal{M}}_{g,(1,\ldots,1)},\qquad
\overline{\mathcal{M}}_{g,0|n}=\overline{\mathcal{M}}_{g,(1/n,\ldots,1/n)}. M g , n = M g , ( 1 , … , 1 ) , M g , 0∣ n = M g , ( 1/ n , … , 1/ n ) . In the heavy space M ‾ g , n \overline{\mathcal{M}}_{g,n} M g , n M ‾ g , 0 ∣ n \overline{\mathcal{M}}_{g,0|n} M g , 0∣ n d n = n + 3 g − 3 d_n=n+3g-3 d n = n + 3 g − 3 g ≥ 2 g\geq2 g ≥ 2 n = 0 n=0 n = 0 M ‾ g \overline{\mathcal{M}}_g M g g = 1 g=1 g = 1
Define the exponential generating series of Hodge–Deligne polynomials
L g ( w ; u , v ) = ∑ n ≥ 0 E ( M ‾ g , 0 ∣ n ; u , v ) w n n ! , H g ( z ; u , v ) = ∑ n ≥ 0 E ( M ‾ g , n ; u , v ) z n n ! . (13) \mathscr{L}_g(w;u,v)=\sum_{n\geq0}E(\overline{\mathcal{M}}_{g,0|n};u,v)\frac{w^n}{n!},\qquad
\mathscr{H}_g(z;u,v)=\sum_{n\geq0}E(\overline{\mathcal{M}}_{g,n};u,v)\frac{z^n}{n!}. \tag*{(13)} L g ( w ; u , v ) = n ≥ 0 ∑ E ( M g , 0∣ n ; u , v ) n ! w n , H g ( z ; u , v ) = n ≥ 0 ∑ E ( M g , n ; u , v ) n ! z n . (13) the even Betti polynomials
L g , n ( t ) = ∑ k = 0 d n b 2 k ( M ‾ g , 0 ∣ n ) t k , H g , n ( t ) = ∑ k = 0 d n b 2 k ( M ‾ g , n ) t k . (14) \mathcal{L}_{g,n}(t)=\sum_{k=0}^{d_n}b_{2k}(\overline{\mathcal{M}}_{g,0|n})t^k,\qquad
\mathcal{H}_{g,n}(t)=\sum_{k=0}^{d_n}b_{2k}(\overline{\mathcal{M}}_{g,n})t^k. \tag*{(14)} L g , n ( t ) = k = 0 ∑ d n b 2 k ( M g , 0∣ n ) t k , H g , n ( t ) = k = 0 ∑ d n b 2 k ( M g , n ) t k . (14) and their exponential generating series
L g ( w , t ) = ∑ n ≥ 0 L g , n ( t ) w n n ! , H g ( z , t ) = ∑ n ≥ 0 H g , n ( t ) z n n ! . \mathcal{L}_g(w,t)=\sum_{n\geq0}\mathcal{L}_{g,n}(t)\frac{w^n}{n!},\qquad
\mathcal{H}_g(z,t)=\sum_{n\geq0}\mathcal{H}_{g,n}(t)\frac{z^n}{n!}. L g ( w , t ) = n ≥ 0 ∑ L g , n ( t ) n ! w n , H g ( z , t ) = n ≥ 0 ∑ H g , n ( t ) n ! z n . Script letters L , H \mathscr{L},\mathscr{H} L , H L , H \mathcal{L},\mathcal{H} L , H
The heavy/light substitution
The heavy and light series are related by a substitution that is independent of the genus and comes from genus zero. The spaces M ‾ 0 , n + 1 \overline{\mathcal{M}}_{0,n+1} M 0 , n + 1 H 2 k ( M ‾ 0 , n + 1 ) H^{2k}(\overline{\mathcal{M}}_{0,n+1}) H 2 k ( M 0 , n + 1 ) ( k , k ) (k,k) ( k , k )
φ ( z , t ) = z + ∑ n ≥ 2 z n n ! ∑ k = 0 n − 2 b 2 k ( M ‾ 0 , n + 1 ) t k . \varphi(z,t)=z+\sum_{n\geq2}\frac{z^n}{n!}\sum_{k=0}^{n-2}b_{2k}(\overline{\mathcal{M}}_{0,n+1})t^k. φ ( z , t ) = z + n ≥ 2 ∑ n ! z n k = 0 ∑ n − 2 b 2 k ( M 0 , n + 1 ) t k . Summing over stable trees gives the functional equation [22 p. 194, Theorem 4.2.1]
( 1 + φ ) t = t 2 φ − t ( t − 1 ) z + 1. (15) (1+\varphi)^t=t^2\varphi-t(t-1)z+1. \tag*{(15)} ( 1 + φ ) t = t 2 φ − t ( t − 1 ) z + 1. (15) Equivalently, by the non-equivariant specialization of Getzler’s Legendre-transform relation [11 Theorem 5.9] , the formal inverse of φ \varphi φ
φ ⟨ − 1 ⟩ ( x , t ) = x − ∑ n ≥ 2 E ( M ‾ 0 , n + 1 ; u , v ) x n n ! = t 2 x − ( 1 + x ) t + 1 t ( t − 1 ) , t = u v . \varphi^{\langle-1\rangle}(x,t)=x-\sum_{n\geq2}E(\overline{\mathcal{M}}_{0,n+1};u,v)\frac{x^n}{n!}
=\frac{t^2x-(1+x)^t+1}{t(t-1)},\qquad t=uv. φ ⟨ − 1 ⟩ ( x , t ) = x − n ≥ 2 ∑ E ( M 0 , n + 1 ; u , v ) n ! x n = t ( t − 1 ) t 2 x − ( 1 + x ) t + 1 , t = uv . Kannan, Serpente and Yun proved that the light Hodge series is obtained by substituting a universal function into the heavy one [16 Corollary B] . In the all-light case it reads
L g ( w ; u , v ) = H g ( R ( w , u v ) ; u , v ) . (16) \mathcal{L}_g(w;u,v)=\mathcal{H}_g(R(w,uv);u,v). \tag*{(16)} L g ( w ; u , v ) = H g ( R ( w , uv ) ; u , v ) . (16) The substitution combines two geometric steps [16 Propositions 4.6–4.7] . First, a heavy curve is obtained from a curve with distinct markings and no rational tails by attaching genus-zero tails at the markings. Since φ \varphi φ φ ⟨ − 1 ⟩ ( x , u v ) \varphi^{\langle-1\rangle}(x,uv) φ ⟨ − 1 ⟩ ( x , uv ) z z z H g \mathcal{H}_g H g x x x e w − 1 e^w-1 e w − 1
R ( w , t ) = φ ⟨ − 1 ⟩ ( e w − 1 , t ) = t 2 e w − t 2 − e t w + 1 t ( t − 1 ) , R ( w , 1 ) = ( 2 − w ) e w − 2. (17) R(w,t)=\varphi^{\langle-1\rangle}(e^w-1,t)=\frac{t^2e^w-t^2-e^{tw}+1}{t(t-1)},\qquad R(w,1)=(2-w)e^w-2. \tag*{(17)} R ( w , t ) = φ ⟨ − 1 ⟩ ( e w − 1 , t ) = t ( t − 1 ) t 2 e w − t 2 − e tw + 1 , R ( w , 1 ) = ( 2 − w ) e w − 2. (17) Since R ( 0 , t ) = 0 R(0,t)=0 R ( 0 , t ) = 0 R w ( 0 , t ) = 1 R_w(0,t)=1 R w ( 0 , t ) = 1 R R R T ( z , t ) T(z,t) T ( z , t ) T ( 0 , t ) = 0 T(0,t)=0 T ( 0 , t ) = 0
T ( z , t ) = log ( 1 + φ ( z , t ) ) , (18) T(z,t)=\log(1+\varphi(z,t)), \tag*{(18)} T ( z , t ) = log ( 1 + φ ( z , t )) , (18) with the branch of the logarithm vanishing at 1. The coefficients of R R R w w w Q [ t ] \mathbb{Q}[t] Q [ t ] T T T z z z t t t T T T z = 0 z=0 z = 0 t t t
Proposition 3.1. For every g ≥ 1 g\ge1 g ≥ 1
H g ( z ; u , v ) = L g ( T ( z , u v ) ; u , v ) , H g ( z , t ) = L g ( T ( z , t ) , t ) . (19) \mathcal{H}_g(z;u,v)=\mathcal{L}_g(T(z,uv);u,v),\qquad\mathcal{H}_g(z,t)=\mathcal{L}_g(T(z,t),t). \tag*{(19)} H g ( z ; u , v ) = L g ( T ( z , uv ) ; u , v ) , H g ( z , t ) = L g ( T ( z , t ) , t ) . (19) Proof. Substituting w = T ( z , u v ) w=T(z,uv) w = T ( z , uv ) R ( T ( z , u v ) , u v ) = z R(T(z,uv),uv)=z R ( T ( z , uv ) , uv ) = z ( u , v ) = ( t , t ) (u,v)=(\sqrt{t},\sqrt{t}) ( u , v ) = ( t , t ) ( − t , − t ) (-\sqrt{t},-\sqrt{t}) ( − t , − t ) u v = t uv=t uv = t T ( z , t ) T(z,t) T ( z , t )
The singularity of T T T
τ ( t ) = log t t − 1 , ρ ( t ) = t 1 t − 1 − 1 − t − 1 , (20) \tau(t)=\frac{\log t}{t-1},\qquad\rho(t)=t^{\frac{1}{t-1}}-1-t^{-1}, \tag*{(20)} τ ( t ) = t − 1 log t , ρ ( t ) = t t − 1 1 − 1 − t − 1 , (20) where log t \log t log t t = 1 t=1 t = 1 τ ( 1 ) = 1 \tau(1)=1 τ ( 1 ) = 1 ρ ( 1 ) = e − 2 \rho(1)=e-2 ρ ( 1 ) = e − 2
Lemma 3.2. For complex t t t T ( z , t ) T(z,t) T ( z , t ) z = ρ ( t ) z=\rho(t) z = ρ ( t ) Δ t = ρ ( t ) Δ 0 \Delta_t=\rho(t)\Delta_0 Δ t = ρ ( t ) Δ 0 Δ 0 \Delta_0 Δ 0 Δ \Delta Δ z → ρ ( t ) z\to\rho(t) z → ρ ( t ) Δ t \Delta_t Δ t
T ( z , t ) = τ ( t ) − ( 2 ρ ( t ) ) 1 2 t − t 2 ( t − 1 ) ( 1 − z ρ ( t ) ) 1 2 + O ( 1 − z ρ ( t ) ) , (21) T(z,t)=\tau(t)-(2\rho(t))^{\frac{1}{2}}t^{-\frac{t}{2(t-1)}}\left(1-\frac{z}{\rho(t)}\right)^{\frac{1}{2}}+O\left(1-\frac{z}{\rho(t)}\right), \tag*{(21)} T ( z , t ) = τ ( t ) − ( 2 ρ ( t ) ) 2 1 t − 2 ( t − 1 ) t ( 1 − ρ ( t ) z ) 2 1 + O ( 1 − ρ ( t ) z ) , (21) uniformly for t t t ( 2 ρ ( t ) ) 1 / 2 (2\rho(t))^{1/2} ( 2 ρ ( t ) ) 1/2 t = 1 t=1 t = 1 t − t / ( 2 ( t − 1 ) ) t^{-t/(2(t-1))} t − t / ( 2 ( t − 1 )) e − 1 / 2 e^{-1/2} e − 1/2
Proof. A direct computation gives
R w ( τ ( t ) , t ) = 0 , R ( τ ( t ) , t ) = ρ ( t ) , R_w(\tau(t),t)=0,\qquad R(\tau(t),t)=\rho(t), R w ( τ ( t ) , t ) = 0 , R ( τ ( t ) , t ) = ρ ( t ) , so w = τ ( t ) w=\tau(t) w = τ ( t ) R ( ⋅ , t ) R(\mathord{\cdot},t) R ( ⋅ , t ) ρ ( t ) \rho(t) ρ ( t ) [5 Equations (3.3)–(3.8)] , the inverse function theorem at this quadratic critical point gives a convergent expansion
1 + φ ( z , t ) = t 1 t − 1 − ( 2 ρ ( t ) t 2 − t t − 1 ) 1 2 ( 1 − z ρ ( t ) ) 1 2 + O ( 1 − z ρ ( t ) ) , 1+\varphi(z,t)=t^{\frac{1}{t-1}}-\left(2\rho(t)t^{\frac{2-t}{t-1}}\right)^{\frac{1}{2}}\left(1-\frac{z}{\rho(t)}\right)^{\frac{1}{2}}+O\left(1-\frac{z}{\rho(t)}\right), 1 + φ ( z , t ) = t t − 1 1 − ( 2 ρ ( t ) t t − 1 2 − t ) 2 1 ( 1 − ρ ( t ) z ) 2 1 + O ( 1 − ρ ( t ) z ) , whose coefficients depend holomorphically on t t t τ ( t ) \tau(t) τ ( t ) t 1 / ( t − 1 ) ≠ 0 t^{1/(t-1)}\ne0 t 1/ ( t − 1 ) = 0
It remains to show that ρ ( t ) \rho(t) ρ ( t ) t = 1 t=1 t = 1 R ( w , 1 ) = w − ∑ n ≥ 3 n − 2 n ! w n R(w,1)=w-\sum_{n\ge3}\frac{n-2}{n!}w^n R ( w , 1 ) = w − ∑ n ≥ 3 n ! n − 2 w n T ( z , 1 ) T(z,1) T ( z , 1 ) R w ( w , 1 ) = ( 1 − w ) e w > 0 R_w(w,1)=(1-w)e^w>0 R w ( w , 1 ) = ( 1 − w ) e w > 0 [ 0 , 1 ) [0,1) [ 0 , 1 ) R ( 0 , 1 ) = 0 R(0,1)=0 R ( 0 , 1 ) = 0 R ( 1 , 1 ) = e − 2 R(1,1)=e-2 R ( 1 , 1 ) = e − 2 [ 0 , e − 2 ) [0,e-2) [ 0 , e − 2 ) 1 1 1 z z z e − 2 e-2 e − 2 [10 Theorem IV.6] , the radius of convergence is e − 2 e-2 e − 2 T ( e − 2 , 1 ) = 1 T(e-2,1)=1 T ( e − 2 , 1 ) = 1 ∣ z ∣ ≤ e − 2 |z|\le e-2 ∣ z ∣ ≤ e − 2 ∣ z ∣ = e − 2 |z|=e-2 ∣ z ∣ = e − 2 R w ( T ( z , 1 ) , 1 ) = 0 R_w(T(z,1),1)=0 R w ( T ( z , 1 ) , 1 ) = 0 T ( z , 1 ) = 1 T(z,1)=1 T ( z , 1 ) = 1 z = R ( 1 , 1 ) = e − 2 z=R(1,1)=e-2 z = R ( 1 , 1 ) = e − 2 T ( ⋅ , 1 ) T(\cdot,1) T ( ⋅ , 1 ) ∣ z ∣ = e − 2 |z|=e-2 ∣ z ∣ = e − 2 e − 2 e-2 e − 2 t t t Δ t = ρ ( t ) Δ 0 \Delta_t=\rho(t)\Delta_0 Δ t = ρ ( t ) Δ 0 t t t 1 1 1 t t t
The square-root singularity of T T T w = τ ( t ) w=\tau(t) w = τ ( t ) R R R R w R_w R w
R w ( w , t ) = t e w − e t w t − 1 = e w ( 1 − e ( t − 1 ) w − 1 t − 1 ) . R_w(w,t)=\frac{te^w-e^{tw}}{t-1}=e^w\left(1-\frac{e^{(t-1)w}-1}{t-1}\right). R w ( w , t ) = t − 1 t e w − e tw = e w ( 1 − t − 1 e ( t − 1 ) w − 1 ) . We write the second factor as 1 − W ( w , t ) 1-W(w,t) 1 − W ( w , t )
W ( w , t ) = e ( t − 1 ) w − 1 t − 1 , W ( w , 1 ) = w . (22) W(w,t)=\frac{e^{(t-1)w}-1}{t-1},\qquad W(w,1)=w. \tag*{(22)} W ( w , t ) = t − 1 e ( t − 1 ) w − 1 , W ( w , 1 ) = w . (22) The function W W W t t t 1 1 1 1 / ( 1 − W ( w , t ) ) 1/(1-W(w,t)) 1/ ( 1 − W ( w , t )) w = τ ( t ) w=\tau(t) w = τ ( t ) W ( τ ( t ) , t ) = 1 W(\tau(t),t)=1 W ( τ ( t ) , t ) = 1 W w ( τ ( t ) , t ) = t ≠ 0 W_w(\tau(t),t)=t\ne0 W w ( τ ( t ) , t ) = t = 0 R R R ρ ( t ) \rho(t) ρ ( t ) T T T 1 / ( 1 − W ) 1/(1-W) 1/ ( 1 − W ) [10 Example III.25] .
The logarithmic singularity in genus one
Following [19 Section 4] , we analyze the genus-one light spaces through the morphisms M ‾ 1 , 0 ∣ n → M ‾ 1 , 1 \overline{\mathcal{M}}_{1,0|n}\to\overline{\mathcal{M}}_{1,1} M 1 , 0∣ n → M 1 , 1 n ≥ 1 n\ge1 n ≥ 1
Lemma 3.3. The full Hodge series has the form
L 1 ( w ; u , v ) = − 1 2 log ( 1 − W ( w , u v ) ) + J 1 ( w ; u , v ) , (23) \mathcal{L}_1(w;u,v)=-\frac{1}{2}\log(1-W(w,uv))+\mathcal{J}_1(w;u,v), \tag*{(23)} L 1 ( w ; u , v ) = − 2 1 log ( 1 − W ( w , uv )) + J 1 ( w ; u , v ) , (23) where the logarithm vanishes at w = 0 w=0 w = 0 J 1 \mathcal{J}_1 J 1 w w w ( u , v ) (u,v) ( u , v )
Proof. We decompose M ‾ 1 , 0 ∣ n \overline{\mathcal{M}}_{1,0|n} M 1 , 0∣ n M ‾ 1 , 1 \overline{\mathcal{M}}_{1,1} M 1 , 1 M ‾ 1 , 1 \overline{\mathcal{M}}_{1,1} M 1 , 1
The boundary fiber. Its curves have only rational components. Since the total weight is one, stability forces every component to have at least two node branches. The genus condition then forces the dual graph to be a cycle, and stability requires each component to carry a marking. For a cycle of j ≥ 3 j\ge3 j ≥ 3
( n j ) ( j − 1 ) ! 2 ( u v − 1 ) n − j \binom{n}{j}\frac{(j-1)!}{2}(uv-1)^{n-j} ( j n ) 2 ( j − 1 )! ( uv − 1 ) n − j by [19 Proposition 4.3] . Here the Stirling number counts the partitions of the n n n j j j ( j − 1 ) ! / 2 (j-1)!/2 ( j − 1 )! /2 k k k 0 0 0 ∞ \infty ∞ ( C ∗ ) k / C ∗ ≅ ( C ∗ ) k − 1 (\mathbb{C}^*)^k/\mathbb{C}^*\cong(\mathbb{C}^*)^{k-1} ( C ∗ ) k / C ∗ ≅ ( C ∗ ) k − 1 ( u v − 1 ) k − 1 (uv-1)^{k-1} ( uv − 1 ) k − 1 ∑ n ( n j ) x n / n ! = ( e x − 1 ) j / j ! \sum_n\binom{n}{j}x^n/n!=(e^x-1)^j/j! ∑ n ( j n ) x n / n ! = ( e x − 1 ) j / j ! [10 Appendix A.8] and writing W = W ( w , u v ) W=W(w,uv) W = W ( w , uv )
∑ j ≥ 3 ∑ n ≥ j ( n j ) ( j − 1 ) ! 2 ( u v − 1 ) n − j w n n ! = 1 2 ∑ j ≥ 3 W j j = − 1 2 log ( 1 − W ) − W 2 − W 2 4 . \sum_{j\ge3}\sum_{n\ge j}\binom{n}{j}\frac{(j-1)!}{2}(uv-1)^{n-j}\frac{w^n}{n!}
=\frac{1}{2}\sum_{j\ge3}\frac{W^j}{j}
=-\frac{1}{2}\log(1-W)-\frac{W}{2}-\frac{W^2}{4}. j ≥ 3 ∑ n ≥ j ∑ ( j n ) 2 ( j − 1 )! ( uv − 1 ) n − j n ! w n = 2 1 j ≥ 3 ∑ j W j = − 2 1 log ( 1 − W ) − 2 W − 4 W 2 . By [19 Proposition 4.3] , the cycles with one and two components contribute
( u v + 1 ) n − 1 + ( u v − 1 ) n − 1 2 ( n ≥ 1 ) , \frac{(uv+1)^{n-1}+(uv-1)^{n-1}}{2}\qquad(n\ge1), 2 ( uv + 1 ) n − 1 + ( uv − 1 ) n − 1 ( n ≥ 1 ) , ( 2 n − 1 − 1 ) ( u v + 1 ) n − 2 + ( u v − 1 ) n − 2 2 ( n ≥ 2 ) . \frac{(2^{n-1}-1)(uv+1)^{n-2}+(uv-1)^{n-2}}{2}\qquad(n\ge2). 2 ( 2 n − 1 − 1 ) ( uv + 1 ) n − 2 + ( uv − 1 ) n − 2 ( n ≥ 2 ) . If ∣ u v ± 1 ∣ ≤ A |uv\pm1|\le A ∣ uv ± 1∣ ≤ A A ≥ 1 A\ge1 A ≥ 1 ( 2 A ) n (2A)^n ( 2 A ) n ∣ w ∣ ≤ R |w|\le R ∣ w ∣ ≤ R e 2 A R e^{2AR} e 2 A R w w w ( w , u , v ) (w,u,v) ( w , u , v )
The interior. Let I n ( u , v ) I_n(u,v) I n ( u , v ) M 1 , 1 \mathcal{M}_{1,1} M 1 , 1 [19 Proposition 4.1] , I n ( u , v ) = I n e v e n ( u v ) + I n o d d ( u , v ) I_n(u,v)=I_n^{\mathrm{even}}(uv)+I_n^{\mathrm{odd}}(u,v) I n ( u , v ) = I n even ( uv ) + I n odd ( u , v )
I n e v e n ( t ) = ∑ j = 0 ⌊ ( n − 1 ) / 2 ⌋ ( n − 1 2 j ) ( 1 + t ) n − 1 − 2 j ( α j j t j + 1 − ∑ i = 0 j − 1 α i j t i ) , I_n^{\mathrm{even}}(t)=\sum_{j=0}^{\lfloor(n-1)/2\rfloor}\binom{n-1}{2j}(1+t)^{n-1-2j}\left(\alpha_{jj}t^{j+1}-\sum_{i=0}^{j-1}\alpha_{ij}t^i\right), I n even ( t ) = j = 0 ∑ ⌊( n − 1 ) /2 ⌋ ( 2 j n − 1 ) ( 1 + t ) n − 1 − 2 j ( α j j t j + 1 − i = 0 ∑ j − 1 α ij t i ) , I n o d d ( u , v ) = − ∑ j = 0 ⌊ ( n − 1 ) / 2 ⌋ ( n − 1 2 j ) ( 1 + u v ) n − 1 − 2 j ∑ i = 0 j − 1 α i j β 2 j − 2 i + 2 ( u v ) i ( u 2 j − 2 i + 1 + v 2 j − 2 i + 1 ) , I_n^{\mathrm{odd}}(u,v)=-\sum_{j=0}^{\lfloor(n-1)/2\rfloor}\binom{n-1}{2j}(1+uv)^{n-1-2j}\sum_{i=0}^{j-1}\alpha_{ij}\beta_{2j-2i+2}(uv)^i\left(u^{2j-2i+1}+v^{2j-2i+1}\right), I n odd ( u , v ) = − j = 0 ∑ ⌊( n − 1 ) /2 ⌋ ( 2 j n − 1 ) ( 1 + uv ) n − 1 − 2 j i = 0 ∑ j − 1 α ij β 2 j − 2 i + 2 ( uv ) i ( u 2 j − 2 i + 1 + v 2 j − 2 i + 1 ) , with
α i j = ( 2 j i ) − ( 2 j i − 1 ) , β ℓ = { ⌊ ℓ / 12 ⌋ − 1 , ℓ ≡ 2 ( m o d 12 ) , ⌊ ℓ / 12 ⌋ , otherwise . \alpha_{ij}=\binom{2j}{i}-\binom{2j}{i-1},\qquad
\beta_\ell=
\begin{cases}
\lfloor\ell/12\rfloor-1, & \ell\equiv2\pmod{12},\\
\lfloor\ell/12\rfloor, & \text{otherwise}.
\end{cases} α ij = ( i 2 j ) − ( i − 1 2 j ) , β ℓ = { ⌊ ℓ /12 ⌋ − 1 , ⌊ ℓ /12 ⌋ , ℓ ≡ 2 ( mod 12 ) , otherwise . For even ℓ ≥ 4 \ell\ge4 ℓ ≥ 4 β ℓ = dim S ℓ ( SL 2 ( Z ) ) \beta_\ell=\dim S_\ell(\operatorname{SL}_2(\mathbb{Z})) β ℓ = dim S ℓ ( SL 2 ( Z )) t = u v t=uv t = uv M = max ( 1 , ∣ u ∣ , ∣ v ∣ ) M=\max(1,|u|,|v|) M = max ( 1 , ∣ u ∣ , ∣ v ∣ ) 0 ≤ α i j ≤ 4 j 0\le\alpha_{ij}\le4^j 0 ≤ α ij ≤ 4 j 0 ≤ i ≤ j 0\le i\le j 0 ≤ i ≤ j 0 ≤ β 2 j − 2 i + 2 ≤ n 0\le\beta_{2j-2i+2}\le n 0 ≤ β 2 j − 2 i + 2 ≤ n 2 j ≤ n 2j\le n 2 j ≤ n
∣ α j j t j + 1 − ∑ i = 0 j − 1 α i j t i ∣ ≤ ( j + 1 ) 4 j M 2 j + 2 ≤ n M 2 ( 2 M ) 2 j , \left|\alpha_{jj}t^{j+1}-\sum_{i=0}^{j-1}\alpha_{ij}t^i\right|
\le(j+1)4^jM^{2j+2}\le nM^2(2M)^{2j}, α j j t j + 1 − i = 0 ∑ j − 1 α ij t i ≤ ( j + 1 ) 4 j M 2 j + 2 ≤ n M 2 ( 2 M ) 2 j , and the inner sum in I n o d d I_n^{\mathrm{odd}} I n odd n 2 M ( 2 M ) 2 j n^2M(2M)^{2j} n 2 M ( 2 M ) 2 j ∣ 1 + u v ∣ ≤ 1 + M 2 |1+uv|\le1+M^2 ∣1 + uv ∣ ≤ 1 + M 2
∣ I n ( u , v ) ∣ ≤ K n 2 M 2 ∑ j ( n − 1 2 j ) ( 1 + M 2 ) n − 1 − 2 j ( 2 M ) 2 j ≤ K n 2 M 2 ( M + 1 ) 2 n − 2 |I_n(u,v)|\le Kn^2M^2\sum_j\binom{n-1}{2j}(1+M^2)^{n-1-2j}(2M)^{2j}
\le Kn^2M^2(M+1)^{2n-2} ∣ I n ( u , v ) ∣ ≤ K n 2 M 2 j ∑ ( 2 j n − 1 ) ( 1 + M 2 ) n − 1 − 2 j ( 2 M ) 2 j ≤ K n 2 M 2 ( M + 1 ) 2 n − 2 for an absolute constant K K K I n e v e n ( u v ) I_n^{\mathrm{even}}(uv) I n even ( uv ) I n o d d ( u , v ) I_n^{\mathrm{odd}}(u,v) I n odd ( u , v ) ( w , u , v ) (w,u,v) ( w , u , v ) − W / 2 − W 2 / 4 -W/2-W^2/4 − W /2 − W 2 /4 J 1 \mathcal{J}_1 J 1
Averaging the specializations ( u , v ) = ( t , t ) (u,v)=(\sqrt{t},\sqrt{t}) ( u , v ) = ( t , t ) ( − t , − t ) (-\sqrt{t},-\sqrt{t}) ( − t , − t ) I n o d d I_n^{\mathrm{odd}} I n odd u v = t uv=t uv = t
L 1 ( w , t ) = − 1 2 log ( 1 − W ( w , t ) ) + J 1 ( w , t ) . (24) \mathcal{L}_1(w,t)=-\frac{1}{2}\log(1-W(w,t))+\mathcal{J}_1(w,t). \tag*{(24)} L 1 ( w , t ) = − 2 1 log ( 1 − W ( w , t )) + J 1 ( w , t ) . (24) with
J 1 ( w , t ) = ∑ n ≥ 1 I n e v e n ( t ) w n n ! + e ( t + 1 ) w − 1 2 ( t + 1 ) + ( e ( t + 1 ) w − 1 ) 2 4 ( t + 1 ) 2 . \mathcal{J}_1(w,t)=\sum_{n\ge1}I_n^{\mathrm{even}}(t)\frac{w^n}{n!}+\frac{e^{(t+1)w}-1}{2(t+1)}+\frac{(e^{(t+1)w}-1)^2}{4(t+1)^2}. J 1 ( w , t ) = n ≥ 1 ∑ I n even ( t ) n ! w n + 2 ( t + 1 ) e ( t + 1 ) w − 1 + 4 ( t + 1 ) 2 ( e ( t + 1 ) w − 1 ) 2 . The last two terms come from the cycles with one and two components after subtraction of W / 2 W/2 W /2 W 2 / 4 W^2/4 W 2 /4 t = − 1 t=-1 t = − 1 J 1 \mathcal{J}_1 J 1 w w w t t t
Since the even part of E ( M 1 , 0 ∣ n ; u , v ) E(\mathcal{M}_{1,0|n};u,v) E ( M 1 , 0∣ n ; u , v ) u v uv uv H 2 k ( M ‾ 1 , 0 ∣ n ) H^{2k}(\overline{\mathcal{M}}_{1,0|n}) H 2 k ( M 1 , 0∣ n ) ( k , k ) (k,k) ( k , k ) T ( z , u v ) T(z,uv) T ( z , uv ) u v uv uv H 2 k ( M ‾ 1 , n ) H^{2k}(\overline{\mathcal{M}}_{1,n}) H 2 k ( M 1 , n ) ( k , k ) (k,k) ( k , k )
Stable cores in higher genus
Now let g ≥ 2 g \ge2 g ≥ 2
Forgetting the markings and stabilizing defines π n : M ‾ g , 0 ∣ n → M ‾ g \pi_n : \overline{\mathcal{M}}_{g,0|n} \to\overline{\mathcal{M}}_g π n : M g , 0∣ n → M g Γ \Gamma Γ C C C g v g_v g v d v d_v d v 2 g v − 2 + d v > 0 2g_v - 2 + d_v > 0 2 g v − 2 + d v > 0 g = ∑ v g v + b 1 ( Γ ) g = \sum_v g_v + b_1(\Gamma) g = ∑ v g v + b 1 ( Γ ) Γ \Gamma Γ
S Γ = ∏ v ∈ V ( Γ ) M g v , d v . S_\Gamma= \prod_{v \in V(\Gamma)} \mathcal{M}_{g_v,d_v}. S Γ = v ∈ V ( Γ ) ∏ M g v , d v . a point of S Γ S_\Gamma S Γ C C C M ‾ g \overline{\mathcal{M}}_g M g Γ \Gamma Γ M Γ = [ S Γ / Aut Γ ] \mathcal{M}_\Gamma= [S_\Gamma/\operatorname{Aut}\Gamma] M Γ = [ S Γ / Aut Γ ]
dim M Γ = ∑ v ( 3 g v − 3 + d v ) = 3 g − 3 − ∣ E ( Γ ) ∣ . (25) \dim\mathcal{M}_\Gamma= \sum_v (3g_v - 3 + d_v) = 3g - 3 - |E(\Gamma)|. \tag*{(25)} dim M Γ = v ∑ ( 3 g v − 3 + d v ) = 3 g − 3 − ∣ E ( Γ ) ∣. (25) A light curve with core C C C n n n C C C π n \pi_n π n k ≥ 1 k \ge1 k ≥ 1 L M k = M ‾ 0 , 2 ∣ k \mathrm{LM}_k = \overline{\mathcal{M}}_{0,2|k} LM k = M 0 , 2∣ k [21 ] , whose two heavy markings are the ordered endpoints and whose light markings carry the assigned labels. For k = 0 k = 0 k = 0 L M 0 = p t \mathrm{LM}_0 = \mathrm{pt} LM 0 = pt
The markings on the smooth locus of the stable core are parametrized as follows. Let F Γ → S Γ \mathcal{F}_\Gamma\to S_\Gamma F Γ → S Γ M g v , d v \mathcal{M}_{g_v,d_v} M g v , d v S Γ S_\Gamma S Γ v v v
X Γ , r = ( F Γ ) S Γ r ( r ≥ 1 ) , X Γ , 0 = S Γ , X_{\Gamma,r} = (\mathcal{F}_\Gamma)^r_{S_\Gamma} \quad(r \ge1), \qquad X_{\Gamma,0} = S_\Gamma, X Γ , r = ( F Γ ) S Γ r ( r ≥ 1 ) , X Γ , 0 = S Γ , the r r r r r r Aut Γ \operatorname{Aut}\Gamma Aut Γ S Γ S_\Gamma S Γ F Γ \mathcal{F}_\Gamma F Γ X Γ , r X_{\Gamma,r} X Γ , r σ ∈ Aut Γ \sigma\in\operatorname{Aut}\Gamma σ ∈ Aut Γ
B Γ , σ ( w ; u , v ) = ∑ r ≥ 0 E σ ( X Γ , r ; u , v ) w r r ! . (26) \mathcal{B}_{\Gamma,\sigma}(w;u,v) = \sum_{r \ge0} E_\sigma(X_{\Gamma,r};u,v)\frac{w^r}{r!}. \tag*{(26)} B Γ , σ ( w ; u , v ) = r ≥ 0 ∑ E σ ( X Γ , r ; u , v ) r ! w r . (26) Only assignments of markings to vertices fixed by σ \sigma σ σ \sigma σ σ \sigma σ
For an edge fixed by σ \sigma σ σ \sigma σ ι \iota ι L M k \mathrm{LM}_k LM k
C ( w , u v ) = ∑ k ≥ 0 E ( L M k ; u , v ) w k k ! , D ( w , u v ) = ∑ k ≥ 0 E ι ( L M k ; u , v ) w k k ! . C(w,uv) = \sum_{k \ge0} E(\mathrm{LM}_k;u,v)\frac{w^k}{k!}, \qquad D(w,uv) = \sum_{k \ge0} E_\iota(\mathrm{LM}_k;u,v)\frac{w^k}{k!}. C ( w , uv ) = k ≥ 0 ∑ E ( LM k ; u , v ) k ! w k , D ( w , uv ) = k ≥ 0 ∑ E ι ( LM k ; u , v ) k ! w k . Both depend on ( u , v ) (u,v) ( u , v ) u v uv uv Lemma 3.6 ).
Proposition 3.4. Let g ≥ 2 g \ge2 g ≥ 2 Γ \Gamma Γ g g g σ ∈ Aut Γ \sigma\in\operatorname{Aut}\Gamma σ ∈ Aut Γ p ( σ ) p(\sigma) p ( σ ) q ( σ ) q(\sigma) q ( σ ) σ \sigma σ
L g ( w ; u , v ) = ∑ Γ 1 ∣ Aut Γ ∣ ∑ σ ∈ Aut Γ B Γ , σ ( w ; u , v ) C ( w , u v ) p ( σ ) D ( w , u v ) q ( σ ) . (27) \mathcal{L}_g(w;u,v) = \sum_\Gamma\frac{1}{|\operatorname{Aut}\Gamma|} \sum_{\sigma\in\operatorname{Aut}\Gamma} \mathcal{B}_{\Gamma,\sigma}(w;u,v) C(w,uv)^{p(\sigma)} D(w,uv)^{q(\sigma)}. \tag*{(27)} L g ( w ; u , v ) = Γ ∑ ∣ Aut Γ∣ 1 σ ∈ Aut Γ ∑ B Γ , σ ( w ; u , v ) C ( w , uv ) p ( σ ) D ( w , uv ) q ( σ ) . (27) where Γ \Gamma Γ
Proof. Fix Γ \Gamma Γ n n n π n − 1 ( M Γ ) \pi_n^{-1}(\mathcal{M}_{\Gamma}) π n − 1 ( M Γ ) S Γ S_{\Gamma} S Γ r r r k e k_e k e e e e
X Γ , r × ∏ e ∈ E ( Γ ) L M k e X_{\Gamma,r} \times\prod_{e\in E(\Gamma)} LM_{k_e} X Γ , r × e ∈ E ( Γ ) ∏ L M k e where, after ordering the half-edges of each edge, a nonempty chain is attached by identifying its ordered endpoints with the two flag points. No further gluing data are needed. The chains have no automorphisms preserving the endpoints and the labels, because each component has two attaching points and at least one marking, and automorphisms of the core components are already accounted for in X Γ , r X_{\Gamma,r} X Γ , r Aut Γ \operatorname{Aut}\Gamma Aut Γ π n − 1 ( M Γ ) \pi_n^{-1}(\mathcal{M}_{\Gamma}) π n − 1 ( M Γ )
Fix σ ∈ Aut Γ \sigma\in\operatorname{Aut}\Gamma σ ∈ Aut Γ σ \sigma σ σ \sigma σ ι \iota ι C C C D D D B Γ , σ B_{\Gamma,\sigma} B Γ , σ n ! / ( r ! ∏ k e ! ) n!/(r!\prod k_e!) n ! / ( r ! ∏ k e !) r r r ( k e ) (k_e) ( k e ) n n n σ \sigma σ B Γ , σ C p ( σ ) D q ( σ ) B_{\Gamma,\sigma}C^{p(\sigma)}D^{q(\sigma)} B Γ , σ C p ( σ ) D q ( σ ) Aut Γ \operatorname{Aut}\Gamma Aut Γ M Γ \mathcal{M}_{\Gamma} M Γ M g \mathcal{M}_g M g (27) .
Example 3.5 (The theta graph). Put t = u v t=uv t = uv Γ θ \Gamma_{\theta} Γ θ v 1 , v 2 v_1,v_2 v 1 , v 2 S Γ θ = M 0 , 3 2 S_{\Gamma_{\theta}}=\mathcal{M}_{0,3}^2 S Γ θ = M 0 , 3 2 U = U 1 ⊔ U 2 U=U_1\sqcup U_2 U = U 1 ⊔ U 2 U i = P 1 ∖ { 0 , 1 , ∞ } U_i=\mathbb{P}^1\setminus\{0,1,\infty\} U i = P 1 ∖ { 0 , 1 , ∞ } ∣ Aut Γ θ ∣ = 2 ⋅ 3 ! = 12 |\operatorname{Aut}\Gamma_{\theta}|=2\cdot3!=12 ∣ Aut Γ θ ∣ = 2 ⋅ 3 ! = 12 E ( U ) = 2 ( t − 2 ) E(U)=2(t-2) E ( U ) = 2 ( t − 2 ) B Γ θ , 1 = e 2 ( t − 2 ) w B_{\Gamma_{\theta},1}=e^{2(t-2)w} B Γ θ , 1 = e 2 ( t − 2 ) w e 2 ( t − 2 ) w C 3 / 12 e^{2(t-2)w}C^3/12 e 2 ( t − 2 ) w C 3 /12
If σ \sigma σ E σ ( U i ) = 1 + t − # { fixed flags at v i } E_{\sigma}(U_i)=1+t-\#\{\text{fixed flags at }v_i\} E σ ( U i ) = 1 + t − # { fixed flags at v i } B Γ θ , σ = e 2 t w B_{\Gamma_{\theta},\sigma}=e^{2tw} B Γ θ , σ = e 2 tw ( p , q ) = ( 1 , 0 ) (p,q)=(1,0) ( p , q ) = ( 1 , 0 ) 3 3 3 e 2 ( t + 1 ) w e^{2(t+1)w} e 2 ( t + 1 ) w ( p , q ) = ( 0 , 0 ) (p,q)=(0,0) ( p , q ) = ( 0 , 0 ) σ \sigma σ r = 0 r=0 r = 0 B Γ θ , σ = 1 B_{\Gamma_{\theta},\sigma}=1 B Γ θ , σ = 1 ( p , q ) = ( 0 , 3 ) (p,q)=(0,3) ( p , q ) = ( 0 , 3 ) ( p , q ) = ( 0 , 1 ) (p,q)=(0,1) ( p , q ) = ( 0 , 1 ) ( p , q ) = ( 0 , 0 ) (p,q)=(0,0) ( p , q ) = ( 0 , 0 ) Γ θ \Gamma_{\theta} Γ θ
1 12 ( e 2 ( t − 2 ) w C 3 + 3 e 2 t w C + 2 e 2 ( t + 1 ) w + D 3 + 3 D + 2 ) \frac{1}{12}\left(e^{2(t-2)w}C^3+3e^{2tw}C+2e^{2(t+1)w}+D^3+3D+2\right) 12 1 ( e 2 ( t − 2 ) w C 3 + 3 e 2 tw C + 2 e 2 ( t + 1 ) w + D 3 + 3 D + 2 ) to L 2 ( w ; u , v ) \mathcal{L}_2(w;u,v) L 2 ( w ; u , v ) C = C ( w , t ) C=C(w,t) C = C ( w , t ) D = D ( w , t ) D=D(w,t) D = D ( w , t )
We next compute the chain factors and show that the core factors are entire. Together, these two facts locate the poles of (27) .
Lemma 3.6. With t = u v t=uv t = uv
C ( w , t ) = t − 1 t − e ( t − 1 ) w = 1 1 − W ( w , t ) , D ( w , t ) = 1 + e ( t + 1 ) w − 1 t + 1 . (28) C(w,t)=\frac{t-1}{t-e^{(t-1)w}}=\frac{1}{1-W(w,t)},\qquad D(w,t)=1+\frac{e^{(t+1)w}-1}{t+1}. \tag*{(28)} C ( w , t ) = t − e ( t − 1 ) w t − 1 = 1 − W ( w , t ) 1 , D ( w , t ) = 1 + t + 1 e ( t + 1 ) w − 1 . (28) In particular, D D D ( w , t ) (w,t) ( w , t )
Proof. The formula for C C C [21 Theorem 2.3] . Bergström and Minabe computed the S 2 × S k S_2\times S_k S 2 × S k L M k LM_k L M k [1 Theorem 2.3] , and both formulas follow from [1 Proposition 4.1] by specialization, as we now explain.
The cohomology of L M k LM_k L M k H 2 j H^{2j} H 2 j ( j , j ) (j,j) ( j , j ) q q q [1 ] is our t t t p m x p_m^x p m x p m y p_m^y p m y 1 / ( 2 ⋅ k ! ) 1/(2\cdot k!) 1/ ( 2 ⋅ k !) p 1 y = w p_1^y=w p 1 y = w p m y = 0 p_m^y=0 p m y = 0 m ≥ 2 m\geq2 m ≥ 2 w k / k ! w^k/k! w k / k !
s ( k − j , 1 j ) y ⟼ ( k − 1 j ) w k k ! , s^{y}_{(k-j,1^j)}\longmapsto\binom{k-1}{j}\frac{w^k}{k!}, s ( k − j , 1 j ) y ⟼ ( j k − 1 ) k ! w k , so the functions of [1 Definition 2.2] , with q = t q=t q = t
f k y = ∑ j = 0 k − 1 ( − 1 ) j s ( k − j , 1 j ) y t k − 1 − j ⟼ ( t − 1 ) k − 1 w k k ! . f_k^y=\sum_{j=0}^{k-1}(-1)^j s_{(k-j,1^j)}^y t^{k-1-j}\longmapsto(t-1)^{k-1}\frac{w^k}{k!}. f k y = j = 0 ∑ k − 1 ( − 1 ) j s ( k − j , 1 j ) y t k − 1 − j ⟼ ( t − 1 ) k − 1 k ! w k . g k y = ∑ j = 0 k − 1 s ( k − j , 1 j ) y t k − 1 − j ⟼ ( t + 1 ) k − 1 w k k ! . g_k^y=\sum_{j=0}^{k-1}s_{(k-j,1^j)}^y t^{k-1-j}\longmapsto(t+1)^{k-1}\frac{w^k}{k!}. g k y = j = 0 ∑ k − 1 s ( k − j , 1 j ) y t k − 1 − j ⟼ ( t + 1 ) k − 1 k ! w k . For the identity of S 2 S_2 S 2 ( p 1 x ) 2 = 2 (p_1^x)^2=2 ( p 1 x ) 2 = 2 p 2 x = 0 p_2^x=0 p 2 x = 0 [1 (4.2)] becomes
C ( w , t ) = ( 1 − ∑ k ≥ 1 ( t − 1 ) k − 1 w k k ! ) − 1 = 1 1 − W ( w , t ) . C(w,t)=\left(1-\sum_{k\geq1}(t-1)^{k-1}\frac{w^k}{k!}\right)^{-1}=\frac{1}{1-W(w,t)}. C ( w , t ) = ( 1 − k ≥ 1 ∑ ( t − 1 ) k − 1 k ! w k ) − 1 = 1 − W ( w , t ) 1 . For the exchange of the endpoints, set p 1 x = 0 p_1^x=0 p 1 x = 0 p 2 x = 2 p_2^x=2 p 2 x = 2 [1 (4.2)] becomes
( 1 + ∑ k ≥ 1 g k y ) ( 1 − ∑ k ≥ 1 p 2 y ∘ f k y ) − 1 . \left(1+\sum_{k\geq1}g_k^y\right)\left(1-\sum_{k\geq1}p_2^y\circ f_k^y\right)^{-1}. ( 1 + k ≥ 1 ∑ g k y ) ( 1 − k ≥ 1 ∑ p 2 y ∘ f k y ) − 1 . The plethysm p 2 y ∘ f k y p_2^y\circ f_k^y p 2 y ∘ f k y p m y p_m^y p m y p 2 m y p_{2m}^y p 2 m y t t t t 2 t^2 t 2
D ( w , t ) = 1 + ∑ k ≥ 1 ( t + 1 ) k − 1 w k k ! = 1 + e ( t + 1 ) w − 1 t + 1 . D(w,t)=1+\sum_{k\geq1}(t+1)^{k-1}\frac{w^k}{k!}=1+\frac{e^{(t+1)w}-1}{t+1}. D ( w , t ) = 1 + k ≥ 1 ∑ ( t + 1 ) k − 1 k ! w k = 1 + t + 1 e ( t + 1 ) w − 1 . The constant terms 1 correspond to L M 0 = p t \mathrm{LM}_0=\mathrm{pt} LM 0 = pt D D D t = − 1 t=-1 t = − 1 1 + w 1+w 1 + w
Remark 3.7. Under the substitution (19) , the chain factor C C C
C ( T ( z , t ) , t ) = e T ( z , t ) T z ( z , t ) = ∂ z φ ( z , t ) = 1 + ∑ n ≥ 1 E ( M ‾ 0 , n + 2 ; u , v ) z n n ! , t = u v . C(T(z,t),t)=e^{T(z,t)}T_z(z,t)=\partial_z\varphi(z,t)=1+\sum_{n\geq1}E(\overline{\mathcal M}_{0,n+2};u,v)\frac{z^n}{n!},\qquad t=uv. C ( T ( z , t ) , t ) = e T ( z , t ) T z ( z , t ) = ∂ z φ ( z , t ) = 1 + n ≥ 1 ∑ E ( M 0 , n + 2 ; u , v ) n ! z n , t = uv . Indeed, (17) and (28) give R w ( w , t ) = e w / C ( w , t ) R_w(w,t)=e^w/C(w,t) R w ( w , t ) = e w / C ( w , t ) R ( T ( z , t ) , t ) = z R(T(z,t),t)=z R ( T ( z , t ) , t ) = z (18) and the definition of φ \varphi φ M ‾ 0 , n + 2 → L M n \overline{\mathcal M}_{0,n+2}\to\mathrm{LM}_n M 0 , n + 2 → LM n
Lemma 3.8. For σ ∈ Aut Γ \sigma\in\operatorname{Aut}\Gamma σ ∈ Aut Γ B Γ , σ B_{\Gamma,\sigma} B Γ , σ (26) is entire in w w w ( u , v ) (u,v) ( u , v )
Proof. Because of the factorial in (26) , it suffices to bound E σ ( X Γ , r ; u , v ) E_\sigma(X_{\Gamma,r};u,v) E σ ( X Γ , r ; u , v ) A B r AB^r A B r ( u , v ) (u,v) ( u , v ) A , B A,B A , B r r r
Replace each positive-genus factor of S Γ S_\Gamma S Γ S ′ S' S ′ [25 Section 2.3.3] in genus at least two and [17 Corollary 4.7.2] in genus one, taking one flag as the origin. Genus-zero factors need no cover, and the cover does not depend on r r r X Γ , r ′ = X Γ , r × S Γ S ′ X'_{\Gamma,r}=X_{\Gamma,r}\times_{S_\Gamma}S' X Γ , r ′ = X Γ , r × S Γ S ′ X Γ , r ′ → S ′ X'_{\Gamma,r}\to S' X Γ , r ′ → S ′
The variety S ′ S' S ′ K K K A \mathcal A A
∑ p dim H p ( S ′ ; A ) ≤ K rank A . (29) \sum_p\dim H^p(S';\mathcal A)\leq K\operatorname{rank}\mathcal A. \tag*{(29)} p ∑ dim H p ( S ′ ; A ) ≤ K rank A . (29) whatever the monodromy of A \mathcal A A B B B F Γ \mathcal F_\Gamma F Γ r r r f r : X Γ , r ′ → S ′ fr:X'_{\Gamma,r}\to S' f r : X Γ , r ′ → S ′ B r B^r B r (29) give
∑ i b i ( X Γ , r ′ ) ≤ ∑ p , q dim H p ( S ′ ; R q f r ∗ Q ) ≤ K B r . (30) \sum_i b_i(X'_{\Gamma,r})\leq\sum_{p,q}\dim H^p(S';R^qf_{r*}\mathbb{Q})\leq KB^r. \tag*{(30)} i ∑ b i ( X Γ , r ′ ) ≤ p , q ∑ dim H p ( S ′ ; R q f r ∗ Q ) ≤ K B r . (30) The cohomological transfer for the finite étale cover X Γ , r ′ → X Γ , r X'_{\Gamma,r}\to X_{\Gamma,r} X Γ , r ′ → X Γ , r X Γ , r X_{\Gamma,r} X Γ , r X Γ , r ′ X'_{\Gamma,r} X Γ , r ′
Finally, the eigenvalues of σ \sigma σ X Γ , r X_{\Gamma,r} X Γ , r dim X Γ , r = dim S Γ + r \dim X_{\Gamma,r}=\dim S_{\Gamma}+r dim X Γ , r = dim S Γ + r K 0 ⊂ C 2 K_0\subset\mathbb{C}^2 K 0 ⊂ C 2 A K 0 , B K 0 A_{K_0},B_{K_0} A K 0 , B K 0 sup ( u , v ) ∈ K 0 ∣ E σ ( X Γ , r ; u , v ) ∣ ≤ A K 0 B K 0 r \sup_{(u,v)\in K_0}|E_\sigma(X_{\Gamma,r};u,v)|\leq A_{K_0}B_{K_0}^r sup ( u , v ) ∈ K 0 ∣ E σ ( X Γ , r ; u , v ) ∣ ≤ A K 0 B K 0 r r r r (26) converges absolutely and uniformly on compact subsets. The level cover is used only for the Betti bound and need not be compatible with σ \sigma σ
The pole of order 3 g − 3 3g-3 3 g − 3
Let G g \mathcal{G}_g G g g g g
c g = ∑ Γ ∈ G g 1 ∣ Aut Γ ∣ > 0. (31) c_g=\sum_{\Gamma\in\mathcal{G}_g}\frac{1}{|\operatorname{Aut}\Gamma|}>0. \tag*{(31)} c g = Γ ∈ G g ∑ ∣ Aut Γ∣ 1 > 0. (31) Proposition 3.9. For g ≥ 2 g\geq2 g ≥ 2 J g ( w ; u , v ) \mathcal{J}_g(w;u,v) J g ( w ; u , v ) w w w ( u , v ) (u,v) ( u , v ) t = u v t=uv t = uv
L g ( w ; u , v ) = c g e 2 ( g − 1 ) ( t − 2 ) w ( 1 − W ( w , t ) ) 3 g − 3 + J g ( w ; u , v ) ( 1 − W ( w , t ) ) 3 g − 4 . (32) \mathcal{L}_g(w;u,v)=\frac{c_g e^{2(g-1)(t-2)w}}{(1-W(w,t))^{3g-3}}+\frac{\mathcal{J}_g(w;u,v)}{(1-W(w,t))^{3g-4}}. \tag*{(32)} L g ( w ; u , v ) = ( 1 − W ( w , t ) ) 3 g − 3 c g e 2 ( g − 1 ) ( t − 2 ) w + ( 1 − W ( w , t ) ) 3 g − 4 J g ( w ; u , v ) . (32) In particular, for t t t 1 1 1 L g \mathcal{L}_g L g 3 g − 3 3g-3 3 g − 3 w = τ ( t ) w=\tau(t) w = τ ( t )
Proof. By Lemmas 3.6 and 3.8, each term of (27) is an entire function divided by ( 1 − W ) p ( σ ) (1-W)^{p(\sigma)} ( 1 − W ) p ( σ ) p ( σ ) ≤ ∣ E ( Γ ) ∣ = 3 g − 3 − dim M Γ p(\sigma)\leq|E(\Gamma)|=3g-3-\dim\mathcal{M}_{\Gamma} p ( σ ) ≤ ∣ E ( Γ ) ∣ = 3 g − 3 − dim M Γ (25) . Equality p ( σ ) = 3 g − 3 p(\sigma)=3g-3 p ( σ ) = 3 g − 3 dim M Γ = 0 \dim\mathcal{M}_{\Gamma}=0 dim M Γ = 0 σ \sigma σ σ = 1 \sigma=1 σ = 1 Γ \Gamma Γ S Γ S_{\Gamma} S Γ 2 g − 2 2g-2 2 g − 2 P 1 \mathbb{P}^1 P 1 2 ( g − 1 ) ( t − 2 ) 2(g-1)(t-2) 2 ( g − 1 ) ( t − 2 ) B Γ , 1 = e 2 ( g − 1 ) ( t − 2 ) w B_{\Gamma,1}=e^{2(g-1)(t-2)w} B Γ , 1 = e 2 ( g − 1 ) ( t − 2 ) w (32) . All other terms have p ( σ ) ≤ 3 g − 4 p(\sigma)\leq3g-4 p ( σ ) ≤ 3 g − 4 ( 1 − W ) 3 g − 4 (1-W)^{3g-4} ( 1 − W ) 3 g − 4 W ( τ ( t ) , t ) = 1 W(\tau(t),t)=1 W ( τ ( t ) , t ) = 1 W w ( τ ( t ) , t ) = t ≠ 0 W_w(\tau(t),t)=t\neq0 W w ( τ ( t ) , t ) = t = 0 1 − W 1-W 1 − W τ ( t ) \tau(t) τ ( t ) 3 g − 4 3g-4 3 g − 4 3 g − 3 3g-3 3 g − 3
The leading term of (32) depends only on u v uv uv (3) share it, and
L g ( w , t ) = c g e 2 ( g − 1 ) ( t − 2 ) w ( 1 − W ( w , t ) ) 3 g − 3 + J g ( w , t ) ( 1 − W ( w , t ) ) 3 g − 4 , (33) \mathcal{L}_g(w,t)=\frac{c_g e^{2(g-1)(t-2)w}}{(1-W(w,t))^{3g-3}}+\frac{\mathcal{J}_g(w,t)}{(1-W(w,t))^{3g-4}}, \tag*{(33)} L g ( w , t ) = ( 1 − W ( w , t ) ) 3 g − 3 c g e 2 ( g − 1 ) ( t − 2 ) w + ( 1 − W ( w , t ) ) 3 g − 4 J g ( w , t ) , (33) where J g \mathcal{J}_g J g w w w t t t t t t 1 1 1 t t t
Remark 3.10 (Genus one as the case of exponent zero). In all genera, the leading singularity comes from rational curves whose dual graphs have only bivalent and trivalent vertices. Such a graph has 2 g − 2 2g-2 2 g − 2 g ≥ 2 g\geq2 g ≥ 2 3 g − 3 3g-3 3 g − 3 ( 1 − W ) − 1 (1-W)^{-1} ( 1 − W ) − 1 g = 1 g=1 g = 1 1 2 ∑ r ≥ 1 W r / r = − 1 2 log ( 1 − W ) \frac{1}{2}\sum_{r\geq1}W^r/r=-\frac{1}{2}\log(1-W) 2 1 ∑ r ≥ 1 W r / r = − 2 1 log ( 1 − W ) 3 g − 3 3g-3 3 g − 3
Proposition 3.11. The constants c g c_g c g
∑ g ≥ 2 c g x g − 1 = log ( ∑ m ≥ 0 ( 6 m ) ! 2 3 m ( 3 m ) ! ( 3 ! ) 2 m ( 2 m ) ! x m ) . \sum_{g\ge2} c_g x^{g-1}=\log\left(\sum_{m\ge0}\frac{(6m)!}{2^{3m}(3m)!(3!)^{2m}(2m)!}x^m\right). g ≥ 2 ∑ c g x g − 1 = log ( m ≥ 0 ∑ 2 3 m ( 3 m )! ( 3 ! ) 2 m ( 2 m )! ( 6 m )! x m ) . Proof. A trivalent graph with 2 m 2m 2 m 3 m 3m 3 m m + 1 m+1 m + 1 2 m 2m 2 m ( 6 m ) ! / ( 2 3 m ( 3 m ) ! ) (6m)!/(2^{3m}(3m)!) ( 6 m )! / ( 2 3 m ( 3 m )!) 6 m 6m 6 m G m G_m G m ( 2 m ) ! ( 3 ! ) 2 m (2m)!(3!)^{2m} ( 2 m )! ( 3 ! ) 2 m 2 m 2m 2 m Γ \Gamma Γ Aut Γ \operatorname{Aut}\Gamma Aut Γ ∣ G m ∣ |G_m| ∣ G m ∣ ∑ Γ 1 / ∣ Aut Γ ∣ \sum_\Gamma1/|\operatorname{Aut}\Gamma| ∑ Γ 1/∣ Aut Γ∣ r r r Γ \Gamma Γ r ! ∣ Aut Γ ∣ r r!|\operatorname{Aut}\Gamma|^r r ! ∣ Aut Γ ∣ r m = 0 m=0 m = 0
Example 3.12 (Low genus). Expanding the logarithm gives
c 2 = 5 24 , c 3 = 5 16 , c 4 = 1105 1152 , c 5 = 565 128 . c_2=\frac{5}{24},\qquad c_3=\frac{5}{16},\qquad c_4=\frac{1105}{1152},\qquad c_5=\frac{565}{128}. c 2 = 24 5 , c 3 = 16 5 , c 4 = 1152 1105 , c 5 = 128 565 . In genus two, G 2 \mathcal{G}_2 G 2
∣ Aut Γ ∣ = 12 ∣ Aut Γ ∣ = 8 |\operatorname{Aut}\Gamma|=12 \qquad|\operatorname{Aut}\Gamma|=8 ∣ Aut Γ∣ = 12 ∣ Aut Γ∣ = 8
The labels indicate the vertex genera. The automorphism groups have orders 3 ! ⋅ 2 = 12 3!\cdot2=12 3 ! ⋅ 2 = 12 2 2 ⋅ 2 = 8 2^2\cdot2=8 2 2 ⋅ 2 = 8 c 2 = 1 12 + 1 8 = 5 24 c_2=\frac{1}{12}+\frac{1}{8}=\frac{5}{24} c 2 = 12 1 + 8 1 = 24 5
Coefficient asymptotics and Gaussian limits
In this section, we extract uniform coefficient asymptotics from the singular expansions of Section 3 and deduce the central limit theorems for the even and full Betti distributions, together with the asymptotics of the Euler characteristics.
Asymptotics of the even Betti polynomials
Theorem 4.1. For every fixed g ≥ 1 g\ge1 g ≥ 1 n → ∞ n\to\infty n → ∞
L g , n ( t ) = A g ( t ) n ! n 3 g − 4 τ ( t ) − n ( 1 + O ( n − 1 ) ) , H g , n ( t ) = C g ( t ) n ! n 3 g − 5 2 ρ ( t ) − n ( 1 + O ( n − 1 2 ) ) . (34) \begin{aligned}
L_{g,n}(t)&=A_g(t)n!n^{3g-4}\tau(t)^{-n}\left(1+O(n^{-1})\right),\\
H_{g,n}(t)&=C_g(t)n!n^{\frac{3g-5}{2}}\rho(t)^{-n}\left(1+O(n^{-\frac12})\right). \tag*{(34)}
\end{aligned} L g , n ( t ) H g , n ( t ) = A g ( t ) n ! n 3 g − 4 τ ( t ) − n ( 1 + O ( n − 1 ) ) , = C g ( t ) n ! n 2 3 g − 5 ρ ( t ) − n ( 1 + O ( n − 2 1 ) ) . (34) uniformly for complex t t t 1 1 1 A 1 ( t ) = 1 / 2 A_1(t)=1/2 A 1 ( t ) = 1/2 C 1 ( t ) = 1 / 4 C_1(t)=1/4 C 1 ( t ) = 1/4 g ≥ 2 g\ge2 g ≥ 2
A g ( t ) = c g e 2 ( g − 1 ) ( t − 2 ) τ ( t ) Γ ( 3 g − 3 ) [ t τ ( t ) ] 3 g − 3 , C g ( t ) = c g e g − 1 2 ( t − 2 ) τ ( t ) Γ ( 3 g − 3 2 ) [ 2 ρ ( t ) ] 3 g − 3 2 . (35) A_g(t)=\frac{c_g e^{2(g-1)(t-2)\tau(t)}}{\Gamma(3g-3)[t\tau(t)]^{3g-3}},\qquad C_g(t)=\frac{c_g e^{\frac{g-1}{2}(t-2)\tau(t)}}{\Gamma\left(\frac{3g-3}{2}\right)[2\rho(t)]^{\frac{3g-3}{2}}}. \tag*{(35)} A g ( t ) = Γ ( 3 g − 3 ) [ t τ ( t ) ] 3 g − 3 c g e 2 ( g − 1 ) ( t − 2 ) τ ( t ) , C g ( t ) = Γ ( 2 3 g − 3 ) [ 2 ρ ( t ) ] 2 3 g − 3 c g e 2 g − 1 ( t − 2 ) τ ( t ) . (35) The amplitudes are holomorphic and nonvanishing near t = 1 t=1 t = 1 A g ( 1 ) , C g ( 1 ) > 0 A_g(1),C_g(1)>0 A g ( 1 ) , C g ( 1 ) > 0
Proof. In each case we determine the singular expansion at the dominant singularity and apply the transfer theorem [10 Theorem VI.4] uniformly in t t t § 2.2 .
Light series. To isolate the zero of 1 − W ( w , t ) 1-W(w,t) 1 − W ( w , t ) w = τ ( t ) w=\tau(t) w = τ ( t )
G ( w , t ) = 1 − W ( w , t ) 1 − w / τ ( t ) = t τ ( t ) e ( t − 1 ) ( w − τ ( t ) ) − 1 ( t − 1 ) ( w − τ ( t ) ) . G(w,t)=\frac{1-W(w,t)}{1-w/\tau(t)}
=t\tau(t)\frac{e^{(t-1)(w-\tau(t))}-1}{(t-1)(w-\tau(t))}. G ( w , t ) = 1 − w / τ ( t ) 1 − W ( w , t ) = t τ ( t ) ( t − 1 ) ( w − τ ( t )) e ( t − 1 ) ( w − τ ( t )) − 1 . The singularity of G G G w = τ ( t ) w=\tau(t) w = τ ( t ) t τ ( t ) t\tau(t) t τ ( t ) G ( w , 1 ) = 1 G(w,1)=1 G ( w , 1 ) = 1 R > 1 R>1 R > 1 t = 1 t=1 t = 1 G G G ∣ w ∣ ≤ R |w|\le R ∣ w ∣ ≤ R w = 0 w=0 w = 0 sup t ∣ τ ( t ) ∣ < R \sup_t|\tau(t)|<R sup t ∣ τ ( t ) ∣ < R Δ \Delta Δ Δ 0 L \Delta_0^L Δ 0 L τ ( t ) Δ 0 L ⊂ { ∣ w ∣ < R } \tau(t)\Delta_0^L\subset\{|w|<R\} τ ( t ) Δ 0 L ⊂ { ∣ w ∣ < R } t t t τ ( t ) Δ 0 L \tau(t)\Delta_0^L τ ( t ) Δ 0 L
For g = 1 g=1 g = 1
L 1 ( w , t ) = − 1 2 log ( 1 − w τ ( t ) ) + B ( τ ( t ) , t ) + O ( 1 − w τ ( t ) ) , (36) \mathcal{L}_1(w,t)=-\frac{1}{2}\log\left(1-\frac{w}{\tau(t)}\right)+B(\tau(t),t)+O\left(1-\frac{w}{\tau(t)}\right), \tag*{(36)} L 1 ( w , t ) = − 2 1 log ( 1 − τ ( t ) w ) + B ( τ ( t ) , t ) + O ( 1 − τ ( t ) w ) , (36) where B ( w , t ) = J 1 ( w , t ) − 1 2 log G ( w , t ) B(w,t)=\mathcal{J}_1(w,t)-\frac{1}{2}\log G(w,t) B ( w , t ) = J 1 ( w , t ) − 2 1 log G ( w , t ) [ w n ] L 1 ( w , t ) = τ ( t ) − n ( 1 2 n + O ( n − 2 ) ) [w^n]\mathcal{L}_1(w,t)=\tau(t)^{-n}\left(\frac{1}{2n}+O(n^{-2})\right) [ w n ] L 1 ( w , t ) = τ ( t ) − n ( 2 n 1 + O ( n − 2 ) ) A 1 = 1 / 2 A_1=1/2 A 1 = 1/2 g ≥ 2 g\ge2 g ≥ 2 L g ( w , t ) \mathcal{L}_g(w,t) L g ( w , t ) τ ( t ) \tau(t) τ ( t )
L g ( w , t ) = c g e 2 ( g − 1 ) ( t − 2 ) τ ( t ) [ t τ ( t ) ] 3 g − 3 ( 1 − w τ ( t ) ) − ( 3 g − 3 ) + O ( ( 1 − w τ ( t ) ) − ( 3 g − 4 ) ) , (37) \mathcal{L}_g(w,t)=\frac{c_g e^{2(g-1)(t-2)\tau(t)}}{[t\tau(t)]^{3g-3}}\left(1-\frac{w}{\tau(t)}\right)^{-(3g-3)}+O\left(\left(1-\frac{w}{\tau(t)}\right)^{-(3g-4)}\right), \tag*{(37)} L g ( w , t ) = [ t τ ( t ) ] 3 g − 3 c g e 2 ( g − 1 ) ( t − 2 ) τ ( t ) ( 1 − τ ( t ) w ) − ( 3 g − 3 ) + O ( ( 1 − τ ( t ) w ) − ( 3 g − 4 ) ) , (37) so transfer gives the first line of (4.1) with relative error O ( n − 1 ) O(n^{-1}) O ( n − 1 )
Heavy series. We first check that composition with T T T Δ t = ρ ( t ) Δ 0 \Delta_t=\rho(t)\Delta_0 Δ t = ρ ( t ) Δ 0
R w ( w , t ) = e w ( 1 − W ( w , t ) ) , W ( τ ( t ) , t ) = 1 , R w ( T ( z , t ) , t ) T z ( z , t ) = 1. (38) R_w(w,t)=e^w(1-W(w,t)),\qquad W(\tau(t),t)=1,\qquad R_w(T(z,t),t)T_z(z,t)=1. \tag*{(38)} R w ( w , t ) = e w ( 1 − W ( w , t )) , W ( τ ( t ) , t ) = 1 , R w ( T ( z , t ) , t ) T z ( z , t ) = 1. (38) Since T T T Δ t \Delta_t Δ t R w ( T ( z , t ) , t ) ≠ 0 R_w(T(z,t),t)\ne0 R w ( T ( z , t ) , t ) = 0 1 − W ( T ( z , t ) , t ) ≠ 0 1-W(T(z,t),t)\ne0 1 − W ( T ( z , t ) , t ) = 0 Δ t \Delta_t Δ t W w ( τ ( t ) , t ) = t W_w(\tau(t),t)=t W w ( τ ( t ) , t ) = t w = τ ( t ) w=\tau(t) w = τ ( t )
1 − W ( T ( z , t ) , t ) = − t ( T ( z , t ) − τ ( t ) ) + O ( ( T ( z , t ) − τ ( t ) ) 2 ) = ( 2 ρ ( t ) ) 1 / 2 e 1 2 ( t − 2 ) τ ( t ) ( 1 − z ρ ( t ) ) 1 / 2 ( 1 + O ( ( 1 − z ρ ( t ) ) 1 / 2 ) ) . (39) \begin{aligned}
1-W(T(z,t),t)&=-t(T(z,t)-\tau(t))+O((T(z,t)-\tau(t))^2)\\
&=(2\rho(t))^{1/2}e^{\frac{1}{2}(t-2)\tau(t)}\left(1-\frac{z}{\rho(t)}\right)^{1/2}\left(1+O\left(\left(1-\frac{z}{\rho(t)}\right)^{1/2}\right)\right). \tag*{(39)}
\end{aligned} 1 − W ( T ( z , t ) , t ) = − t ( T ( z , t ) − τ ( t )) + O (( T ( z , t ) − τ ( t ) ) 2 ) = ( 2 ρ ( t ) ) 1/2 e 2 1 ( t − 2 ) τ ( t ) ( 1 − ρ ( t ) z ) 1/2 ( 1 + O ( ( 1 − ρ ( t ) z ) 1/2 ) ) . (39) For g = 1 g=1 g = 1
H 1 ( z , t ) = − 1 4 log ( 1 − z ρ ( t ) ) + B 0 ( t ) + O ( ( 1 − z ρ ( t ) ) 1 / 2 ) , (40) \mathcal{H}_1(z,t)=-\frac{1}{4}\log\left(1-\frac{z}{\rho(t)}\right)+B_0(t)+O\left(\left(1-\frac{z}{\rho(t)}\right)^{1/2}\right), \tag*{(40)} H 1 ( z , t ) = − 4 1 log ( 1 − ρ ( t ) z ) + B 0 ( t ) + O ( ( 1 − ρ ( t ) z ) 1/2 ) , (40) with B 0 ( t ) = J 1 ( τ ( t ) , t ) − 1 4 log ( 2 ρ ( t ) ) − 1 4 ( t − 2 ) τ ( t ) B_0(t)=\mathcal{J}_1(\tau(t),t)-\frac{1}{4}\log(2\rho(t))-\frac{1}{4}(t-2)\tau(t) B 0 ( t ) = J 1 ( τ ( t ) , t ) − 4 1 log ( 2 ρ ( t )) − 4 1 ( t − 2 ) τ ( t ) t = 1 t=1 t = 1 [ z n ] H 1 ( z , t ) = ρ ( t ) − n ( 1 4 n + O ( n − 3 / 2 ) ) [z^n]\mathcal{H}_1(z,t)=\rho(t)^{-n}\left(\frac{1}{4n}+O(n^{-3/2})\right) [ z n ] H 1 ( z , t ) = ρ ( t ) − n ( 4 n 1 + O ( n − 3/2 ) ) C 1 = 1 / 4 C_1=1/4 C 1 = 1/4 g ≥ 2 g\ge2 g ≥ 2
H g ( z , t ) = c g e g − 1 2 ( t − 2 ) τ ( t ) [ 2 ρ ( t ) ] 3 g − 3 2 ( 1 − z ρ ( t ) ) − 3 g − 3 2 + O ( ( 1 − z ρ ( t ) ) − 3 g − 4 2 ) , (41) \mathcal{H}_g(z,t)=\frac{c_g e^{\frac{g-1}{2}(t-2)\tau(t)}}{[2\rho(t)]^{\frac{3g-3}{2}}}\left(1-\frac{z}{\rho(t)}\right)^{-\frac{3g-3}{2}}+O\left(\left(1-\frac{z}{\rho(t)}\right)^{-\frac{3g-4}{2}}\right), \tag*{(41)} H g ( z , t ) = [ 2 ρ ( t ) ] 2 3 g − 3 c g e 2 g − 1 ( t − 2 ) τ ( t ) ( 1 − ρ ( t ) z ) − 2 3 g − 3 + O ( ( 1 − ρ ( t ) z ) − 2 3 g − 4 ) , (41) and transfer gives the second line of (4.1) with relative error O ( n − 1 / 2 ) O(n^{-1/2}) O ( n − 1/2 ) t = 1 t=1 t = 1
Central limit theorems
Let X g , n H X^H_{g,n} X g , n H X g , n L X^L_{g,n} X g , n L H g , n ( t ) / H g , n ( 1 ) H_{g,n}(t)/H_{g,n}(1) H g , n ( t ) / H g , n ( 1 ) L g , n ( t ) / L g , n ( 1 ) L_{g,n}(t)/L_{g,n}(1) L g , n ( t ) / L g , n ( 1 ) d n / 2 d_n/2 d n /2
Definition 4.2. We set
ν = 3 − e 6 ( e − 2 ) = 0.0653685 … , r H = 1 4 ν = 3 ( e − 2 ) 2 ( 3 − e ) = 3.824470 … . \nu=\frac{3-e}{6(e-2)}=0.0653685\ldots,\qquad r_H=\frac{1}{4\nu}=\frac{3(e-2)}{2(3-e)}=3.824470\ldots. ν = 6 ( e − 2 ) 3 − e = 0.0653685 … , r H = 4 ν 1 = 2 ( 3 − e ) 3 ( e − 2 ) = 3.824470 … . The constant ν \nu ν M ‾ 0 , n \overline{\mathcal{M}}_{0,n} M 0 , n [5 Theorem 3.1] and of the Fulton–MacPherson spaces of smooth projective curves [6 Section 4] . The constant r H r_H r H M g , n \mathcal{M}_{g,n} M g , n Theorem 5.6 ).
Theorem 4.3. Fix g ≥ 1 g \ge1 g ≥ 1 ( σ g , n J ) 2 = Var ( X g , n J ) (\sigma_{g,n}^{J})^2 = \operatorname{Var}(X_{g,n}^{J}) ( σ g , n J ) 2 = Var ( X g , n J ) J ∈ { H , L } J \in\{H,L\} J ∈ { H , L } E X g , n H = E X g , n L = d n / 2 \mathbb{E}X_{g,n}^{H} = \mathbb{E}X_{g,n}^{L} = d_n/2 E X g , n H = E X g , n L = d n /2
( σ g , n H ) 2 = ν n + ( g − 1 ) ( 11 − 4 e ) 12 ( e − 2 ) + O ( n − 1 / 2 ) , ( σ g , n L ) 2 = n − g + 1 12 + O ( n − 1 ) , (\sigma_{g,n}^{H})^2 = \nu n + \frac{(g-1)(11-4e)}{12(e-2)} + O(n^{-1/2}), \qquad(\sigma_{g,n}^{L})^2 = \frac{n-g+1}{12} + O(n^{-1}), ( σ g , n H ) 2 = ν n + 12 ( e − 2 ) ( g − 1 ) ( 11 − 4 e ) + O ( n − 1/2 ) , ( σ g , n L ) 2 = 12 n − g + 1 + O ( n − 1 ) , and, for J ∈ { H , L } J \in\{H,L\} J ∈ { H , L }
sup x ∈ R ∣ P ( X g , n J − d n / 2 σ g , n J ≤ x ) − Φ ( x ) ∣ = O ( n − 1 / 2 ) . \sup_{x \in\mathbb{R}} \left| \mathbb{P}\left(\frac{X_{g,n}^{J}-d_n/2}{\sigma_{g,n}^{J}} \le x\right) - \Phi(x) \right| = O(n^{-1/2}). x ∈ R sup P ( σ g , n J X g , n J − d n /2 ≤ x ) − Φ ( x ) = O ( n − 1/2 ) . where Φ ( x ) \Phi(x) Φ ( x )
Proof. Dividing the estimates of Theorem 4.1 by their values at t = 1 t=1 t = 1
H g , n ( t ) H g , n ( 1 ) = C g ( t ) C g ( 1 ) ( e − 2 ρ ( t ) ) n ( 1 + O ( n − 1 / 2 ) ) , (42) \frac{H_{g,n}(t)}{H_{g,n}(1)} = \frac{C_g(t)}{C_g(1)}\left(\frac{e-2}{\rho(t)}\right)^n\left(1+O(n^{-1/2})\right), \tag*{(42)} H g , n ( 1 ) H g , n ( t ) = C g ( 1 ) C g ( t ) ( ρ ( t ) e − 2 ) n ( 1 + O ( n − 1/2 ) ) , (42) L g , n ( t ) L g , n ( 1 ) = A g ( t ) A g ( 1 ) ( 1 τ ( t ) ) n ( 1 + O ( n − 1 ) ) , \frac{L_{g,n}(t)}{L_{g,n}(1)} = \frac{A_g(t)}{A_g(1)}\left(\frac{1}{\tau(t)}\right)^n\left(1+O(n^{-1})\right), L g , n ( 1 ) L g , n ( t ) = A g ( 1 ) A g ( t ) ( τ ( t ) 1 ) n ( 1 + O ( n − 1 ) ) , uniformly near t = 1 t=1 t = 1
e − 2 ρ ( t ) = 1 + t − 1 2 − 5 e − 12 24 ( e − 2 ) ( t − 1 ) 2 + O ( ( t − 1 ) 3 ) , \frac{e-2}{\rho(t)} = 1 + \frac{t-1}{2} - \frac{5e-12}{24(e-2)}(t-1)^2 + O((t-1)^3), ρ ( t ) e − 2 = 1 + 2 t − 1 − 24 ( e − 2 ) 5 e − 12 ( t − 1 ) 2 + O (( t − 1 ) 3 ) , 1 τ ( t ) = 1 + t − 1 2 − ( t − 1 ) 2 12 + O ( ( t − 1 ) 3 ) , \frac{1}{\tau(t)} = 1 + \frac{t-1}{2} - \frac{(t-1)^2}{12} + O((t-1)^3), τ ( t ) 1 = 1 + 2 t − 1 − 12 ( t − 1 ) 2 + O (( t − 1 ) 3 ) , so that
m ( e − 2 ρ ) = m ( 1 τ ) = 1 2 , v ( e − 2 ρ ) = ν > 0 , v ( 1 τ ) = 1 12 > 0. \mathfrak{m}\left(\frac{e-2}{\rho}\right) = \mathfrak{m}\left(\frac{1}{\tau}\right) = \frac{1}{2}, \qquad\mathfrak{v}\left(\frac{e-2}{\rho}\right) = \nu> 0, \qquad\mathfrak{v}\left(\frac{1}{\tau}\right) = \frac{1}{12} > 0. m ( ρ e − 2 ) = m ( τ 1 ) = 2 1 , v ( ρ e − 2 ) = ν > 0 , v ( τ 1 ) = 12 1 > 0. For g ≥ 2 g \ge2 g ≥ 2 (35) ) and using ( t − 2 ) τ ( t ) = log t − τ ( t ) (t-2)\tau(t)=\log t-\tau(t) ( t − 2 ) τ ( t ) = log t − τ ( t )
log C g ( t ) = const + g − 1 2 ( log t − τ ( t ) − 3 log ρ ( t ) ) , (43) \log C_g(t) = \operatorname{const} + \frac{g-1}{2}\left(\log t-\tau(t)-3\log\rho(t)\right), \tag*{(43)} log C g ( t ) = const + 2 g − 1 ( log t − τ ( t ) − 3 log ρ ( t ) ) , (43) log A g ( t ) = const − ( g − 1 ) ( log t + 2 τ ( t ) + 3 log τ ( t ) ) , \log A_g(t) = \operatorname{const} - (g-1)\left(\log t+2\tau(t)+3\log\tau(t)\right), log A g ( t ) = const − ( g − 1 ) ( log t + 2 τ ( t ) + 3 log τ ( t ) ) , and differentiating at t = 1 t=1 t = 1
m ( C g ) = m ( A g ) = 3 g − 3 2 , v ( C g ) = ( g − 1 ) ( 11 − 4 e ) 12 ( e − 2 ) , v ( A g ) = − g − 1 12 . \mathfrak{m}(C_g)=\mathfrak{m}(A_g)=\frac{3g-3}{2}, \qquad\mathfrak{v}(C_g)=\frac{(g-1)(11-4e)}{12(e-2)}, \qquad\mathfrak{v}(A_g)=-\frac{g-1}{12}. m ( C g ) = m ( A g ) = 2 3 g − 3 , v ( C g ) = 12 ( e − 2 ) ( g − 1 ) ( 11 − 4 e ) , v ( A g ) = − 12 g − 1 . For g = 1 g=1 g = 1 Theorem 2.1 , applied with β n = n \beta_n=n β n = n ε n = n − 1 / 2 \varepsilon_n=n^{-1/2} ε n = n − 1/2 ε n = n − 1 \varepsilon_n=n^{-1} ε n = n − 1
Corollary 4.4. For each fixed g ≥ 1 g \ge1 g ≥ 1 M ‾ g , n \overline{\mathcal{M}}_{g,n} M g , n M ‾ g , 0 ∣ n \overline{\mathcal{M}}_{g,0|n} M g , 0∣ n d n = n + 3 g − 3 d_n=n+3g-3 d n = n + 3 g − 3 Theorem 4.3 , with the same error orders. After centering and scaling their distribution functions converge uniformly to Φ \Phi Φ O ( n − 1 / 2 ) O(n^{-1/2}) O ( n − 1/2 )
Proof. By ((3) ), the Poincaré polynomials are obtained by setting u = v = − q u=v=-q u = v = − q (23) ) and ((32) ) depend only on u v = q 2 uv=q^2 uv = q 2 (19) ) and the argument of Theorem 4.1 , the Poincaré polynomials therefore have the same leading terms and relative errors as the even Betti polynomials at t = q 2 t=q^2 t = q 2 q q q 1 1 1 q = 1 q=1 q = 1 (42) ) with t = q 2 t=q^2 t = q 2 m ( f ( q 2 ) ) = 2 m ( f ) \mathfrak{m}(f(q^2))=2\mathfrak{m}(f) m ( f ( q 2 )) = 2 m ( f ) v ( f ( q 2 ) ) = 4 v ( f ) \mathfrak{v}(f(q^2))=4\mathfrak{v}(f) v ( f ( q 2 )) = 4 v ( f ) f f f 1 1 1 f ( 1 ) ≠ 0 f(1) \ne0 f ( 1 ) = 0 Theorem 2.1 gives the claim. The means are given by Poincaré duality.
Example 4.5. Figure 1 compares the exact normalized even Betti numbers of M ‾ 2 , 50 \overline{\mathcal{M}}_{2,50} M 2 , 50 M ‾ 2 , 0 ∣ 50 \overline{\mathcal{M}}_{2,0|50} M 2 , 0∣50 53 / 2 53/2 53/2 3.885 3.885 3.885 4.521 4.521 4.521 k = 26 , 27 k=26,27 k = 26 , 27 0.0913 % 0.0913\% 0.0913% 0.1426 % 0.1426\% 0.1426%
Figure 1. Even Betti distributions for g = 2 g = 2 g = 2 n = 50 n = 50 n = 50 k k k b 2 k / ∑ j b 2 j b_{2k}/\sum_j b_{2j} b 2 k / ∑ j b 2 j
Euler characteristics
Kannan, Serpente and Yun asked for the asymptotics of χ ( M ‾ g , 0 ∣ n ) / χ ( M ‾ g , n ) \chi(\overline{\mathcal{M}}_{g,0|n})/\chi(\overline{\mathcal{M}}_{g,n}) χ ( M g , 0∣ n ) / χ ( M g , n ) g ≥ 2 g \ge2 g ≥ 2 [16 Question 5.3] . For g = 1 g=1 g = 1 2 ( e − 2 ) n 2(e-2)^n 2 ( e − 2 ) n [16 Corollary 5.2] . Here χ \chi χ
Corollary 4.6. For every fixed g ≥ 2 g \ge2 g ≥ 2
χ ( M ‾ g , 0 ∣ n ) χ ( M ‾ g , n ) = ( 2 ( e − 2 ) e ) 3 g − 3 2 Γ ( 3 g − 3 2 ) Γ ( 3 g − 3 ) n 3 g − 3 2 ( e − 2 ) n ( 1 + O ( n − 1 2 ) ) . (44) \frac{\chi(\overline{\mathcal{M}}_{g,0|n})}{\chi(\overline{\mathcal{M}}_{g,n})}
=
\left(\frac{2(e-2)}{e}\right)^{\frac{3g-3}{2}}
\frac{\Gamma\left(\frac{3g-3}{2}\right)}{\Gamma(3g-3)}
n^{\frac{3g-3}{2}}(e-2)^n\left(1+O\left(n^{-\frac{1}{2}}\right)\right). \tag*{(44)} χ ( M g , n ) χ ( M g , 0∣ n ) = ( e 2 ( e − 2 ) ) 2 3 g − 3 Γ ( 3 g − 3 ) Γ ( 2 3 g − 3 ) n 2 3 g − 3 ( e − 2 ) n ( 1 + O ( n − 2 1 ) ) . (44) In particular, the ratio tends to zero. For g = 1 g=1 g = 1 2 ( e − 2 ) n 2(e-2)^n 2 ( e − 2 ) n
Proof. The estimates in the proof of Corollary 4.4, evaluated at q = 1 q=1 q = 1
∑ j b j ( M ‾ g , n ) H g , n ( 1 ) = 1 + O ( n − 1 2 ) , ∑ j b j ( M ‾ g , 0 ∣ n ) L g , n ( 1 ) = 1 + O ( n − 1 ) . (45) \frac{\sum_j b_j(\overline{\mathcal{M}}_{g,n})}{H_{g,n}(1)}
=1+O\left(n^{-\frac{1}{2}}\right),
\qquad
\frac{\sum_j b_j(\overline{\mathcal{M}}_{g,0|n})}{L_{g,n}(1)}
=1+O\left(n^{-1}\right). \tag*{(45)} H g , n ( 1 ) ∑ j b j ( M g , n ) = 1 + O ( n − 2 1 ) , L g , n ( 1 ) ∑ j b j ( M g , 0∣ n ) = 1 + O ( n − 1 ) . (45) Since the Euler characteristic is twice the even Betti total minus the total Betti number,
χ ( M ‾ g , n ) = H g , n ( 1 ) ( 1 + O ( n − 1 2 ) ) and χ ( M ‾ g , 0 ∣ n ) = L g , n ( 1 ) ( 1 + O ( n − 1 ) ) . (46) \chi(\overline{\mathcal{M}}_{g,n})=H_{g,n}(1)\left(1+O\left(n^{-\frac{1}{2}}\right)\right)
\quad\text{and}\quad
\chi(\overline{\mathcal{M}}_{g,0|n})=L_{g,n}(1)\left(1+O\left(n^{-1}\right)\right). \tag*{(46)} χ ( M g , n ) = H g , n ( 1 ) ( 1 + O ( n − 2 1 ) ) and χ ( M g , 0∣ n ) = L g , n ( 1 ) ( 1 + O ( n − 1 ) ) . (46) Theorem 4.1 at t = 1 t=1 t = 1
χ ( M ‾ g , 0 ∣ n ) χ ( M ‾ g , n ) = A g ( 1 ) C g ( 1 ) n 3 g − 3 2 ( e − 2 ) n ( 1 + O ( n − 1 2 ) ) , A g ( 1 ) C g ( 1 ) = ( 2 ( e − 2 ) e ) 3 g − 3 2 Γ ( 3 g − 3 2 ) Γ ( 3 g − 3 ) , \frac{\chi(\overline{\mathcal{M}}_{g,0|n})}{\chi(\overline{\mathcal{M}}_{g,n})}
=
\frac{A_g(1)}{C_g(1)}
n^{\frac{3g-3}{2}}(e-2)^n\left(1+O\left(n^{-\frac{1}{2}}\right)\right),
\qquad
\frac{A_g(1)}{C_g(1)}
=
\left(\frac{2(e-2)}{e}\right)^{\frac{3g-3}{2}}
\frac{\Gamma\left(\frac{3g-3}{2}\right)}{\Gamma(3g-3)}, χ ( M g , n ) χ ( M g , 0∣ n ) = C g ( 1 ) A g ( 1 ) n 2 3 g − 3 ( e − 2 ) n ( 1 + O ( n − 2 1 ) ) , C g ( 1 ) A g ( 1 ) = ( e 2 ( e − 2 ) ) 2 3 g − 3 Γ ( 3 g − 3 ) Γ ( 2 3 g − 3 ) , which is (4.11). The ratio tends to zero because 0 < e − 2 < 1 0<e-2<1 0 < e − 2 < 1 g = 1 g=1 g = 1 A 1 / C 1 = 2 A_1/C_1=2 A 1 / C 1 = 2 2 ( e − 2 ) n ( 1 + O ( n − 1 2 ) ) 2(e-2)^n\left(1+O\left(n^{-\frac{1}{2}}\right)\right) 2 ( e − 2 ) n ( 1 + O ( n − 2 1 ) )
By (45) , odd cohomology forms a proportion O ( n − 1 / 2 ) O(n^{-1/2}) O ( n − 1/2 ) M ‾ g , n \overline{\mathcal{M}}_{g,n} M g , n O ( n − 1 ) O(n^{-1}) O ( n − 1 ) M ‾ g , 0 ∣ n \overline{\mathcal{M}}_{g,0|n} M g , 0∣ n g ≥ 2 g \ge2 g ≥ 2
Local limits and central ultra-log-concavity
The central limit theorems of Section 4 control cumulative probabilities but not individual Betti numbers. In this section, we prove a local limit theorem with Edgeworth-type corrections, deduce asymptotic formulas for the individual even Betti numbers, and then compare neighboring Betti numbers precisely enough to determine the central ultra-log-concavity thresholds. The local limit theorem is a refinement of Hwang’s Fourier method [14 Theorem 1 and Section 2] that exploits the symmetry provided by Poincaré duality.
A symmetric local limit theorem
A Gaussian approximation to individual probabilities suffices for the Betti number asymptotics, but ultra-log-concavity compares neighboring probabilities and requires more precision. Since the distributions are symmetric, the odd centered cumulants vanish. We therefore establish a symmetric version of the local limit theorem with higher-order corrections involving only even cumulants. As usual, He j ( z ) = ( − 1 ) j e z 2 / 2 d j d z j e − z 2 / 2 \operatorname{He}_{j}(z) = (-1)^{j}e^{z^{2}/2}\frac{d^{j}}{dz^{j}}e^{-z^{2}/2} He j ( z ) = ( − 1 ) j e z 2 /2 d z j d j e − z 2 /2 j j j
Theorem 5.1 (Symmetric quasi-powers local limit theorem). Let ( Y n ) (Y_n) ( Y n ) M n ( s ) = E e s Y n M_n(s) = \mathbb{E}e^{sY_n} M n ( s ) = E e s Y n
(1) Quasi-powers. There are r 0 > 0 r_0 > 0 r 0 > 0 ε n → 0 \varepsilon_n \to0 ε n → 0 λ , ψ \lambda,\psi λ , ψ n n n ∣ s ∣ ≤ r 0 |s| \le r_0 ∣ s ∣ ≤ r 0 λ ( 0 ) = ψ ( 0 ) = 0 \lambda(0) = \psi(0) = 0 λ ( 0 ) = ψ ( 0 ) = 0 λ ′ ′ ( 0 ) > 0 \lambda''(0) > 0 λ ′′ ( 0 ) > 0 M n M_n M n
M n ( s ) = exp ( n λ ( s ) + ψ ( s ) ) ( 1 + O ( ε n ) ) (47) M_n(s) = \exp\bigl(n\lambda(s) + \psi(s)\bigr)\bigl(1 + O(\varepsilon_n)\bigr) \tag*{(47)} M n ( s ) = exp ( nλ ( s ) + ψ ( s ) ) ( 1 + O ( ε n ) ) (47) uniformly for complex ∣ s ∣ ≤ r 0 |s| \le r_0 ∣ s ∣ ≤ r 0
(2) Symmetry. With μ n = E Y n \mu_n = \mathbb{E}Y_n μ n = E Y n P ( Y n = μ n + x ) = P ( Y n = μ n − x ) \mathbb{P}(Y_n = \mu_n + x) = \mathbb{P}(Y_n = \mu_n - x) P ( Y n = μ n + x ) = P ( Y n = μ n − x ) x ∈ R x \in\mathbb{R} x ∈ R
(3) Decay away from zero. For every sufficiently small δ > 0 \delta> 0 δ > 0 c δ > 0 c_\delta> 0 c δ > 0
sup δ ≤ ∣ θ ∣ ≤ π ∣ M n ( i θ ) ∣ = O ( e − c δ n ) . (48) \sup_{\delta\le|\theta| \le\pi} |M_n(i\theta)| = O(e^{-c_\delta n}). \tag*{(48)} δ ≤ ∣ θ ∣ ≤ π sup ∣ M n ( i θ ) ∣ = O ( e − c δ n ) . (48) Write p n , k = P ( Y n = k ) p_{n,k} = \mathbb{P}(Y_n = k) p n , k = P ( Y n = k ) σ n 2 = Var ( Y n ) \sigma_n^2 = \operatorname{Var}(Y_n) σ n 2 = Var ( Y n ) κ j , n = κ j ( Y n ) \kappa_{j,n} = \kappa_j(Y_n) κ j , n = κ j ( Y n ) z n , k = ( k − μ n ) / σ n z_{n,k} = (k - \mu_n)/\sigma_n z n , k = ( k − μ n ) / σ n
σ n 2 = n λ ′ ′ ( 0 ) + ψ ′ ′ ( 0 ) + O ( ε n ) , κ j , n = n λ ( j ) ( 0 ) + ψ ( j ) ( 0 ) + O ( ε n ) ( j = 4 , 6 ) , (49) \sigma_n^2 = n\lambda''(0) + \psi''(0) + O(\varepsilon_n), \qquad\kappa_{j,n} = n\lambda^{(j)}(0) + \psi^{(j)}(0) + O(\varepsilon_n) \quad(j = 4,6), \tag*{(49)} σ n 2 = n λ ′′ ( 0 ) + ψ ′′ ( 0 ) + O ( ε n ) , κ j , n = n λ ( j ) ( 0 ) + ψ ( j ) ( 0 ) + O ( ε n ) ( j = 4 , 6 ) , (49) and, uniformly for all k ∈ Z k \in\mathbb{Z} k ∈ Z
σ n p n , k = e − z n , k 2 / 2 2 π [ 1 + κ 4 , n 4 ! σ n 4 He 4 ( z n , k ) + κ 6 , n 6 ! σ n 6 He 6 ( z n , k ) + κ 4 , n 2 2 ( 4 ! ) 2 σ n 8 He 8 ( z n , k ) ] + O ( n − 3 ) . (50) \sigma_n p_{n,k} =
\frac{e^{-z_{n,k}^{2}/2}}{\sqrt{2\pi}}
\left[
1 + \frac{\kappa_{4,n}}{4!\,\sigma_n^4}\operatorname{He}_4(z_{n,k})
+ \frac{\kappa_{6,n}}{6!\,\sigma_n^6}\operatorname{He}_6(z_{n,k})
+ \frac{\kappa_{4,n}^{2}}{2(4!)^2\sigma_n^8}\operatorname{He}_8(z_{n,k})
\right] + O(n^{-3}). \tag*{(50)} σ n p n , k = 2 π e − z n , k 2 /2 [ 1 + 4 ! σ n 4 κ 4 , n He 4 ( z n , k ) + 6 ! σ n 6 κ 6 , n He 6 ( z n , k ) + 2 ( 4 ! ) 2 σ n 8 κ 4 , n 2 He 8 ( z n , k ) ] + O ( n − 3 ) . (50) Proof. The cumulant estimates (49) are (11) . For the local expansion, fix a small δ ∈ ( 0 , r 0 / 2 ) \delta\in(0,r_0/2) δ ∈ ( 0 , r 0 /2 ) (12) as p n , k = I 1 + I 2 p_{n,k} = I_1 + I_2 p n , k = I 1 + I 2
I 1 = 1 2 π ∫ − δ δ e − i k θ M n ( i θ ) d θ , I 2 = 1 2 π ∫ δ ≤ ∣ θ ∣ ≤ π e − i k θ M n ( i θ ) d θ . I_1 = \frac{1}{2\pi}\int_{-\delta}^{\delta} e^{-ik\theta}M_n(i\theta)\,d\theta,\qquad
I_2 = \frac{1}{2\pi}\int_{\delta\le|\theta| \le\pi} e^{-ik\theta}M_n(i\theta)\,d\theta. I 1 = 2 π 1 ∫ − δ δ e − ik θ M n ( i θ ) d θ , I 2 = 2 π 1 ∫ δ ≤ ∣ θ ∣ ≤ π e − ik θ M n ( i θ ) d θ . By condition (3), ∣ I 2 ∣ = O ( e − c δ n ) |I_2| = O(e^{-c_\delta n}) ∣ I 2 ∣ = O ( e − c δ n ) k k k I 1 I_1 I 1 K n ( s ) = log M n ( s ) − μ n s K_n(s) = \log M_n(s) - \mu_n s K n ( s ) = log M n ( s ) − μ n s (11) with j = 8 j = 8 j = 8 K n ( 8 ) ( s ) = O ( n ) K_n^{(8)}(s) = O(n) K n ( 8 ) ( s ) = O ( n ) ∣ s ∣ ≤ r 0 / 2 |s| \le r_0/2 ∣ s ∣ ≤ r 0 /2 K n K_n K n
K n ( s ) = σ n 2 s 2 2 + κ 4 , n s 4 4 ! + κ 6 , n s 6 6 ! + O ( n ∣ s ∣ 8 ) , ∣ s ∣ ≤ r 0 / 2. (51) K_n(s) = \frac{\sigma_n^2s^2}{2} + \frac{\kappa_{4,n}s^4}{4!} + \frac{\kappa_{6,n}s^6}{6!} + O(n|s|^8),\qquad|s| \le r_0/2. \tag*{(51)} K n ( s ) = 2 σ n 2 s 2 + 4 ! κ 4 , n s 4 + 6 ! κ 6 , n s 6 + O ( n ∣ s ∣ 8 ) , ∣ s ∣ ≤ r 0 /2. (51) Substituting s = i τ / σ n s = i\tau/\sigma_n s = i τ / σ n
σ n I 1 = 1 2 π ∫ − δ σ n δ σ n e − i τ z n , k e K n ( i τ / σ n ) d τ , K n ( i τ / σ n ) = − τ 2 2 + κ 4 , n τ 4 4 ! σ n 4 − κ 6 , n τ 6 6 ! σ n 6 + O ( n − 3 τ 8 ) . \sigma_n I_1 = \frac{1}{2\pi}\int_{-\delta\sigma_n}^{\delta\sigma_n} e^{-i\tau z_{n,k}}e^{K_n(i\tau/\sigma_n)}\,d\tau,\qquad
K_n(i\tau/\sigma_n) = -\frac{\tau^2}{2} + \frac{\kappa_{4,n}\tau^4}{4!\,\sigma_n^4} - \frac{\kappa_{6,n}\tau^6}{6!\,\sigma_n^6} + O(n^{-3}\tau^8). σ n I 1 = 2 π 1 ∫ − δ σ n δ σ n e − i τ z n , k e K n ( i τ / σ n ) d τ , K n ( i τ / σ n ) = − 2 τ 2 + 4 ! σ n 4 κ 4 , n τ 4 − 6 ! σ n 6 κ 6 , n τ 6 + O ( n − 3 τ 8 ) . Fix 0 < α < 1 / 4 0<\alpha<1/4 0 < α < 1/4 ∣ τ ∣ ≤ n α |\tau|\le n^\alpha ∣ τ ∣ ≤ n α − τ 2 / 2 -\tau^2/2 − τ 2 /2 o ( 1 ) o(1) o ( 1 ) τ 4 / n ≤ n 4 α − 1 \tau^4/n\le n^{4\alpha-1} τ 4 / n ≤ n 4 α − 1
e K n ( i τ / σ n ) = e − τ 2 / 2 P n ( τ ) + O ( n − 3 e − τ 2 / 2 ( τ 8 + τ 10 + τ 12 ) ) , (52) e^{K_n(i\tau/\sigma_n)}=e^{-\tau^2/2}P_n(\tau)+O\left(n^{-3}e^{-\tau^2/2}(\tau^8+\tau^{10}+\tau^{12})\right), \tag*{(52)} e K n ( i τ / σ n ) = e − τ 2 /2 P n ( τ ) + O ( n − 3 e − τ 2 /2 ( τ 8 + τ 10 + τ 12 ) ) , (52) where
P n ( τ ) = 1 + κ 4 , n τ 4 4 ! σ n 4 − κ 6 , n τ 6 6 ! σ n 6 + κ 4 , n 2 τ 8 2 ( 4 ! ) 2 σ n 8 . P_n(\tau)=1+\frac{\kappa_{4,n}\tau^4}{4!\,\sigma_n^4}-\frac{\kappa_{6,n}\tau^6}{6!\,\sigma_n^6}+\frac{\kappa_{4,n}^2\tau^8}{2(4!)^2\sigma_n^8}. P n ( τ ) = 1 + 4 ! σ n 4 κ 4 , n τ 4 − 6 ! σ n 6 κ 6 , n τ 6 + 2 ( 4 ! ) 2 σ n 8 κ 4 , n 2 τ 8 . The coefficients of τ 4 \tau^4 τ 4 τ 6 \tau^6 τ 6 τ 8 \tau^8 τ 8 P n P_n P n O ( n − 1 ) O(n^{-1}) O ( n − 1 ) O ( n − 2 ) O(n^{-2}) O ( n − 2 ) O ( n − 2 ) O(n^{-2}) O ( n − 2 ) (52) integrates to O ( n − 3 ) O(n^{-3}) O ( n − 3 ) k k k ∣ e − i τ z n , k ∣ = 1 |e^{-i\tau z_{n,k}}|=1 ∣ e − i τ z n , k ∣ = 1
On the range n α < ∣ τ ∣ ≤ δ σ n n^\alpha<|\tau|\le\delta\sigma_n n α < ∣ τ ∣ ≤ δ σ n (49) give ℜ K n ( i τ / σ n ) = − τ 2 / 2 + O ( τ 4 / n ) \Re K_n(i\tau/\sigma_n)=-\tau^2/2+O(\tau^4/n) ℜ K n ( i τ / σ n ) = − τ 2 /2 + O ( τ 4 / n ) τ 4 / n ≤ δ 2 ( σ n 2 / n ) τ 2 \tau^4/n\le\delta^2(\sigma_n^2/n)\tau^2 τ 4 / n ≤ δ 2 ( σ n 2 / n ) τ 2 δ \delta δ ∣ e K n ( i τ / σ n ) ∣ ≤ e − τ 2 / 4 |e^{K_n(i\tau/\sigma_n)}|\le e^{-\tau^2/4} ∣ e K n ( i τ / σ n ) ∣ ≤ e − τ 2 /4 ∣ τ ∣ ≤ δ σ n |\tau|\le\delta\sigma_n ∣ τ ∣ ≤ δ σ n O ( e − c n 2 α ) O(e^{-cn^{2\alpha}}) O ( e − c n 2 α ) c > 0 c>0 c > 0 e − τ 2 / 2 P n ( τ ) e^{-\tau^2/2}P_n(\tau) e − τ 2 /2 P n ( τ ) ∣ τ ∣ ≤ n α |\tau|\le n^\alpha ∣ τ ∣ ≤ n α R \mathbb{R} R O ( e − n 2 α / 4 ) O(e^{-n^{2\alpha}/4}) O ( e − n 2 α /4 ) P n P_n P n
σ n I 1 = 1 2 π ∫ R e − i τ z n , k e − τ 2 / 2 P n ( τ ) d τ + O ( n − 3 ) . \sigma_n I_1=\frac{1}{2\pi}\int_{\mathbb{R}}e^{-i\tau z_{n,k}}e^{-\tau^2/2}P_n(\tau)\,d\tau+O(n^{-3}). σ n I 1 = 2 π 1 ∫ R e − i τ z n , k e − τ 2 /2 P n ( τ ) d τ + O ( n − 3 ) . Differentiating the Fourier transform of the Gaussian gives
1 2 π ∫ R e − i τ z τ 2 j e − τ 2 / 2 d τ = ( − 1 ) j He 2 j ( z ) e − z 2 / 2 2 π ( j ≥ 0 ) , (53) \frac{1}{2\pi}\int_{\mathbb{R}}e^{-i\tau z}\tau^{2j}e^{-\tau^2/2}\,d\tau=(-1)^j\operatorname{He}_{2j}(z)\frac{e^{-z^2/2}}{\sqrt{2\pi}}\qquad(j\ge0), \tag*{(53)} 2 π 1 ∫ R e − i τ z τ 2 j e − τ 2 /2 d τ = ( − 1 ) j He 2 j ( z ) 2 π e − z 2 /2 ( j ≥ 0 ) , (53) so τ 4 , τ 6 , τ 8 \tau^4,\tau^6,\tau^8 τ 4 , τ 6 , τ 8 He 4 , − He 6 , He 8 \operatorname{He}_4,-\operatorname{He}_6,\operatorname{He}_8 He 4 , − He 6 , He 8 P n P_n P n He 6 \operatorname{He}_6 He 6 σ n I 2 = O ( n e − c δ n ) = o ( n − 3 ) \sigma_n I_2=O(\sqrt{n}e^{-c\delta n})=o(n^{-3}) σ n I 2 = O ( n e − cδ n ) = o ( n − 3 ) (50) . □
Local limits for the even Betti numbers
We verify the hypotheses of Theorem 5.1 for X g , n H X^H_{g,n} X g , n H X g , n L X^L_{g,n} X g , n L t = e s t=e^s t = e s (42) gives
E e s X g , n H = e n λ H ( s ) + ψ H ( s ) ( 1 + O ( n − 1 / 2 ) ) , E e s X g , n L = e n λ L ( s ) + ψ L ( s ) ( 1 + O ( n − 1 ) ) , \mathbb{E}e^{sX^H_{g,n}}=e^{n\lambda_H(s)+\psi_H(s)}\left(1+O(n^{-1/2})\right),\qquad
\mathbb{E}e^{sX^L_{g,n}}=e^{n\lambda_L(s)+\psi_L(s)}\left(1+O(n^{-1})\right), E e s X g , n H = e n λ H ( s ) + ψ H ( s ) ( 1 + O ( n − 1/2 ) ) , E e s X g , n L = e n λ L ( s ) + ψ L ( s ) ( 1 + O ( n − 1 ) ) , uniformly on a fixed complex disc, where
λ H ( s ) = log e s − 2 ρ ( e s ) , λ L ( s ) = − log τ ( e s ) = log e s − 1 s , ψ H ( s ) = log C g ( e s ) C g ( 1 ) , ψ L ( s ) = log A g ( e s ) A g ( 1 ) . (54) \begin{aligned}
\lambda_H(s)&=\log\frac{e^s-2}{\rho(e^s)},&
\lambda_L(s)&=-\log\tau(e^s)=\log\frac{e^s-1}{s},\\
\psi_H(s)&=\log\frac{C_g(e^s)}{C_g(1)},&
\psi_L(s)&=\log\frac{A_g(e^s)}{A_g(1)}. \tag*{(54)}
\end{aligned} λ H ( s ) ψ H ( s ) = log ρ ( e s ) e s − 2 , = log C g ( 1 ) C g ( e s ) , λ L ( s ) ψ L ( s ) = − log τ ( e s ) = log s e s − 1 , = log A g ( 1 ) A g ( e s ) . (54) all logarithms vanishing at s = 0 s=0 s = 0 (7) and the proof of Theorem 4.3 ,
λ H ′ ( 0 ) = λ L ′ ( 0 ) = 1 2 , ψ H ′ ( 0 ) = ψ L ′ ( 0 ) = 3 g − 3 2 , λ H ′ ′ ( 0 ) = ν , λ L ′ ′ ( 0 ) = 1 12 , ψ H ′ ′ ( 0 ) = ( g − 1 ) ( 11 − 4 e ) 12 ( e − 2 ) , ψ L ′ ′ ( 0 ) = − g − 1 12 . (55) \begin{aligned}
\lambda_H'(0)&=\lambda_L'(0)=\frac12,&
\psi_H'(0)&=\psi_L'(0)=\frac{3g-3}{2},&
\lambda_H''(0)&=\nu,&
\lambda_L''(0)&=\frac1{12},\\
\psi_H''(0)&=\frac{(g-1)(11-4e)}{12(e-2)},&
\psi_L''(0)&=-\frac{g-1}{12}. \tag*{(55)}
\end{aligned} λ H ′ ( 0 ) ψ H ′′ ( 0 ) = λ L ′ ( 0 ) = 2 1 , = 12 ( e − 2 ) ( g − 1 ) ( 11 − 4 e ) , ψ H ′ ( 0 ) ψ L ′′ ( 0 ) = ψ L ′ ( 0 ) = 2 3 g − 3 , = − 12 g − 1 . λ H ′′ ( 0 ) = ν , λ L ′′ ( 0 ) = 12 1 , (55) This is condition (1), with ε n = n − 1 / 2 \varepsilon_n=n^{-1/2} ε n = n − 1/2 ε n = n − 1 \varepsilon_n=n^{-1} ε n = n − 1 d n / 2 d_n/2 d n /2
Proposition 5.2. For every fixed g ≥ 1 g\ge1 g ≥ 1 δ ∈ ( 0 , π ) \delta\in(0,\pi) δ ∈ ( 0 , π ) c 1 , c 2 > 0 c_1,c_2>0 c 1 , c 2 > 0
sup δ ≤ ∣ θ ∣ ≤ π ∣ E e i θ X g , n H ∣ = O ( e − c 1 n ) , sup δ ≤ ∣ θ ∣ ≤ π ∣ E e i θ X g , n L ∣ = O ( e − c 2 n ) . (56) \sup_{\delta\le|\theta|\le\pi}\left|\mathbb{E}e^{i\theta X^H_{g,n}}\right|=O(e^{-c_1n}),\qquad
\sup_{\delta\le|\theta|\le\pi}\left|\mathbb{E}e^{i\theta X^L_{g,n}}\right|=O(e^{-c_2n}). \tag*{(56)} δ ≤ ∣ θ ∣ ≤ π sup E e i θ X g , n H = O ( e − c 1 n ) , δ ≤ ∣ θ ∣ ≤ π sup E e i θ X g , n L = O ( e − c 2 n ) . (56) Proof. Write t = e i θ t=e^{i\theta} t = e i θ δ ≤ ∣ θ ∣ ≤ π \delta\le|\theta|\le\pi δ ≤ ∣ θ ∣ ≤ π log t = i θ \log t=i\theta log t = i θ
Light case. The zeros of $1-W(w,t)$ closest to the origin have modulus ∣ τ ( t ) ∣ = ∣ θ ∣ / ( 2 ∣ sin ( θ / 2 ) ∣ ) > 1 |\tau(t)|=|\theta|/(2|\sin(\theta/2)|)>1 ∣ τ ( t ) ∣ = ∣ θ ∣/ ( 2∣ sin ( θ /2 ) ∣ ) > 1 1 < R L < min δ ≤ ∣ θ ∣ ≤ π ∣ τ ( e i θ ) ∣ 1<R_L<\min_{\delta\le|\theta|\le\pi}|\tau(e^{i\theta})| 1 < R L < min δ ≤ ∣ θ ∣ ≤ π ∣ τ ( e i θ ) ∣ (24) and (33) , L g ( w , t ) L_g(w,t) L g ( w , t ) ∣ w ∣ ≤ R L |w|\le R_L ∣ w ∣ ≤ R L ∣ w ∣ = R L |w|=R_L ∣ w ∣ = R L
Heavy case. Expanding (17) gives
R ( w , t ) = w − ∑ m ≥ 3 t ( 1 + t + ⋯ + t m − 3 ) w m m ! . R(w,t)=w-\sum_{m\geq3}t(1+t+\cdots+t^{m-3})\frac{w^m}{m!}. R ( w , t ) = w − m ≥ 3 ∑ t ( 1 + t + ⋯ + t m − 3 ) m ! w m . The coefficient polynomial has absolute value at most m − 2 m-2 m − 2 m = 4 m=4 m = 4 ∣ t + t 2 ∣ = 2 ∣ cos ( θ / 2 ) ∣ ≤ 2 cos ( δ / 2 ) |t+t^2|=2|\cos(\theta/2)|\leq2\cos(\delta/2) ∣ t + t 2 ∣ = 2∣ cos ( θ /2 ) ∣ ≤ 2 cos ( δ /2 ) ∑ m ≥ 3 ( m − 2 ) / m ! = 3 − e \sum_{m\geq3}(m-2)/m!=3-e ∑ m ≥ 3 ( m − 2 ) / m ! = 3 − e ∣ w ∣ ≤ 1 |w|\leq1 ∣ w ∣ ≤ 1
∣ R ( w , t ) − w ∣ ≤ 3 − e − 1 − cos ( δ / 2 ) 12 . |R(w,t)-w|\leq3-e-\frac{1-\cos(\delta/2)}{12}. ∣ R ( w , t ) − w ∣ ≤ 3 − e − 12 1 − cos ( δ /2 ) . Let R H = e − 2 + ( 1 − cos ( δ / 2 ) ) / 24 > e − 2 R_H=e-2+(1-\cos(\delta/2))/24>e-2 R H = e − 2 + ( 1 − cos ( δ /2 )) /24 > e − 2 ∣ z ∣ ≤ R H |z|\leq R_H ∣ z ∣ ≤ R H ∣ w ∣ = 1 |w|=1 ∣ w ∣ = 1 ∣ ( R ( w , t ) − z ) − w ∣ ≤ 1 − ( 1 − cos ( δ / 2 ) ) / 24 < 1 |(R(w,t)-z)-w|\leq1-(1-\cos(\delta/2))/24<1 ∣ ( R ( w , t ) − z ) − w ∣ ≤ 1 − ( 1 − cos ( δ /2 )) /24 < 1 R ( w , t ) = z R(w,t)=z R ( w , t ) = z ∣ w ∣ < 1 |w|<1 ∣ w ∣ < 1 T ( z , t ) T(z,t) T ( z , t ) ∣ z ∣ ≤ R H |z|\leq R_H ∣ z ∣ ≤ R H
∣ T ( z , t ) ∣ ≤ ∣ z ∣ + ∣ R ( T ( z , t ) , t ) − T ( z , t ) ∣ ≤ 1 − 1 − cos ( δ / 2 ) 24 , |T(z,t)|\leq|z|+|R(T(z,t),t)-T(z,t)|\leq1-\frac{1-\cos(\delta/2)}{24}, ∣ T ( z , t ) ∣ ≤ ∣ z ∣ + ∣ R ( T ( z , t ) , t ) − T ( z , t ) ∣ ≤ 1 − 24 1 − cos ( δ /2 ) , so T ( z , t ) T(z,t) T ( z , t ) L g ( w , t ) \mathcal{L}_g(w,t) L g ( w , t ) (19) , H g ( z , t ) \mathcal{H}_g(z,t) H g ( z , t ) ∣ z ∣ ≤ R H |z|\leq R_H ∣ z ∣ ≤ R H ∣ z ∣ = R H |z|=R_H ∣ z ∣ = R H
Conclusion. Cauchy’s estimates give ∣ H g , n ( e i θ ) ∣ / n ! = O ( R H − n ) |\mathcal{H}_{g,n}(e^{i\theta})|/n!=O(R_H^{-n}) ∣ H g , n ( e i θ ) ∣/ n ! = O ( R H − n ) ∣ L g , n ( e i θ ) ∣ / n ! = O ( R L − n ) |\mathcal{L}_{g,n}(e^{i\theta})|/n!=O(R_L^{-n}) ∣ L g , n ( e i θ ) ∣/ n ! = O ( R L − n ) θ \theta θ Theorem 4.1 , H g , n ( 1 ) / n ! \mathcal{H}_{g,n}(1)/n! H g , n ( 1 ) / n ! L g , n ( 1 ) / n ! \mathcal{L}_{g,n}(1)/n! L g , n ( 1 ) / n ! n − 1 ( e − 2 ) − n n^{-1}(e-2)^{-n} n − 1 ( e − 2 ) − n n − 1 n^{-1} n − 1 (56) are O ( n ( ( e − 2 ) / R H ) n ) O(n((e-2)/R_H)^n) O ( n (( e − 2 ) / R H ) n ) O ( n R L − n ) O(nR_L^{-n}) O ( n R L − n )
Theorem 5.3. Fix g ≥ 1 g\geq1 g ≥ 1 J ∈ { H , L } J\in\{H,L\} J ∈ { H , L } p n , k = P ( X g , n J = k ) p_{n,k}=\mathbb{P}(X^J_{g,n}=k) p n , k = P ( X g , n J = k ) σ n 2 = Var ( X g , n J ) \sigma_n^2=\operatorname{Var}(X^J_{g,n}) σ n 2 = Var ( X g , n J ) z n , k = ( k − d n / 2 ) / σ n z_{n,k}=(k-d_n/2)/\sigma_n z n , k = ( k − d n /2 ) / σ n κ j , n \kappa_{j,n} κ j , n j j j X g , n J X^J_{g,n} X g , n J (50) holds uniformly for all k ∈ Z k\in\mathbb{Z} k ∈ Z
Proof. The three hypotheses of Theorem 5.1 were verified above. □
Multiplying the local probabilities by the even Betti totals of Theorem 4.1 gives the individual Betti numbers.
Corollary 5.4. Fix g ≥ 1 g\geq1 g ≥ 1 x n , k = k − d n / 2 x_{n,k}=k-d_n/2 x n , k = k − d n /2 k k k
b 2 k ( M ‾ g , n ) = C g ( 1 ) 2 π ν n ! ( e − 2 ) − n n 3 g − 6 2 [ exp ( − x n , k 2 2 ν n ) + O ( n − 1 / 2 ) ] . b_{2k}(\overline{\mathcal{M}}_{g,n})=\frac{C_g(1)}{\sqrt{2\pi\nu}}\,n!(e-2)^{-n}n^{\frac{3g-6}{2}}\left[\exp\left(-\frac{x_{n,k}^2}{2\nu n}\right)+O(n^{-1/2})\right]. b 2 k ( M g , n ) = 2 π ν C g ( 1 ) n ! ( e − 2 ) − n n 2 3 g − 6 [ exp ( − 2 ν n x n , k 2 ) + O ( n − 1/2 ) ] . b 2 k ( M ‾ g , 0 ∣ n ) = 6 π A g ( 1 ) n ! n 6 g − 9 2 [ exp ( − 6 x n , k 2 n ) + O ( n − 1 ) ] . (57) b_{2k}(\overline{\mathcal{M}}_{g,0|n})=\sqrt{\frac{6}{\pi}}\,A_g(1)n!n^{\frac{6g-9}{2}}\left[\exp\left(-\frac{6x_{n,k}^2}{n}\right)+O(n^{-1})\right]. \tag*{(57)} b 2 k ( M g , 0∣ n ) = π 6 A g ( 1 ) n ! n 2 6 g − 9 [ exp ( − n 6 x n , k 2 ) + O ( n − 1 ) ] . (57) where A 1 ( 1 ) = 1 2 A_1(1)=\frac{1}{2} A 1 ( 1 ) = 2 1 C 1 ( 1 ) = 1 4 C_1(1)=\frac{1}{4} C 1 ( 1 ) = 4 1 g ≥ 2 g\geq2 g ≥ 2
A g ( 1 ) = c g e − 2 ( g − 1 ) Γ ( 3 g − 3 ) , C g ( 1 ) = c g e − g − 1 2 Γ ( 3 g − 3 2 ) [ 2 ( e − 2 ) ] 3 g − 3 2 . A_g(1)=\frac{c_g e^{-2(g-1)}}{\Gamma(3g-3)},\qquad C_g(1)=\frac{c_g e^{-\frac{g-1}{2}}}{\Gamma\left(\frac{3g-3}{2}\right)[2(e-2)]^{\frac{3g-3}{2}}}. A g ( 1 ) = Γ ( 3 g − 3 ) c g e − 2 ( g − 1 ) , C g ( 1 ) = Γ ( 2 3 g − 3 ) [ 2 ( e − 2 ) ] 2 3 g − 3 c g e − 2 g − 1 . Proof. Since κ 4 , n , κ 6 , n = O ( n ) \kappa_{4,n},\kappa_{6,n}=O(n) κ 4 , n , κ 6 , n = O ( n ) e − y 2 / 2 He j ( y ) e^{-y^2/2}\operatorname{He}_j(y) e − y 2 /2 He j ( y ) R \mathbb{R} R Theorem 5.3 gives
P ( X g , n J = k ) = 1 2 π σ g , n J [ exp ( − x n , k 2 2 ( σ g , n J ) 2 ) + O ( n − 1 ) ] . \mathbb{P}(X^J_{g,n}=k)=\frac{1}{\sqrt{2\pi}\sigma^J_{g,n}}\left[\exp\left(-\frac{x_{n,k}^2}{2(\sigma^J_{g,n})^2}\right)+O(n^{-1})\right]. P ( X g , n J = k ) = 2 π σ g , n J 1 [ exp ( − 2 ( σ g , n J ) 2 x n , k 2 ) + O ( n − 1 ) ] . uniformly in k k k Theorem 4.3 , ( σ g , n J ) 2 = v J n + O ( 1 ) (\sigma^J_{g,n})^2=v_Jn+O(1) ( σ g , n J ) 2 = v J n + O ( 1 ) v H = ν v_H=\nu v H = ν v L = 1 / 12 v_L=1/12 v L = 1/12 ( σ g , n J ) 2 (\sigma^J_{g,n})^2 ( σ g , n J ) 2 v J n v_Jn v J n O ( n − 1 ) O(n^{-1}) O ( n − 1 ) k k k 1 / σ g , n J = ( v J n ) − 1 / 2 ( 1 + O ( n − 1 ) ) 1/\sigma^J_{g,n}=(v_Jn)^{-1/2}(1+O(n^{-1})) 1/ σ g , n J = ( v J n ) − 1/2 ( 1 + O ( n − 1 )) H g , n ( 1 ) \mathcal{H}_{g,n}(1) H g , n ( 1 ) L g , n ( 1 ) \mathcal{L}_{g,n}(1) L g , n ( 1 ) Theorem 4.1 and evaluating (35) at t = 1 t=1 t = 1
Central ultra-log-concavity
For a function f f f Δ 2 f ( k ) = f ( k − 1 ) − 2 f ( k ) + f ( k + 1 ) \Delta^2 f(k)=f(k-1)-2f(k)+f(k+1) Δ 2 f ( k ) = f ( k − 1 ) − 2 f ( k ) + f ( k + 1 ) ( a k ) k = 0 d (a_k)_{k=0}^{d} ( a k ) k = 0 d 1 ≤ k ≤ d − 1 1\le k\le d-1 1 ≤ k ≤ d − 1 r r r k k k
Δ 2 log ( a k ( d k ) r ) ≤ 0 , (58) \Delta^2\log\left(\frac{a_k}{\binom{d}{k}^{r}}\right)\le0, \tag*{(58)} Δ 2 log ( ( k d ) r a k ) ≤ 0 , (58) and strict r r r
The strategy is to compare Δ 2 log p n , k \Delta^2\log p_{n,k} Δ 2 log p n , k r Δ 2 log ( d n k ) r\Delta^2\log\binom{d_n}{k} r Δ 2 log ( k d n ) n − 1 n^{-1} n − 1 d n = n + O ( 1 ) d_n=n+O(1) d n = n + O ( 1 ) r r r 1 / ( 4 v ) 1/(4v) 1/ ( 4 v ) v n v_n v n n − 1 n^{-1} n − 1 n − 2 n^{-2} n − 2 (50) then contribute only o ( n − 2 ) o(n^{-2}) o ( n − 2 ) n − 2 n^{-2} n − 2
Lemma 5.5 (Central ultra-log-concavity criterion). Suppose that ( Y n ) (Y_n) ( Y n ) satisfies the hypotheses of Theorem 5.1 , that Y n Y_n Y n takes values in { 0 , … , d n } \{0,\ldots,d_n\} { 0 , … , d n } with d n = n + O ( 1 ) d_n=n+O(1) d n = n + O ( 1 ) and that μ n = d n / 2 \mu_n=d_n/2 μ n = d n /2 Put v = λ ′ ′ ( 0 ) > 0 v=\lambda''(0)>0 v = λ ′′ ( 0 ) > 0 so that
σ n 2 = v n + O ( 1 ) . (59) \sigma_n^2=vn+O(1). \tag*{(59)} σ n 2 = v n + O ( 1 ) . (59) Fix 0 < c 0 < 2 0<c_0<\sqrt{2} 0 < c 0 < 2 The following statements hold uniformly for all k k k such that ∣ z n , k + j ∣ ≤ c 0 log n |z_{n,k+j}|\le c_0\sqrt{\log n} ∣ z n , k + j ∣ ≤ c 0 log n for j = − 1 , 0 , 1 j=-1,0,1 j = − 1 , 0 , 1
Put r ∗ = 1 / ( 4 v ) r_* = 1/(4v) r ∗ = 1/ ( 4 v ) 0 ≤ r < r ∗ 0\le r<r_* 0 ≤ r < r ∗ ( p n , k ) k (p_{n,k})_k ( p n , k ) k r r r k k k n n n r > r ∗ r>r_* r > r ∗ (58) holds at k k k n n n
The sequence ( p n , k ) k (p_{n,k})_k ( p n , k ) k r ∗ r_* r ∗ k k k n n n
λ ( 4 ) ( 0 ) ≤ − 8 v 3 and ψ ′ ′ ( 0 ) < v ( 2 ψ ′ ( 0 ) + 1 ) + λ ( 4 ) ( 0 ) 2 v . (60) \lambda^{(4)}(0)\le-8v^3
\qquad\text{and}\qquad
\psi''(0)<v\bigl(2\psi'(0)+1\bigr)+\frac{\lambda^{(4)}(0)}{2v}. \tag*{(60)} λ ( 4 ) ( 0 ) ≤ − 8 v 3 and ψ ′′ ( 0 ) < v ( 2 ψ ′ ( 0 ) + 1 ) + 2 v λ ( 4 ) ( 0 ) . (60) Proof. Put L n = c 0 log n L_n=c_0\sqrt{\log n} L n = c 0 log n ∣ z n , k ∣ ≤ L n |z_{n,k}|\le L_n ∣ z n , k ∣ ≤ L n e − z n , k 2 / 2 / 2 π e^{-z_{n,k}^2/2}/\sqrt{2\pi} e − z n , k 2 /2 / 2 π n − c 0 2 / 2 / 2 π n^{-c_0^2/2}/\sqrt{2\pi} n − c 0 2 /2 / 2 π c 0 2 / 2 < 1 c_0^2/2<1 c 0 2 /2 < 1 k k k z n , k z_{n,k} z n , k σ n − 1 = O ( n − 1 / 2 ) \sigma_n^{-1}=O(n^{-1/2}) σ n − 1 = O ( n − 1/2 )
(1) Keeping only the fourth-cumulant term in (50) gives
σ n p n , k = e − z n , k 2 / 2 2 π [ 1 + κ 4 , n 4 ! σ n 4 He 4 ( z n , k ) ] + O ( n − 2 ) . \sigma_n p_{n,k}=\frac{e^{-z_{n,k}^2/2}}{\sqrt{2\pi}}\left[1+\frac{\kappa_{4,n}}{4!\sigma_n^4}\operatorname{He}_4(z_{n,k})\right]+O(n^{-2}). σ n p n , k = 2 π e − z n , k 2 /2 [ 1 + 4 ! σ n 4 κ 4 , n He 4 ( z n , k ) ] + O ( n − 2 ) . After division by the Gaussian factor, the remainder becomes O ( n − 2 e L n 2 / 2 ) O(n^{-2}e^{L_n^2/2}) O ( n − 2 e L n 2 /2 ) O ( n − 1 ( 1 + L n 4 ) ) O(n^{-1}(1+L_n^4)) O ( n − 1 ( 1 + L n 4 )) p n , k > 0 p_{n,k}>0 p n , k > 0 n n n
log p n , k = − log ( 2 π σ n ) − z n , k 2 2 + κ 4 , n 4 ! σ n 4 He 4 ( z n , k ) + O ( n − 2 ( 1 + L n 8 + e L n 2 / 2 ) ) . \log p_{n,k}=-\log(\sqrt{2\pi}\sigma_n)-\frac{z_{n,k}^2}{2}+\frac{\kappa_{4,n}}{4!\sigma_n^4}\operatorname{He}_4(z_{n,k})+O\left(n^{-2}(1+L_n^8+e^{L_n^2/2})\right). log p n , k = − log ( 2 π σ n ) − 2 z n , k 2 + 4 ! σ n 4 κ 4 , n He 4 ( z n , k ) + O ( n − 2 ( 1 + L n 8 + e L n 2 /2 ) ) . The second difference of the fourth-cumulant term is O ( n − 2 ( 1 + L n 2 ) ) O(n^{-2}(1+L_n^2)) O ( n − 2 ( 1 + L n 2 )) O ( n − 2 e L n 2 / 2 ) = o ( n − 1 ) O(n^{-2}e^{L_n^2/2})=o(n^{-1}) O ( n − 2 e L n 2 /2 ) = o ( n − 1 ) (59) ,
Δ 2 log p n , k = − σ n − 2 + o ( n − 1 ) = − 1 v n + o ( n − 1 ) . \Delta^2\log p_{n,k}=-\sigma_n^{-2}+o(n^{-1})=-\frac{1}{vn}+o(n^{-1}). Δ 2 log p n , k = − σ n − 2 + o ( n − 1 ) = − v n 1 + o ( n − 1 ) . On the other hand, since d n = n + O ( 1 ) d_n=n+O(1) d n = n + O ( 1 ) ∣ k − d n / 2 ∣ ≤ σ n L n |k-d_n/2|\le\sigma_nL_n ∣ k − d n /2∣ ≤ σ n L n
Δ 2 log ( d n k ) = log k ( d n − k ) ( k + 1 ) ( d n − k + 1 ) = − 4 n + O ( n − 2 ( 1 + L n 2 ) ) . \Delta^2\log\binom{d_n}{k}
=\log\frac{k(d_n-k)}{(k+1)(d_n-k+1)}
=-\frac{4}{n}+O\left(n^{-2}(1+L_n^2)\right). Δ 2 log ( k d n ) = log ( k + 1 ) ( d n − k + 1 ) k ( d n − k ) = − n 4 + O ( n − 2 ( 1 + L n 2 ) ) . Therefore
Δ 2 log ( p n , k ( d n k ) r ) = 4 r − v − 1 n + o r ( n − 1 ) , (61) \Delta^2\log\left(\frac{p_{n,k}}{\binom{d_n}{k}^{r}}\right)
=\frac{4r-v^{-1}}{n}+o_r(n^{-1}), \tag*{(61)} Δ 2 log ( ( k d n ) r p n , k ) = n 4 r − v − 1 + o r ( n − 1 ) , (61) and for r ≠ r ∗ r \ne r_* r = r ∗ 4 r − v − 1 4r-v^{-1} 4 r − v − 1
(2) We now use the full expansion (50) . The additional terms in log p n , k \log p_{n,k} log p n , k He 6 \operatorname{He}_6 He 6 He 8 \operatorname{He}_8 He 8 He 2 4 \operatorname{He}_2^4 He 2 4 O ( n − 2 ) O(n^{-2}) O ( n − 2 ) O ( n − 3 ( 1 + L n 8 ) ) O(n^{-3}(1+L_n^8)) O ( n − 3 ( 1 + L n 8 )) (50) divided by the Gaussian factor are O ( n − 3 ( 1 + L n 12 + e L n / 2 ) ) = o ( n − 2 ) O(n^{-3}(1+L_n^{12}+e^{L_n/2}))=o(n^{-2}) O ( n − 3 ( 1 + L n 12 + e L n /2 )) = o ( n − 2 ) He 4 ′ ′ ( z ) = 12 ( z 2 − 1 ) \operatorname{He}_4''(z)=12(z^2-1) He 4 ′′ ( z ) = 12 ( z 2 − 1 )
Δ 2 log p n , k = − σ n − 2 + κ 4 , n 2 σ n 6 ( z n , k 2 − 1 ) + o ( n − 2 ) . (62) \Delta^2\log p_{n,k}=-\sigma_n^{-2}+\frac{\kappa_{4,n}}{2\sigma_n^6}(z_{n,k}^2-1)+o(n^{-2}). \tag*{(62)} Δ 2 log p n , k = − σ n − 2 + 2 σ n 6 κ 4 , n ( z n , k 2 − 1 ) + o ( n − 2 ) . (62) Since μ n = n λ ′ ( 0 ) + ψ ′ ( 0 ) + O ( ϵ n ) \mu_n=n\lambda'(0)+\psi'(0)+O(\epsilon_n) μ n = n λ ′ ( 0 ) + ψ ′ ( 0 ) + O ( ϵ n ) d n = 2 μ n = n + O ( 1 ) d_n=2\mu_n=n+O(1) d n = 2 μ n = n + O ( 1 ) λ ′ ( 0 ) = 1 / 2 \lambda'(0)=1/2 λ ′ ( 0 ) = 1/2 d n = n + 2 ψ ′ ( 0 ) + O ( ϵ n ) d_n=n+2\psi'(0)+O(\epsilon_n) d n = n + 2 ψ ′ ( 0 ) + O ( ϵ n ) (49) ,
σ n 2 = v n + ψ ′ ′ ( 0 ) + O ( ϵ n ) , κ 4 , n = n λ ( 4 ) ( 0 ) + O ( 1 ) . (63) \sigma_n^2=vn+\psi''(0)+O(\epsilon_n),\qquad\kappa_{4,n}=n\lambda^{(4)}(0)+O(1). \tag*{(63)} σ n 2 = v n + ψ ′′ ( 0 ) + O ( ϵ n ) , κ 4 , n = n λ ( 4 ) ( 0 ) + O ( 1 ) . (63) Expanding the binomial ratio one order further, with k − d n / 2 = σ n z n , k k-d_n/2=\sigma_n z_{n,k} k − d n /2 = σ n z n , k
Δ 2 log ( d n k ) = − 4 n + 4 ( 2 ψ ′ ( 0 ) + 1 ) − 16 v z n , k 2 n 2 + O ( n − 3 ( 1 + L n 4 ) + ϵ n n − 2 ) . (64) \Delta^2\log\binom{d_n}{k}=-\frac{4}{n}+\frac{4(2\psi'(0)+1)-16vz_{n,k}^2}{n^2}+O(n^{-3}(1+L_n^4)+\epsilon_n n^{-2}). \tag*{(64)} Δ 2 log ( k d n ) = − n 4 + n 2 4 ( 2 ψ ′ ( 0 ) + 1 ) − 16 v z n , k 2 + O ( n − 3 ( 1 + L n 4 ) + ϵ n n − 2 ) . (64) Substituting (63) into (62) and subtracting r ∗ r_* r ∗ (64) , we find
Δ 2 log p n , k ( d n k ) r ∗ = 1 n 2 [ ψ ′ ′ ( 0 ) − v ( 2 ψ ′ ( 0 ) + 1 ) v 2 + 4 z n , k 2 + λ ( 4 ) ( 0 ) ( z n , k 2 − 1 ) 2 v 3 ] + o ( n − 2 ) , (65) \Delta^2\log\frac{p_{n,k}}{\binom{d_n}{k}^{r_*}}=\frac{1}{n^2}\left[\frac{\psi''(0)-v(2\psi'(0)+1)}{v^2}+4z_{n,k}^2+\frac{\lambda^{(4)}(0)(z_{n,k}^2-1)}{2v^3}\right]+o(n^{-2}), \tag*{(65)} Δ 2 log ( k d n ) r ∗ p n , k = n 2 1 [ v 2 ψ ′′ ( 0 ) − v ( 2 ψ ′ ( 0 ) + 1 ) + 4 z n , k 2 + 2 v 3 λ ( 4 ) ( 0 ) ( z n , k 2 − 1 ) ] + o ( n − 2 ) , (65) where the substitutions cost O ( n − 3 ( 1 + L n 2 ) + ϵ n n − 2 ) = o ( n − 2 ) O(n^{-3}(1+L_n^2)+\epsilon_n n^{-2})=o(n^{-2}) O ( n − 3 ( 1 + L n 2 ) + ϵ n n − 2 ) = o ( n − 2 ) (60) , the constant term of the bracket is negative and the coefficient of z n , k 2 z_{n,k}^2 z n , k 2
Theorem 5.6. Fix g ≥ 1 g\ge1 g ≥ 1 0 < c 0 < 2 0<c_0<\sqrt{2} 0 < c 0 < 2 k k k ∣ k + j − d n / 2 ∣ ≤ c 0 σ g , n log n \lvert k+j-d_n/2\rvert\le c_0\sigma_{g,n}\sqrt{\log n} ∣ k + j − d n /2 ∣ ≤ c 0 σ g , n log n j = − 1 , 0 , 1 j=-1,0,1 j = − 1 , 0 , 1
(1) For each fixed 0 ≤ r < r H 0\le r<r_H 0 ≤ r < r H ( b 2 k ( M ‾ g , n ) ) k = 0 d n (b_{2k}(\overline{\mathcal{M}}_{g,n}))_{k=0}^{d_n} ( b 2 k ( M g , n ) ) k = 0 d n r r r k k k n n n r > r H r>r_H r > r H (58) holds at k k k n n n
(2) At r = r H r=r_H r = r H ( b 2 k ( M ‾ g , n ) ) k = 0 d n (b_{2k}(\overline{\mathcal{M}}_{g,n}))_{k=0}^{d_n} ( b 2 k ( M g , n ) ) k = 0 d n r H r_H r H k k k
Proof. The hypotheses of Theorem 5.1 were verified in Section 5.2, with v = ν v=\nu v = ν (55) . Moreover d n = n + O ( 1 ) d_n=n+O(1) d n = n + O ( 1 ) d n / 2 d_n/2 d n /2 r ∗ = 1 / ( 4 ν ) = r H r_*=1/(4\nu)=r_H r ∗ = 1/ ( 4 ν ) = r H (54) and (55) gives
λ H ( 4 ) ( 0 ) + 8 ν 3 = − e ( 4 e 2 + 9 e − 54 ) 1080 ( e − 2 ) 3 = − 0.000141 … , \lambda_H^{(4)}(0)+8\nu^3=-\frac{e(4e^2+9e-54)}{1080(e-2)^3}=-0.000141\ldots, λ H ( 4 ) ( 0 ) + 8 ν 3 = − 1080 ( e − 2 ) 3 e ( 4 e 2 + 9 e − 54 ) = − 0.000141 … , ψ H ′ ′ ( 0 ) − ν ( 2 ψ H ′ ( 0 ) + 1 ) − λ H ( 4 ) ( 0 ) 2 ν = − 10 ( g − 1 ) ( 7 − 2 e ) ( 3 − e ) + 32 e 2 − 219 e + 360 120 ( e − 2 ) ( 3 − e ) . \psi_H''(0)-\nu(2\psi_H'(0)+1)-\frac{\lambda_H^{(4)}(0)}{2\nu}=-\frac{10(g-1)(7-2e)(3-e)+32e^2-219e+360}{120(e-2)(3-e)}. ψ H ′′ ( 0 ) − ν ( 2 ψ H ′ ( 0 ) + 1 ) − 2 ν λ H ( 4 ) ( 0 ) = − 120 ( e − 2 ) ( 3 − e ) 10 ( g − 1 ) ( 7 − 2 e ) ( 3 − e ) + 32 e 2 − 219 e + 360 . Since 4 e 2 + 9 e − 54 = 0.0207 … > 0 4e^2+9e-54=0.0207\ldots>0 4 e 2 + 9 e − 54 = 0.0207 … > 0 7 − 2 e > 0 7-2e>0 7 − 2 e > 0 3 − e > 0 3-e>0 3 − e > 0 32 e 2 − 219 e + 360 = 1.14 … > 0 32e^2-219e+360=1.14\ldots>0 32 e 2 − 219 e + 360 = 1.14 … > 0 g ≥ 1 g\ge1 g ≥ 1 (60) holds, and Lemma 5.5(2) proves (2). □
Theorem 5.7. Fix g ≥ 1 g\ge1 g ≥ 1 0 < c 0 < 2 0<c_0<\sqrt{2} 0 < c 0 < 2 k k k ∣ k + j − d n / 2 ∣ ≤ c 0 σ g , n log n \lvert k+j-d_n/2\rvert\le c_0\sigma_{g,n}\sqrt{\log n} ∣ k + j − d n /2 ∣ ≤ c 0 σ g , n log n j = − 1 , 0 , 1 j=-1,0,1 j = − 1 , 0 , 1
(1) For each fixed 0 ≤ r < 3 0\le r<3 0 ≤ r < 3 ( b 2 k ( M ‾ g , 0 ∣ n ) ) k = 0 d n (b_{2k}(\overline{\mathcal{M}}_{g,0|n}))_{k=0}^{d_n} ( b 2 k ( M g , 0∣ n ) ) k = 0 d n r r r k k k n n n r > 3 r>3 r > 3 (58) holds at k k k n n n
(2) At r = 3 r=3 r = 3 ( b 2 k ( M ‾ g , 0 ∣ n ) ) k = 0 d n (b_{2k}(\overline{\mathcal{M}}_{g,0|n}))_{k=0}^{d_n} ( b 2 k ( M g , 0∣ n ) ) k = 0 d n k k k
Proof. The proof of (1) is the same as in the proof of Theorem 5.6 , now with v = 1 / 12 v=1/12 v = 1/12 r ∗ = 3 r_* = 3 r ∗ = 3 λ L ( s ) = log ( ( e s − 1 ) / s ) = s 2 + s 2 24 − s 4 2880 + O ( s 6 ) \lambda_L(s)=\log((e^s-1)/s)=\frac{s}{2}+\frac{s^2}{24}-\frac{s^4}{2880}+O(s^6) λ L ( s ) = log (( e s − 1 ) / s ) = 2 s + 24 s 2 − 2880 s 4 + O ( s 6 ) λ L ( 4 ) ( 0 ) = − 1 / 120 \lambda_L^{(4)}(0)=-1/120 λ L ( 4 ) ( 0 ) = − 1/120
λ L ( 4 ) ( 0 ) + 8 v 3 = − 1 270 , ψ L ′ ′ ( 0 ) − v ( 2 ψ L ′ ( 0 ) + 1 ) − λ L ( 4 ) ( 0 ) 2 v = − 10 g − 9 30 . \lambda_L^{(4)}(0)+8v^3=-\frac{1}{270},\qquad\psi_L''(0)-v(2\psi_L'(0)+1)-\frac{\lambda_L^{(4)}(0)}{2v}=-\frac{10g-9}{30}. λ L ( 4 ) ( 0 ) + 8 v 3 = − 270 1 , ψ L ′′ ( 0 ) − v ( 2 ψ L ′ ( 0 ) + 1 ) − 2 v λ L ( 4 ) ( 0 ) = − 30 10 g − 9 . Both are negative for g ≥ 1 g\ge1 g ≥ 1
Example 5.8 (Losev–Manin spaces). The critical case in Lemma 5.5(2) is not automatic. Let Y n Y_n Y n M ‾ 0 , 2 ∣ n \overline{\mathcal M}_{0,2\mid n} M 0 , 2 ∣ n d n = n − 1 d_n=n-1 d n = n − 1 [21 Theorem 2.3] , and μ n = ( n − 1 ) / 2 \mu_n=(n-1)/2 μ n = ( n − 1 ) /2 C \mathcal C C w = τ ( t ) w=\tau(t) w = τ ( t ) [ t τ ( t ) ] − 1 ( 1 − w / τ ( t ) ) − 1 [t\tau(t)]^{-1}(1-w/\tau(t))^{-1} [ t τ ( t ) ] − 1 ( 1 − w / τ ( t ) ) − 1 t → 1 t\to1 t → 1
λ ( s ) = log e s − 1 s , ψ ( s ) = λ ( s ) − s . \lambda(s)=\log\frac{e^s-1}{s},\qquad\psi(s)=\lambda(s)-s. λ ( s ) = log s e s − 1 , ψ ( s ) = λ ( s ) − s . For t = e i θ t=e^{i\theta} t = e i θ δ ≤ ∣ θ ∣ ≤ π \delta\le|\theta|\le\pi δ ≤ ∣ θ ∣ ≤ π ∣ θ ∣ / ( 2 ∣ sin ( θ / 2 ) ∣ ) > 1 |\theta|/(2|\sin(\theta/2)|)>1 ∣ θ ∣/ ( 2∣ sin ( θ /2 ) ∣ ) > 1 n ! n! n !
v = 1 12 , ψ ′ ( 0 ) = − 1 2 , ψ ′ ′ ( 0 ) = 1 12 , λ ( 4 ) ( 0 ) = − 1 120 , v=\frac{1}{12},\qquad\psi'(0)=-\frac{1}{2},\qquad\psi''(0)=\frac{1}{12},\qquad\lambda^{(4)}(0)=-\frac{1}{120}, v = 12 1 , ψ ′ ( 0 ) = − 2 1 , ψ ′′ ( 0 ) = 12 1 , λ ( 4 ) ( 0 ) = − 120 1 , and Lemma 5.5(1) gives the threshold r ∗ = 3 r_*=3 r ∗ = 3 r = 3 r=3 r = 3 z n , k = ( k − ( n − 1 ) / 2 ) / σ n z_{n,k}=(k-(n-1)/2)/\sigma_n z n , k = ( k − ( n − 1 ) /2 ) / σ n
Δ 2 log b 2 k ( M ‾ 0 , 2 ∣ n ) ( n − 1 k ) 3 = 16 ( 6 − z n , k 2 ) 5 n 2 + O ( n − 3 ) , \Delta^2\log\frac{b_{2k}(\overline{\mathcal M}_{0,2\mid n})}{\binom{n-1}{k}^{3}}=\frac{16(6-z_{n,k}^2)}{5n^2}+O(n^{-3}), Δ 2 log ( k n − 1 ) 3 b 2 k ( M 0 , 2 ∣ n ) = 5 n 2 16 ( 6 − z n , k 2 ) + O ( n − 3 ) , which is positive near the middle. Hence central 3-ultra-log-concavity fails for M ‾ 0 , 2 ∣ n \overline{\mathcal M}_{0,2\mid n} M 0 , 2 ∣ n M ‾ g , 0 ∣ n \overline{\mathcal M}_{g,0\mid n} M g , 0 ∣ n ψ ′ ( 0 ) \psi'(0) ψ ′ ( 0 ) ψ ′ ′ ( 0 ) \psi''(0) ψ ′′ ( 0 )
Hodge types off the diagonal
Throughout this section, let g ≥ 2 g\ge2 g ≥ 2 u v uv uv M g \mathcal M_g M g
The Hodge-refined series
Substitute ( u , v ) = ( − s ξ , − s ξ − 1 ) (u,v)=(-s\xi,-s\xi^{-1}) ( u , v ) = ( − s ξ , − s ξ − 1 )
L ~ g ( w ; s , ξ ) : = L g ( w ; − s ξ , − s ξ − 1 ) , H ~ g ( z ; s , ξ ) : = H g ( z ; − s ξ , − s ξ − 1 ) . (66) \widetilde{\mathcal L}_g(w;s,\xi):=\mathcal L_g(w;-s\xi,-s\xi^{-1}),\qquad\widetilde{\mathcal H}_g(z;s,\xi):=\mathcal H_g(z;-s\xi,-s\xi^{-1}). \tag*{(66)} L g ( w ; s , ξ ) := L g ( w ; − s ξ , − s ξ − 1 ) , H g ( z ; s , ξ ) := H g ( z ; − s ξ , − s ξ − 1 ) . (66) By purity,
L ~ g ( w ; s , ξ ) = ∑ n ≥ 0 w n n ! ∑ p , q h p , q ( M ‾ g , 0 ∣ n ) s p + q ξ p − q , \widetilde{\mathcal L}_g(w;s,\xi)=\sum_{n\ge0}\frac{w^n}{n!}\sum_{p,q}h^{p,q}(\overline{\mathcal M}_{g,0\mid n})s^{p+q}\xi^{p-q}, L g ( w ; s , ξ ) = n ≥ 0 ∑ n ! w n p , q ∑ h p , q ( M g , 0 ∣ n ) s p + q ξ p − q , and similarly for H ~ g \widetilde{\mathcal H}_g H g s s s ξ \xi ξ u v = s 2 uv=s^2 uv = s 2
H ~ g ( z ; s , ξ ) = L ~ g ( T ( z , s 2 ) ; s , ξ ) . \widetilde{\mathcal H}_g(z;s,\xi)=\widetilde{\mathcal L}_g(T(z,s^2);s,\xi). H g ( z ; s , ξ ) = L g ( T ( z , s 2 ) ; s , ξ ) . For a function F F F ξ \xi ξ
F o d d ( ξ ) = F ( ξ ) − F ( − ξ ) 2 , F e v e n ( ξ ) = F ( ξ ) + F ( − ξ ) 2 . F^{\mathrm{odd}}(\xi)=\frac{F(\xi)-F(-\xi)}{2},\qquad F^{\mathrm{even}}(\xi)=\frac{F(\xi)+F(-\xi)}{2}. F odd ( ξ ) = 2 F ( ξ ) − F ( − ξ ) , F even ( ξ ) = 2 F ( ξ ) + F ( − ξ ) . Since p + q p+q p + q p − q p-q p − q [ ξ 0 ] [\xi^0] [ ξ 0 ]
For Y = M ‾ g , n Y=\overline{\mathcal{M}}_{g,n} Y = M g , n Y = M ‾ g , 0 ∣ n Y=\overline{\mathcal{M}}_{g,0|n} Y = M g , 0∣ n ( p , q ) (p,q) ( p , q )
h p , q ( Y ) P Y ( 1 ) = h p , q ( Y ) ∑ j b j ( Y ) . (67) \frac{h^{p,q}(Y)}{P_Y(1)}=\frac{h^{p,q}(Y)}{\sum_j b_j(Y)}. \tag*{(67)} P Y ( 1 ) h p , q ( Y ) = ∑ j b j ( Y ) h p , q ( Y ) . (67) and let D g , n H D^H_{g,n} D g , n H D g , n L D^L_{g,n} D g , n L p − q p-q p − q
∑ p , q h p , q ( Y ) P Y ( 1 ) s p + q ξ p − q = E ( Y ; − s ξ , − s ξ − 1 ) P Y ( 1 ) . \sum_{p,q}\frac{h^{p,q}(Y)}{P_Y(1)}s^{p+q}\xi^{p-q}=\frac{E(Y; -s\xi,-s\xi^{-1})}{P_Y(1)}. p , q ∑ P Y ( 1 ) h p , q ( Y ) s p + q ξ p − q = P Y ( 1 ) E ( Y ; − s ξ , − s ξ − 1 ) . By Hodge symmetry the distribution of the Hodge difference is symmetric about 0. Every class of odd degree lies off the diagonal, and
∑ p ≠ q h p , q ( Y ) = ∑ k b 2 k + 1 ( Y ) + ∑ p ≠ q p + q e v e n h p , q ( Y ) . (68) \sum_{p\ne q}h^{p,q}(Y)=\sum_k b_{2k+1}(Y)+\sum_{\substack{p\ne q\\p+q\ \mathrm{even}}}h^{p,q}(Y). \tag*{(68)} p = q ∑ h p , q ( Y ) = k ∑ b 2 k + 1 ( Y ) + p = q p + q even ∑ h p , q ( Y ) . (68) Proposition 3.9 already shows that off-diagonal classes are rare. The leading term of L ~ g ( w ; 1 , ξ ) \widetilde{\mathcal{L}}_g(w;1,\xi) L g ( w ; 1 , ξ ) ξ \xi ξ ξ 0 \xi^0 ξ 0 ( 1 − w ) 3 g − 4 (1-w)^{3g-4} ( 1 − w ) 3 g − 4 r − 1 ≤ ∣ ξ ∣ ≤ r r^{-1}\leq|\xi|\leq r r − 1 ≤ ∣ ξ ∣ ≤ r
P ( D g , n H ≠ 0 ) = O ( n − 1 / 2 ) , P ( D g , n L ≠ 0 ) = O ( n − 1 ) . (69) \mathbb{P}(D^H_{g,n}\ne0)=O(n^{-1/2}),\qquad\mathbb{P}(D^L_{g,n}\ne0)=O(n^{-1}). \tag*{(69)} P ( D g , n H = 0 ) = O ( n − 1/2 ) , P ( D g , n L = 0 ) = O ( n − 1 ) . (69) To find the exact orders and the conditional distribution of the Hodge difference, we must compute the leading off-diagonal contribution.
The elliptic vertex
Off-diagonal classes come from vertices of positive genus. To determine their leading contribution, we first compute the vertex series in genus one. For d = 1 , 2 d=1,2 d = 1 , 2 d d d r r r A 1 , d ( w ; s , ξ ) A_{1,d}(w;s,\xi) A 1 , d ( w ; s , ξ ) r r r ( u , v ) = ( − s ξ , − s ξ − 1 ) (u,v)=(-s\xi,-s\xi^{-1}) ( u , v ) = ( − s ξ , − s ξ − 1 ) r r r M 1 , d \mathcal{M}_{1,d} M 1 , d d d d
For the identity automorphism, the vertex series B Γ , 1 B_{\Gamma,1} B Γ , 1 A 1 , d A_{1,d} A 1 , d d d d ( u , v ) = ( − s ξ , − s ξ − 1 ) (u,v)=(-s\xi,-s\xi^{-1}) ( u , v ) = ( − s ξ , − s ξ − 1 ) Γ \Gamma Γ
B Γ , 1 ( w ; − s ξ , − s ξ − 1 ) = A 1 , 1 ( w ; s , ξ ) e ( 2 g − 3 ) ( s 2 − 2 ) w , B_{\Gamma,1}(w;-s\xi,-s\xi^{-1})=A_{1,1}(w;s,\xi)e^{(2g-3)(s^2-2)w}, B Γ , 1 ( w ; − s ξ , − s ξ − 1 ) = A 1 , 1 ( w ; s , ξ ) e ( 2 g − 3 ) ( s 2 − 2 ) w , since each of the 2 g − 3 2g-3 2 g − 3 e ( s 2 − 2 ) w e^{(s^2-2)w} e ( s 2 − 2 ) w
Recall from the proof of Lemma 3.3 that I n ( u , v ) I_n(u,v) I n ( u , v ) n n n ( n − 1 ) (n-1) ( n − 1 ) M 1 , 1 \mathcal{M}_{1,1} M 1 , 1
For d = 1 d=1 d = 1 I r + 1 I_{r+1} I r + 1 r r r k k k I r − k + 1 I_{r-k+1} I r − k + 1
r ! [ w r ] A 1 , 1 ( w ; s , ξ ) = ∑ k = 0 r ( − 1 ) k ( r k ) I r − k + 1 ( − s ξ , − s ξ − 1 ) . r![w^r]A_{1,1}(w;s,\xi)=\sum_{k=0}^{r}(-1)^k\binom{r}{k}I_{r-k+1}(-s\xi,-s\xi^{-1}). r ! [ w r ] A 1 , 1 ( w ; s , ξ ) = k = 0 ∑ r ( − 1 ) k ( k r ) I r − k + 1 ( − s ξ , − s ξ − 1 ) . Passing to exponential generating series yields the factor ∑ k ≥ 0 ( − w ) k / k ! = e − w \sum_{k\geq0}(-w)^k/k!=e^{-w} ∑ k ≥ 0 ( − w ) k / k ! = e − w d = 2 d=2 d = 2 r r r I r + 2 I_{r+2} I r + 2 I r + 1 I_{r+1} I r + 1 I r + 2 − I r + 1 I_{r+2}-I_{r+1} I r + 2 − I r + 1 e − 2 w e^{-2w} e − 2 w
A 1 , 1 ( w ; s , ξ ) = e − w ∑ r ≥ 0 I r + 1 ( − s ξ , − s ξ − 1 ) w r r ! , (70) A_{1,1}(w;s,\xi)=e^{-w}\sum_{r\geq0}I_{r+1}(-s\xi,-s\xi^{-1})\frac{w^r}{r!}, \tag*{(70)} A 1 , 1 ( w ; s , ξ ) = e − w r ≥ 0 ∑ I r + 1 ( − s ξ , − s ξ − 1 ) r ! w r , (70) A 1 , 2 ( w ; s , ξ ) = e − 2 w ∑ r ≥ 0 ( I r + 2 ( − s ξ , − s ξ − 1 ) − I r + 1 ( − s ξ , − s ξ − 1 ) ) w r r ! . A_{1,2}(w;s,\xi)=e^{-2w}\sum_{r\geq0}\left(I_{r+2}(-s\xi,-s\xi^{-1})-I_{r+1}(-s\xi,-s\xi^{-1})\right)\frac{w^r}{r!}. A 1 , 2 ( w ; s , ξ ) = e − 2 w r ≥ 0 ∑ ( I r + 2 ( − s ξ , − s ξ − 1 ) − I r + 1 ( − s ξ , − s ξ − 1 ) ) r ! w r . Since I n e v e n I_n^{\mathrm{even}} I n even u v = s 2 uv=s^2 uv = s 2 ξ \xi ξ w w w ( s , ξ ) ∈ C × C ∗ (s,\xi)\in\mathbb{C}\times\mathbb{C}^* ( s , ξ ) ∈ C × C ∗
The odd part of A 1 , 1 A_{1,1} A 1 , 1
I m ( 2 w ) = ∑ j ≥ 0 w m + 2 j j ! ( m + j ) ! , m ≥ 0 , \mathbb{I}_m(2w)=\sum_{j\geq0}\frac{w^{m+2j}}{j!(m+j)!},\qquad m\geq0, I m ( 2 w ) = j ≥ 0 ∑ j ! ( m + j )! w m + 2 j , m ≥ 0 , and the cusp-form dimensions β k = dim S k ( S L 2 ( Z ) ) \beta_k=\dim S_k(\mathrm{SL}_2(\mathbb{Z})) β k = dim S k ( SL 2 ( Z )) k ≥ 4 k\geq4 k ≥ 4
E ( w ; s , ξ ) = ∑ m ≥ 3 m o d d β m + 1 m w I m ( 2 s w ) ( ξ m + ξ − m ) , (71) \mathcal{E}(w;s,\xi)=\sum_{\substack{m\geq3\\m\ \mathrm{odd}}}\beta_{m+1}\frac{m}{w}\mathbb{I}_m(2sw)(\xi^m+\xi^{-m}), \tag*{(71)} E ( w ; s , ξ ) = m ≥ 3 m odd ∑ β m + 1 w m I m ( 2 s w ) ( ξ m + ξ − m ) , (71) with the value at w = 0 w=0 w = 0
Lemma 6.1. The series E \mathcal{E} E ( w , s ) (w,s) ( w , s ) ξ ≠ 0 \xi\ne0 ξ = 0 t = s 2 t=s^2 t = s 2 α i j = ( 2 j i ) − ( 2 j i − 1 ) \alpha_{ij}=\binom{2j}{i}-\binom{2j}{i-1} α ij = ( i 2 j ) − ( i − 1 2 j )
A 1 , 1 o d d ( w ; s , ξ ) = e t w ∑ j ≥ 1 s 2 j + 1 w 2 j ( 2 j ) ! ∑ i = 0 j − 1 α i j β 2 j − 2 i + 2 ( ξ 2 j − 2 i + 1 + ξ − ( 2 j − 2 i + 1 ) ) = e t w E ( w ; s , ξ ) . A_{1,1}^{\mathrm{odd}}(w;s,\xi)=e^{tw}\sum_{j\geq1}\frac{s^{2j+1}w^{2j}}{(2j)!}\sum_{i=0}^{j-1}\alpha_{ij}\beta_{2j-2i+2}\left(\xi^{2j-2i+1}+\xi^{-(2j-2i+1)}\right)=e^{tw}\mathcal{E}(w;s,\xi). A 1 , 1 odd ( w ; s , ξ ) = e tw j ≥ 1 ∑ ( 2 j )! s 2 j + 1 w 2 j i = 0 ∑ j − 1 α ij β 2 j − 2 i + 2 ( ξ 2 j − 2 i + 1 + ξ − ( 2 j − 2 i + 1 ) ) = e tw E ( w ; s , ξ ) . In particular, A 1 , 1 o d d A_{1,1}^{\mathrm{odd}} A 1 , 1 odd ξ \xi ξ s = 1 s=1 s = 1 w > 0 w>0 w > 0
[ ξ 11 ] A 1 , 1 o d d ( w ; 1 , ξ ) ≥ e w w 10 10 ! , [ ξ 22 ] ( A 1 , 1 o d d ( 1 ; 1 , ξ ) ) 2 ≥ e 2 ( 10 ! ) 2 > 0. (72) [\xi^{11}]A_{1,1}^{\mathrm{odd}}(w;1,\xi)\geq e^w\frac{w^{10}}{10!},\qquad[\xi^{22}]\left(A_{1,1}^{\mathrm{odd}}(1;1,\xi)\right)^2\geq\frac{e^2}{(10!)^2}>0. \tag*{(72)} [ ξ 11 ] A 1 , 1 odd ( w ; 1 , ξ ) ≥ e w 10 ! w 10 , [ ξ 22 ] ( A 1 , 1 odd ( 1 ; 1 , ξ ) ) 2 ≥ ( 10 ! ) 2 e 2 > 0. (72) Proof. At ( u , v ) = ( − s ξ , − s ξ − 1 ) (u,v)=(-s\xi,-s\xi^{-1}) ( u , v ) = ( − s ξ , − s ξ − 1 ) u v = t uv=t uv = t u m + v m = − s m ( ξ m + ξ − m ) u^m+v^m=-s^m(\xi^m+\xi^{-m}) u m + v m = − s m ( ξ m + ξ − m ) m m m I n o d d I_n^{\mathrm{odd}} I n odd
∑ r ≥ 2 j ( r 2 j ) ( 1 + t ) r − 2 j w r r ! = e ( 1 + t ) w w 2 j ( 2 j ) ! \sum_{r\geq2j}\binom{r}{2j}(1+t)^{r-2j}\frac{w^r}{r!}=e^{(1+t)w}\frac{w^{2j}}{(2j)!} r ≥ 2 j ∑ ( 2 j r ) ( 1 + t ) r − 2 j r ! w r = e ( 1 + t ) w ( 2 j )! w 2 j gives the first expression, the factor e − w e^{-w} e − w (70) turning e ( 1 + t ) w e^{(1+t)w} e ( 1 + t ) w e t w e^{tw} e tw
For the second, fix an odd m ≥ 3 m\geq3 m ≥ 3 2 j − 2 i + 1 = m 2j-2i+1=m 2 j − 2 i + 1 = m i = j − ( m − 1 ) / 2 i=j-(m-1)/2 i = j − ( m − 1 ) /2 j = ( m − 1 ) / 2 + l j=(m-1)/2+l j = ( m − 1 ) /2 + l α i j \alpha_{ij} α ij
α l , ( m − 1 ) / 2 + l = ( m − 1 + 2 l l ) − ( m − 1 + 2 l l − 1 ) = m ( m − 1 + 2 l ) ! l ! ( m + l ) ! , \alpha_{l,(m-1)/2+l}=\binom{m-1+2l}{l}-\binom{m-1+2l}{l-1}=\frac{m(m-1+2l)!}{l!(m+l)!}, α l , ( m − 1 ) /2 + l = ( l m − 1 + 2 l ) − ( l − 1 m − 1 + 2 l ) = l ! ( m + l )! m ( m − 1 + 2 l )! , so that
∑ j ≥ ( m − 1 ) / 2 α j − ( m − 1 ) / 2 , j s 2 j + 1 w 2 j ( 2 j ) ! = ∑ l ≥ 0 m s m + 2 l w m − 1 + 2 l l ! ( m + l ) ! = m w I m ( 2 s w ) . \sum_{j\geq(m-1)/2}\alpha_{j-(m-1)/2,j}\frac{s^{2j+1}w^{2j}}{(2j)!}=\sum_{l\geq0}\frac{m\,s^{m+2l}w^{m-1+2l}}{l!(m+l)!}=\frac{m}{w}\mathbb{I}_m(2sw). j ≥ ( m − 1 ) /2 ∑ α j − ( m − 1 ) /2 , j ( 2 j )! s 2 j + 1 w 2 j = l ≥ 0 ∑ l ! ( m + l )! m s m + 2 l w m − 1 + 2 l = w m I m ( 2 s w ) . For ∣ w ∣ ≤ R |w|\leq R ∣ w ∣ ≤ R ∣ s ∣ ≤ S |s|\leq S ∣ s ∣ ≤ S R , S ≥ 1 R,S\geq1 R , S ≥ 1
∣ m w I m ( 2 s w ) ∣ ≤ m S m R m − 1 m ! e S 2 R 2 . \left|\frac{m}{w}\mathbb{I}_m(2sw)\right|\leq\frac{mS^mR^{m-1}}{m!}e^{S^2R^2}. w m I m ( 2 s w ) ≤ m ! m S m R m − 1 e S 2 R 2 . Since β m + 1 = O ( m ) \beta_{m+1}=O(m) β m + 1 = O ( m ) (71) converges normally on compact subsets of C 2 × C ∗ \mathbb{C}^2\times\mathbb{C}^* C 2 × C ∗ E \mathcal{E} E ( w , s ) (w,s) ( w , s ) ξ ≠ 0 \xi\ne0 ξ = 0 α i j ≥ 0 \alpha_{ij} \ge0 α ij ≥ 0 i ≤ j i \le j i ≤ j β k ≥ 0 \beta_k \ge0 β k ≥ 0 s = 1 s = 1 s = 1 w > 0 w > 0 w > 0 i = 0 i = 0 i = 0 j = 5 j = 5 j = 5 α 0 , 5 = β 12 = 1 \alpha_{0,5} = \beta_{12} = 1 α 0 , 5 = β 12 = 1 (72) . □ \square □
Strata of small dimension
Recall from Proposition 3.9 that the full light Hodge series has a pole of order 3 g − 3 3g - 3 3 g − 3 3 g − 4 3g - 4 3 g − 4 3 g − 5 3g - 5 3 g − 5 ξ 22 \xi^{22} ξ 22
Lemma 6.2. Let Γ \Gamma Γ g ≥ 2 g \ge2 g ≥ 2 σ ∈ Aut Γ \sigma\in\operatorname{Aut}\Gamma σ ∈ Aut Γ ( u , v ) = ( − s ξ , − s ξ − 1 ) (u,v) = (-s\xi,-s\xi^{-1}) ( u , v ) = ( − s ξ , − s ξ − 1 ) t = s 2 t = s^2 t = s 2 A 1 , d ( w ; s , ξ ) A_{1,d}(w;s,\xi) A 1 , d ( w ; s , ξ ) E ( w ; s , ξ ) \mathcal{E}(w;s,\xi) E ( w ; s , ξ ) C ( w , t ) C(w,t) C ( w , t ) 1 / ∣ Aut Γ ∣ 1/|\operatorname{Aut}\Gamma| 1/∣ Aut Γ∣
(1) The term of ( Γ , σ ) (\Gamma,\sigma) ( Γ , σ ) (27) is an entire function divided by ( 1 − W ) p ( σ ) (1-W)^{p(\sigma)} ( 1 − W ) p ( σ )
p ( σ ) ≤ ∣ E ( Γ ) ∣ = 3 g − 3 − dim M Γ , p(\sigma) \le|E(\Gamma)| = 3g - 3 - \dim\mathcal{M}_{\Gamma}, p ( σ ) ≤ ∣ E ( Γ ) ∣ = 3 g − 3 − dim M Γ , with equality only if σ = 1 \sigma= 1 σ = 1
(2) If every vertex of Γ \Gamma Γ ( u , v ) (u,v) ( u , v ) u v uv uv (66) , it is independent of ξ \xi ξ
(3) If Γ \Gamma Γ dim M Γ = 1 \dim\mathcal{M}_{\Gamma} = 1 dim M Γ = 1 Γ \Gamma Γ 3 g − 4 3g - 4 3 g − 4 2 g − 3 2g - 3 2 g − 3
Table 1.
Moreover, for every σ ∈ Aut Γ \sigma\in\operatorname{Aut}\Gamma σ ∈ Aut Γ (66) has the form A 1 , 1 ( w ; s , ξ ) F σ ( w ; s ) A_{1,1}(w;s,\xi)F_{\sigma}(w;s) A 1 , 1 ( w ; s , ξ ) F σ ( w ; s ) F σ F_{\sigma} F σ ξ \xi ξ ξ \xi ξ
(4) If Γ \Gamma Γ dim M Γ = 2 \dim\mathcal{M}_{\Gamma} = 2 dim M Γ = 2 Γ \Gamma Γ 3 g − 5 3g - 5 3 g − 5 2 g − 4 2g - 4 2 g − 4
e ( 2 g − 4 ) ( t − 2 ) w C 3 g − 5 e^{(2g-4)(t-2)w}C^{3g-5} e ( 2 g − 4 ) ( t − 2 ) w C 3 g − 5 times the factor in the second column.
Table 2.
Here the vertex types are the pairs ( g v , d v ) (g_v,d_v) ( g v , d v ) ξ 2 k \xi^{2k} ξ 2 k k ≠ 0 k \ne0 k = 0 2 g − 4 2g - 4 2 g − 4 2 g − 5 2g - 5 2 g − 5
Proof. Assertion (1) follows from Lemmas 3.6 and 3.8 and (25) . If p ( σ ) = ∣ E ( Γ ) ∣ p(\sigma) = |E(\Gamma)| p ( σ ) = ∣ E ( Γ ) ∣ σ \sigma σ σ = 1 \sigma= 1 σ = 1
For (2), stratifying the configurations of markings on a rational vertex by their coincidence partition expresses the vertex spaces in terms of products of moduli spaces M 0 , k \mathcal{M}_{0,k} M 0 , k E σ E_{\sigma} E σ u v uv uv u v uv uv
dim M Γ = ∑ v ( 3 g v − 3 + d v ) . \dim\mathcal{M}_{\Gamma}=\sum_v(3g_v-3+d_v). dim M Γ = v ∑ ( 3 g v − 3 + d v ) . A rational vertex contributes d v − 3 ≥ 0 d_v-3\ge0 d v − 3 ≥ 0 d v ≥ 1 d_v\ge1 d v ≥ 1 (25) , and the rational vertex counts follow by summing the valencies.
For the identity contributions, the vertex series multiply and each edge contributes C C C e ( t − 2 ) w e^{(t-2)w} e ( t − 2 ) w 0 , 1 , ∞ 0,1,\infty 0 , 1 , ∞ r r r M 0 , 4 × ( P 1 ) r \mathcal{M}_{0,4}\times(\mathbb{P}^{1})^r M 0 , 4 × ( P 1 ) r ( t − 2 ) ( t − 3 ) r (t-2)(t-3)^r ( t − 2 ) ( t − 3 ) r
( t − 2 ) e ( t − 3 ) w . (t-2)e^{(t-3)w}. ( t − 2 ) e ( t − 3 ) w . Multiplying these rational factors by the elliptic factors A 1 , 1 A_{1,1} A 1 , 1 A 1 , 2 A_{1,2} A 1 , 2 2 g − 5 2g-5 2 g − 5
e ( 2 g − 5 ) ( t − 2 ) w ( t − 2 ) e ( t − 3 ) w = e ( 2 g − 4 ) ( t − 2 ) w ( t − 2 ) e − w . e^{(2g-5)(t-2)w}(t-2)e^{(t-3)w}=e^{(2g-4)(t-2)w}(t-2)e^{-w}. e ( 2 g − 5 ) ( t − 2 ) w ( t − 2 ) e ( t − 3 ) w = e ( 2 g − 4 ) ( t − 2 ) w ( t − 2 ) e − w . The tables omit only the graph weight from (27) . The cohomology defining A 1 , d A_{1,d} A 1 , d
By Lemma 6.1 , A 1 , 1 o d d = e t w E A_{1,1}^{\mathrm{odd}}=e^{tw}\mathcal{E} A 1 , 1 odd = e tw E
( 2 g − 3 ) ( t − 2 ) + t = 2 ( ( g − 1 ) ( t − 2 ) + 1 ) , (2g-3)(t-2)+t=2((g-1)(t-2)+1), ( 2 g − 3 ) ( t − 2 ) + t = 2 (( g − 1 ) ( t − 2 ) + 1 ) , this gives the odd contribution in (3). By (70) , the even parts of A 1 , 1 A_{1,1} A 1 , 1 A 1 , 2 A_{1,2} A 1 , 2 ξ \xi ξ ( A 1 , 1 e v e n ) 2 (A_{1,1}^{\mathrm{even}})^2 ( A 1 , 1 even ) 2 ξ \xi ξ A 1 , 1 e v e n A 1 , 1 o d d A_{1,1}^{\mathrm{even}}A_{1,1}^{\mathrm{odd}} A 1 , 1 even A 1 , 1 odd ξ \xi ξ ( A 1 , 1 o d d ) 2 (A_{1,1}^{\mathrm{odd}})^2 ( A 1 , 1 odd ) 2
Finally, every automorphism fixes the unique elliptic leaf and its attaching flag, so A 1 , 1 A_{1,1} A 1 , 1 ξ \xi ξ Lemma 3.6 . This proves the assertion in (3) for arbitrary σ \sigma σ ξ \xi ξ
Lemma 6.3. The sum of 1 / ∣ Aut Γ ∣ 1/|\operatorname{Aut}\Gamma| 1/∣ Aut Γ∣ g g g
ℓ 2 = 1 2 , ℓ g = ( 3 g − 6 ) c g − 1 ( g ≥ 3 ) , (73) \ell_2=\frac{1}{2},\qquad\ell_g=(3g-6)c_{g-1}\quad(g\ge3), \tag*{(73)} ℓ 2 = 2 1 , ℓ g = ( 3 g − 6 ) c g − 1 ( g ≥ 3 ) , (73) where c g c_g c g (31) .
Proof. For g = 2 g=2 g = 2 ∣ Aut Γ ∣ = 2 |\operatorname{Aut}\Gamma|=2 ∣ Aut Γ∣ = 2 g ≥ 3 g\ge3 g ≥ 3 g − 1 g-1 g − 1 g − 1 g-1 g − 1 ℓ g = ∑ Γ ∈ G g − 1 ∣ E ( Γ ) ∣ / ∣ Aut Γ ∣ = ( 3 g − 6 ) c g − 1 \ell_g=\sum_{\Gamma\in\mathcal{G}_{g-1}}|E(\Gamma)|/|\operatorname{Aut}\Gamma|=(3g-6)c_{g-1} ℓ g = ∑ Γ ∈ G g − 1 ∣ E ( Γ ) ∣/∣ Aut Γ∣ = ( 3 g − 6 ) c g − 1
Proposition 6.4. With t = s 2 t=s^2 t = s 2 K g o d d \mathcal{K}^{\mathrm{odd}}_g K g odd K g e v e n \mathcal{K}^{\mathrm{even}}_g K g even w w w ( s , ξ ) ∈ C × C ∗ (s,\xi)\in\mathbb{C}\times\mathbb{C}^{*} ( s , ξ ) ∈ C × C ∗
L ~ g o d d ( w ; s , ξ ) = ℓ g e 2 ( ( g − 1 ) ( t − 2 ) + 1 ) w E ( w ; s , ξ ) ( 1 − W ( w , t ) ) 3 g − 4 + K g o d d ( w ; s , ξ ) ( 1 − W ( w , t ) ) 3 g − 5 (74) \widetilde{\mathcal{L}}^{\mathrm{odd}}_g(w;s,\xi)=\frac{\ell_g e^{2((g-1)(t-2)+1)w}\mathcal{E}(w;s,\xi)}{(1-W(w,t))^{3g-4}}+\frac{\mathcal{K}^{\mathrm{odd}}_g(w;s,\xi)}{(1-W(w,t))^{3g-5}} \tag*{(74)} L g odd ( w ; s , ξ ) = ( 1 − W ( w , t ) ) 3 g − 4 ℓ g e 2 (( g − 1 ) ( t − 2 ) + 1 ) w E ( w ; s , ξ ) + ( 1 − W ( w , t ) ) 3 g − 5 K g odd ( w ; s , ξ ) (74) and
L ~ g e v e n ( w ; s , ξ ) − [ ξ 0 ] L ~ g e v e n ( w ; s , ξ ) = K g e v e n ( w ; s , ξ ) ( 1 − W ( w , t ) ) 3 g − 5 . (75) \widetilde{\mathcal{L}}^{\mathrm{even}}_g(w;s,\xi)-[\xi^0]\widetilde{\mathcal{L}}^{\mathrm{even}}_g(w;s,\xi)=\frac{\mathcal{K}^{\mathrm{even}}_g(w;s,\xi)}{(1-W(w,t))^{3g-5}}. \tag*{(75)} L g even ( w ; s , ξ ) − [ ξ 0 ] L g even ( w ; s , ξ ) = ( 1 − W ( w , t ) ) 3 g − 5 K g even ( w ; s , ξ ) . (75) Moreover, there are a g > 0 a_g > 0 a g > 0 N g \mathcal{N}_g N g
[ ξ 22 ] L ~ g ( w ; 1 , ξ ) = a g ( 1 − w ) 3 g − 5 + N g ( w ) ( 1 − w ) 3 g − 6 . (76) [\xi^{22}]\widetilde{\mathcal{L}}_g(w;1,\xi)=\frac{a_g}{(1-w)^{3g-5}}+\frac{\mathcal{N}_g(w)}{(1-w)^{3g-6}}. \tag*{(76)} [ ξ 22 ] L g ( w ; 1 , ξ ) = ( 1 − w ) 3 g − 5 a g + ( 1 − w ) 3 g − 6 N g ( w ) . (76) Proof. By Lemma 6.2 (2), rational cores contribute neither to the odd part nor to the off-diagonal part. Among the cores with a vertex of positive genus, Lemma 6.2 (1) and (3) shows that only the identity terms of elliptic-leaf graphs can contribute at pole order 3 g − 4 3g-4 3 g − 4 3 g − 5 3g-5 3 g − 5 Lemma 6.2 (3), with the graph weights computed in Lemma 6.3 , gives the leading term of (6.9). Putting the remaining odd terms over the common denominator ( 1 − W ) 3 g − 5 (1-W)^{3g-5} ( 1 − W ) 3 g − 5
By Lemma 6.2 (3), every term of an elliptic-leaf graph, including the nonidentity terms, has an even part independent of ξ \xi ξ 3 g − 5 3g-5 3 g − 5 Lemma 6.2 (1). Putting these terms over the common denominator ( 1 − W ) 3 g − 5 (1-W)^{3g-5} ( 1 − W ) 3 g − 5
For (6.11), take s = 1 s=1 s = 1 t = 1 t=1 t = 1 C = ( 1 − w ) − 1 C=(1-w)^{-1} C = ( 1 − w ) − 1 ξ 22 \xi^{22} ξ 22 Lemma 6.2 (2) and (3). Terms from strata of dimension at least three, and nonidentity terms from strata of dimension two, have pole order at most 3 g − 6 3g-6 3 g − 6 Lemma 6.2 (1). The table in Lemma 6.2 (4) therefore shows that only double-leaf identity terms remain at order 3 g − 5 3g-5 3 g − 5 w = 1 w=1 w = 1
a g = e − ( 2 g − 4 ) [ ξ 22 ] ( A 1 , 1 o d d ( 1 ; 1 , ξ ) ) 2 ∑ Γ d o u b l e - l e a f 1 ∣ Aut Γ ∣ . a_g=e^{-(2g-4)}[\xi^{22}]\left(A_{1,1}^{\mathrm{odd}}(1;1,\xi)\right)^2\sum_{\Gamma\ \mathrm{double\text{-}leaf}}\frac{1}{|\operatorname{Aut}\Gamma|}. a g = e − ( 2 g − 4 ) [ ξ 22 ] ( A 1 , 1 odd ( 1 ; 1 , ξ ) ) 2 Γ double - leaf ∑ ∣ Aut Γ∣ 1 . where the sum runs over the isomorphism classes of double-leaf graphs of genus g g g
Double-leaf graphs exist for every g ≥ 2 g\ge2 g ≥ 2 g = 2 g=2 g = 2 a g a_g a g a g > 0 a_g>0 a g > 0
Expanding the double-leaf numerators at w = 1 w=1 w = 1 ( 1 − w ) 3 g − 6 (1-w)^{3g-6} ( 1 − w ) 3 g − 6 ∣ ξ ∣ = 1 |\xi|=1 ∣ ξ ∣ = 1 w = 1 w=1 w = 1 1 − w 1-w 1 − w N g \mathcal{N}_g N g
The Hodge difference among off-diagonal classes
Let
Z Q = ∑ m ≥ 3 , m o d d β m + 1 m Im ( 2 ) , q m = q − m = β m + 1 m Im ( 2 ) 2 Z Q ( m ≥ 3 o d d ) , (77) Z_Q=\sum_{\substack{m\ge3,\ m\ \mathrm{odd}}}\beta_{m+1}m\,\operatorname{Im}(2),\qquad q_m=q_{-m}=\frac{\beta_{m+1}m\,\operatorname{Im}(2)}{2Z_Q}\quad(m\ge3\ \mathrm{odd}), \tag*{(77)} Z Q = m ≥ 3 , m odd ∑ β m + 1 m Im ( 2 ) , q m = q − m = 2 Z Q β m + 1 m Im ( 2 ) ( m ≥ 3 odd ) , (77) and q d = 0 q_d=0 q d = 0 d d d Z Q > 0 Z_Q>0 Z Q > 0 β 12 = 1 \beta_{12}=1 β 12 = 1 ( q d ) d ∈ Z (q_d)_{d\in\mathbb{Z}} ( q d ) d ∈ Z Q Q Q Z \mathbb{Z} Z
Q ^ ( ξ ) = ∑ d ∈ Z q d ξ d = E ( 1 ; 1 , ξ ) 2 Z Q . \widehat{Q}(\xi)=\sum_{d\in\mathbb{Z}}q_d\xi^d=\frac{\mathcal{E}(1;1,\xi)}{2Z_Q}. Q ( ξ ) = d ∈ Z ∑ q d ξ d = 2 Z Q E ( 1 ; 1 , ξ ) . Finally, let
γ g H = 2 e 2 ℓ g Z Q c g 2 ( e − 2 ) e Γ ( ( 3 g − 3 ) / 2 ) Γ ( ( 3 g − 4 ) / 2 ) , γ g L = 2 e 2 ℓ g Z Q c g ( 3 g − 4 ) . (78) \gamma_g^H=\frac{2e^2\ell_gZ_Q}{c_g}\sqrt{\frac{2(e-2)}{e}\frac{\Gamma((3g-3)/2)}{\Gamma((3g-4)/2)}},\qquad
\gamma_g^L=\frac{2e^2\ell_gZ_Q}{c_g}(3g-4). \tag*{(78)} γ g H = c g 2 e 2 ℓ g Z Q e 2 ( e − 2 ) Γ (( 3 g − 4 ) /2 ) Γ (( 3 g − 3 ) /2 ) , γ g L = c g 2 e 2 ℓ g Z Q ( 3 g − 4 ) . (78) Theorem 6.5. Fix g ≥ 2 g\ge2 g ≥ 2
(1) The off-diagonal masses satisfy
P ( D g , n H ≠ 0 ) = γ g H n − 1 / 2 + O ( n − 1 ) , P ( D g , n L ≠ 0 ) = γ g L n − 1 + O ( n − 2 ) . (79) \mathbb{P}(D_{g,n}^H\ne0)=\gamma_g^H n^{-1/2}+O(n^{-1}),\qquad
\mathbb{P}(D_{g,n}^L\ne0)=\gamma_g^L n^{-1}+O(n^{-2}). \tag*{(79)} P ( D g , n H = 0 ) = γ g H n − 1/2 + O ( n − 1 ) , P ( D g , n L = 0 ) = γ g L n − 1 + O ( n − 2 ) . (79) The same formulas hold for the probability that p + q p+q p + q
P ( D g , n H ≠ 0 , p + q e v e n ) = O ( n − 1 ) , P ( D g , n L ≠ 0 , p + q e v e n ) = O ( n − 2 ) . (80) \mathbb{P}(D_{g,n}^H\ne0,\ p+q\ \mathrm{even})=O(n^{-1}),\qquad
\mathbb{P}(D_{g,n}^L\ne0,\ p+q\ \mathrm{even})=O(n^{-2}). \tag*{(80)} P ( D g , n H = 0 , p + q even ) = O ( n − 1 ) , P ( D g , n L = 0 , p + q even ) = O ( n − 2 ) . (80) (2) For J ∈ { H , L } J \in\{H,L\} J ∈ { H , L } D g , n J D^J_{g,n} D g , n J D g , n J ≠ 0 D^J_{g,n} \ne0 D g , n J = 0 Q Q Q r > 1 r > 1 r > 1
∑ d ≠ 0 r ∣ d ∣ ∣ P ( D g , n J = d ∣ D g , n J ≠ 0 ) − q d ∣ = { O r ( n − 1 / 2 ) , J = H , O r ( n − 1 ) , J = L . (81) \sum_{d \ne0} r^{|d|}\left|\mathbb{P}(D^J_{g,n}=d \mid D^J_{g,n}\ne0)-q_d\right|=
\begin{cases}
O_r(n^{-1/2}), & J=H,\\
O_r(n^{-1}), & J=L.
\end{cases} \tag*{(81)} d = 0 ∑ r ∣ d ∣ P ( D g , n J = d ∣ D g , n J = 0 ) − q d = { O r ( n − 1/2 ) , O r ( n − 1 ) , J = H , J = L . (81) Moreover, for all sufficiently large n n n k ≥ 1 k \ge1 k ≥ 1
P ( ∣ D g , n H ∣ ≥ k ) ≤ C r n − 1 / 2 r − k , P ( ∣ D g , n L ∣ ≥ k ) ≤ C r n − 1 r − k . (82) \mathbb{P}(|D^H_{g,n}|\ge k)\le C_r n^{-1/2}r^{-k},\qquad
\mathbb{P}(|D^L_{g,n}|\ge k)\le C_r n^{-1}r^{-k}. \tag*{(82)} P ( ∣ D g , n H ∣ ≥ k ) ≤ C r n − 1/2 r − k , P ( ∣ D g , n L ∣ ≥ k ) ≤ C r n − 1 r − k . (82) (3) For every x ∈ R x \in\mathbb{R} x ∈ R d ∈ Z d \in\mathbb{Z} d ∈ Z
P ( p + q − d n 4 ν n ≤ x , D g , n H = d | D g , n H ≠ 0 ) ⟶ Φ ( x ) q d , (83) \mathbb{P}\left(\frac{p+q-d_n}{\sqrt{4\nu n}}\le x,\ D^H_{g,n}=d\ \middle|\ D^H_{g,n}\ne0\right)\longrightarrow\Phi(x)q_d, \tag*{(83)} P ( 4 ν n p + q − d n ≤ x , D g , n H = d D g , n H = 0 ) ⟶ Φ ( x ) q d , (83) where Φ \Phi Φ 4 ν n \sqrt{4\nu n} 4 ν n n / 3 \sqrt{n/3} n /3 D g , n H D^H_{g,n} D g , n H D g , n L D^L_{g,n} D g , n L
Proof. Coefficient asymptotics at s = 1 s=1 s = 1 s = 1 s=1 s = 1 ℓ g e − ( 2 g − 4 ) w E ( w ; 1 , ξ ) \ell_g e^{-(2g-4)w}\mathcal{E}(w;1,\xi) ℓ g e − ( 2 g − 4 ) w E ( w ; 1 , ξ ) ℓ g e − ( 2 g − 4 ) E ( 1 ; 1 , ξ ) \ell_g e^{-(2g-4)}\mathcal{E}(1;1,\xi) ℓ g e − ( 2 g − 4 ) E ( 1 ; 1 , ξ ) w = 1 w=1 w = 1 ( 1 − w ) 3 g − 5 (1-w)^{3g-5} ( 1 − w ) 3 g − 5 [10 Theorem VI.4] , together with the uniformity lemma [10 Lemma IX.2] , gives, uniformly on each annulus r − 1 ≤ ∣ ξ ∣ ≤ r r^{-1}\le|\xi|\le r r − 1 ≤ ∣ ξ ∣ ≤ r
[ w n ] L g o d d ( w ; 1 , ξ ) = ℓ g e − ( 2 g − 4 ) Γ ( 3 g − 4 ) E ( 1 ; 1 , ξ ) n 3 g − 5 + O r ( n 3 g − 6 ) . (84) [w^n]\mathcal{L}^{\mathrm{odd}}_g(w;1,\xi)
=\frac{\ell_g e^{-(2g-4)}}{\Gamma(3g-4)}\mathcal{E}(1;1,\xi)n^{3g-5}+O_r(n^{3g-6}). \tag*{(84)} [ w n ] L g odd ( w ; 1 , ξ ) = Γ ( 3 g − 4 ) ℓ g e − ( 2 g − 4 ) E ( 1 ; 1 , ξ ) n 3 g − 5 + O r ( n 3 g − 6 ) . (84) For the heavy series, Lemma 3.2 gives
1 − T ( z , 1 ) = 2 ( e − 2 ) e ( 1 − z e − 2 ) 1 / 2 + O ( 1 − z e − 2 ) , 1-T(z,1)=\sqrt{\frac{2(e-2)}{e}}\left(1-\frac{z}{e-2}\right)^{1/2}+O\left(1-\frac{z}{e-2}\right), 1 − T ( z , 1 ) = e 2 ( e − 2 ) ( 1 − e − 2 z ) 1/2 + O ( 1 − e − 2 z ) , and by (4.5) the function 1 − T ( z , 1 ) 1-T(z,1) 1 − T ( z , 1 ) Δ \Delta Δ T ( z , 1 ) T(z,1) T ( z , 1 ) [10 Theorem VI.4] , with uniformity as in [10 Lemma IX.2] , gives
[ z n ] H g o d d ( z ; 1 , ξ ) = ( e − 2 ) − n ( ℓ g e − ( 2 g − 4 ) Γ ( ( 3 g − 4 ) / 2 ) ( 2 ( e − 2 ) e ) − ( 3 g − 4 ) / 2 E ( 1 ; 1 , ξ ) n ( 3 g − 6 ) / 2 + O r ( n ( 3 g − 7 ) / 2 ) ) . (85) [z^n]\mathcal{H}^{\mathrm{odd}}_g(z;1,\xi)
=(e-2)^{-n}\left(\frac{\ell_g e^{-(2g-4)}}{\Gamma((3g-4)/2)}
\left(\frac{2(e-2)}{e}\right)^{-(3g-4)/2}
\mathcal{E}(1;1,\xi)n^{(3g-6)/2}
+O_r\left(n^{(3g-7)/2}\right)\right). \tag*{(85)} [ z n ] H g odd ( z ; 1 , ξ ) = ( e − 2 ) − n ( Γ (( 3 g − 4 ) /2 ) ℓ g e − ( 2 g − 4 ) ( e 2 ( e − 2 ) ) − ( 3 g − 4 ) /2 E ( 1 ; 1 , ξ ) n ( 3 g − 6 ) /2 + O r ( n ( 3 g − 7 ) /2 ) ) . (85) The numerators are bounded on compact sets, and the continuation domain of T ( z , 1 ) T(z,1) T ( z , 1 ) ξ \xi ξ E ( 1 ; 1 , ξ ) \mathcal{E}(1;1,\xi) E ( 1 ; 1 , ξ ) g = 2 g=2 g = 2 − 1 / 2 -1/2 − 1/2
Proof of (1). By Theorem 4.1 and (4.12), the total Betti numbers are A g ( 1 ) n ! n 3 g − 4 ( 1 + O ( n − 1 ) ) A_g(1)n!n^{3g-4}(1+O(n^{-1})) A g ( 1 ) n ! n 3 g − 4 ( 1 + O ( n − 1 )) C g ( 1 ) n ! n ( 3 g − 5 ) / 2 ( e − 2 ) − n ( 1 + O ( n − 1 / 2 ) ) C_g(1)n!n^{(3g-5)/2}(e-2)^{-n}(1+O(n^{-1/2})) C g ( 1 ) n ! n ( 3 g − 5 ) /2 ( e − 2 ) − n ( 1 + O ( n − 1/2 )) n ! n! n !
E [ ξ D g , n H 1 { p + q o d d } ] = γ g H n − 1 / 2 Q ^ ( ξ ) + O r ( n − 1 ) , \mathbb{E}\left[\xi^{D^H_{g,n}}\mathbf{1}_{\{p+q\ \mathrm{odd}\}}\right]
=\gamma^H_g n^{-1/2}\widehat{Q}(\xi)+O_r(n^{-1}), E [ ξ D g , n H 1 { p + q odd } ] = γ g H n − 1/2 Q ( ξ ) + O r ( n − 1 ) , E [ ξ D g , n L 1 { p + q o d d } ] = γ g L n − 1 Q ^ ( ξ ) + O r ( n − 2 ) . (86) \mathbb{E}\left[\xi^{D^L_{g,n}}\mathbf{1}_{\{p+q\ \mathrm{odd}\}}\right]
=\gamma^L_g n^{-1}\widehat{Q}(\xi)+O_r(n^{-2}). \tag*{(86)} E [ ξ D g , n L 1 { p + q odd } ] = γ g L n − 1 Q ( ξ ) + O r ( n − 2 ) . (86) Applying the same transfer to (6.10) and dividing by the corresponding total Betti numbers shows that, for J ∈ { H , L } J\in\{H,L\} J ∈ { H , L } E [ ξ D g , n J 1 { p + q e v e n , p ≠ q } ] \mathbb{E}\left[\xi^{D^J_{g,n}}\mathbf{1}_{\{p+q\ \mathrm{even},\ p\ne q\}}\right] E [ ξ D g , n J 1 { p + q even , p = q } ] O r ( n − 1 ) O_r(n^{-1}) O r ( n − 1 ) O r ( n − 2 ) O_r(n^{-2}) O r ( n − 2 ) p + q p+q p + q p ≠ q p\ne q p = q 1 { p + q o d d } \mathbf{1}_{\{p+q\ \mathrm{odd}\}} 1 { p + q odd } 1 { D g , n J ≠ 0 } \mathbf{1}_{\{D^J_{g,n}\ne0\}} 1 { D g , n J = 0 } Q ^ ( 1 ) = 1 \widehat{Q}(1)=1 Q ( 1 ) = 1 ξ = 1 \xi=1 ξ = 1 n n n
Proof of (2). Dividing the off-diagonal Laurent generating functions obtained in (1) by their values at ξ = 1 \xi=1 ξ = 1 Q ^ \widehat{Q} Q r − 1 ≤ ∣ ξ ∣ ≤ r r^{-1}\leq|\xi|\leq r r − 1 ≤ ∣ ξ ∣ ≤ r O r ( n − 1 / 2 ) O_r(n^{-1/2}) O r ( n − 1/2 ) O r ( n − 1 ) O_r(n^{-1}) O r ( n − 1 ) d ≠ 0 d\neq0 d = 0 ξ d \xi^d ξ d P ( D g , n J = d ∣ D g , n J ≠ 0 ) − q d \mathbb{P}(D^J_{g,n}=d\mid D^J_{g,n}\neq0)-q_d P ( D g , n J = d ∣ D g , n J = 0 ) − q d r > 1 r>1 r > 1 R > r R>r R > r ∣ ξ ∣ = R |\xi|=R ∣ ξ ∣ = R ∣ ξ ∣ = R − 1 |\xi|=R^{-1} ∣ ξ ∣ = R − 1 C R n − 1 / 2 R − ∣ d ∣ C_Rn^{-1/2}R^{-|d|} C R n − 1/2 R − ∣ d ∣ C R n − 1 R − ∣ d ∣ C_Rn^{-1}R^{-|d|} C R n − 1 R − ∣ d ∣ r ∣ d ∣ r^{|d|} r ∣ d ∣ ∑ d ≠ 0 ( r / R ) ∣ d ∣ \sum_{d\neq0}(r/R)^{|d|} ∑ d = 0 ( r / R ) ∣ d ∣ ξ = r \xi=r ξ = r ξ = r − 1 \xi=r^{-1} ξ = r − 1
Proof of (3). Let s s s 1 1 1 t = s 2 t=s^2 t = s 2 w = τ ( t ) w=\tau(t) w = τ ( t ) ℓ g e 2 ( ( g − 1 ) ( t − 2 ) + 1 ) τ ( t ) E ( τ ( t ) ; s , ξ ) \ell_g e^{2((g-1)(t-2)+1)}\tau(t)\mathcal{E}(\tau(t);s,\xi) ℓ g e 2 (( g − 1 ) ( t − 2 ) + 1 ) τ ( t ) E ( τ ( t ) ; s , ξ ) t τ ( t ) ( 1 − w / τ ( t ) ) t\tau(t)(1-w/\tau(t)) t τ ( t ) ( 1 − w / τ ( t )) ( 2 ρ ( t ) ) 1 / 2 ( t − 2 ) τ ( t ) / 2 ( 1 − z / ρ ( t ) ) 1 / 2 (2\rho(t))^{1/2}(t-2)\tau(t)/2(1-z/\rho(t))^{1/2} ( 2 ρ ( t ) ) 1/2 ( t − 2 ) τ ( t ) /2 ( 1 − z / ρ ( t ) ) 1/2 s s s ξ \xi ξ n n n ( s , ξ ) = ( 1 , 1 ) (s,\xi)=(1,1) ( s , ξ ) = ( 1 , 1 )
E [ s p + q ξ D g , n H ∣ D g , n H ≠ 0 ] = ( e − 2 ρ ( s 2 ) ) n ( a g H ( s , ξ ) + O r ( n − 1 / 2 ) ) , E [ s p + q ξ D g , n L ∣ D g , n L ≠ 0 ] = τ ( s 2 ) − n ( a g L ( s , ξ ) + O r ( n − 1 ) ) , (87) \begin{aligned}
\mathbb{E}\left[s^{p+q}\xi^{D^H_{g,n}}\mid D^H_{g,n}\neq0\right]&=\left(\frac{e-2}{\rho(s^2)}\right)^n\left(a^H_g(s,\xi)+O_r(n^{-1/2})\right),\\
\mathbb{E}\left[s^{p+q}\xi^{D^L_{g,n}}\mid D^L_{g,n}\neq0\right]&=\tau(s^2)^{-n}\left(a^L_g(s,\xi)+O_r(n^{-1})\right), \tag*{(87)}
\end{aligned} E [ s p + q ξ D g , n H ∣ D g , n H = 0 ] E [ s p + q ξ D g , n L ∣ D g , n L = 0 ] = ( ρ ( s 2 ) e − 2 ) n ( a g H ( s , ξ ) + O r ( n − 1/2 ) ) , = τ ( s 2 ) − n ( a g L ( s , ξ ) + O r ( n − 1 ) ) , (87) where a g H a^H_g a g H a g L a^L_g a g L s s s 1 1 1 ξ ≠ 0 \xi\neq0 ξ = 0 a g J ( 1 , ξ ) = Q ^ ( ξ ) a^J_g(1,\xi)=\widehat{Q}(\xi) a g J ( 1 , ξ ) = Q ( ξ ) J ∈ { H , L } J\in\{H,L\} J ∈ { H , L }
log e − 2 ρ ( e 2 h ) = h + 2 ν h 2 + O ( h 3 ) , − log τ ( e 2 h ) = h + h 2 6 + O ( h 3 ) . \log\frac{e-2}{\rho(e^{2h})}=h+2\nu h^2+O(h^3),\qquad-\log\tau(e^{2h})=h+\frac{h^2}{6}+O(h^3). log ρ ( e 2 h ) e − 2 = h + 2 ν h 2 + O ( h 3 ) , − log τ ( e 2 h ) = h + 6 h 2 + O ( h 3 ) . For fixed real a , b a,b a , b ξ = e i b \xi=e^{ib} ξ = e ib s = exp ( i a / 4 ν n ) s=\exp(ia/\sqrt{4\nu n}) s = exp ( ia / 4 ν n ) s = exp ( i a / n / 3 ) s=\exp(ia/\sqrt{n/3}) s = exp ( ia / n /3 ) s − d n s^{-d_n} s − d n d n = n + O ( 1 ) d_n=n+O(1) d n = n + O ( 1 ) e − a 2 / 2 Q ^ ( e i b ) e^{-a^2/2}\widehat{Q}(e^{ib}) e − a 2 /2 Q ( e ib ) Q Q Q ( − ∞ , x ] × ( d − 1 2 , d + 1 2 ) (-\infty,x]\times(d-\frac{1}{2},d+\frac{1}{2}) ( − ∞ , x ] × ( d − 2 1 , d + 2 1 ) x ∈ R x\in\mathbb{R} x ∈ R d ∈ Z d\in\mathbb{Z} d ∈ Z
Theorem 6.5 describes both the rarity and the Hodge distribution of off-diagonal classes. Their total mass tends to zero at rate n − 1 / 2 n^{-1/2} n − 1/2 M ‾ g , n \overline{\mathcal{M}}_{g,n} M g , n n − 1 n^{-1} n − 1 M ‾ g , 0 ∣ n \overline{\mathcal{M}}_{g,0|n} M g , 0∣ n Q Q Q p − q = d p-q=d p − q = d q d > 0 q_d>0 q d > 0 Q Q Q p p p q q q Q Q Q
Remark 6.6 (Interpretation of Q Q Q Sk ( j ) = e − 2 I ∣ j ∣ ( 2 ) \operatorname{Sk}(j)=e^{-2}\mathbb{I}_{|j|}(2) Sk ( j ) = e − 2 I ∣ j ∣ ( 2 ) 1 1 1 I m − 1 ( 2 ) − I m + 1 ( 2 ) = m I m ( 2 ) \mathbb{I}_{m-1}(2)-\mathbb{I}_{m+1}(2)=m\mathbb{I}_m(2) I m − 1 ( 2 ) − I m + 1 ( 2 ) = m I m ( 2 ) q m q_m q m
q m = q − m = e 2 β m + 1 2 Z Q ( Sk ( m − 1 ) − Sk ( m + 1 ) ) , m ≥ 3 odd . q_m=q_{-m}=\frac{e^2\beta_{m+1}}{2Z_Q}\left(\operatorname{Sk}(m-1)-\operatorname{Sk}(m+1)\right),\qquad m\geq3\text{ odd}. q m = q − m = 2 Z Q e 2 β m + 1 ( Sk ( m − 1 ) − Sk ( m + 1 ) ) , m ≥ 3 odd . The difference of Skellam probabilities in this formula has a representation-theoretic interpretation. Let V V V S L 2 \mathrm{SL}_2 SL 2 ch ( V ) ( ξ ) = ξ + ξ − 1 \operatorname{ch}(V)(\xi)=\xi+\xi^{-1} ch ( V ) ( ξ ) = ξ + ξ − 1
e − 2 ∑ k ≥ 0 ch ( V ⊗ k ) ( ξ ) k ! = exp ( ξ + ξ − 1 − 2 ) = ∑ j ∈ Z Sk ( j ) ξ j . e^{-2}\sum_{k\geq0}\frac{\operatorname{ch}(V^{\otimes k})(\xi)}{k!}=\exp(\xi+\xi^{-1}-2)=\sum_{j\in\mathbb{Z}}\operatorname{Sk}(j)\xi^j. e − 2 k ≥ 0 ∑ k ! ch ( V ⊗ k ) ( ξ ) = exp ( ξ + ξ − 1 − 2 ) = j ∈ Z ∑ Sk ( j ) ξ j . In an S L 2 \mathrm{SL}_2 SL 2 m + 1 m+1 m + 1 m − 1 m-1 m − 1 Sym m − 1 V \operatorname{Sym}^{m-1}V Sym m − 1 V Sk ( m − 1 ) − Sk ( m + 1 ) \operatorname{Sk}(m-1)-\operatorname{Sk}(m+1) Sk ( m − 1 ) − Sk ( m + 1 ) Sym m − 1 V \operatorname{Sym}^{m-1}V Sym m − 1 V
For odd m ≥ 3 m\geq3 m ≥ 3 β m + 1 \beta_{m+1} β m + 1 ± m \pm m ± m Sym m − 1 V \operatorname{Sym}^{m-1}V Sym m − 1 V M 1 , 1 \mathcal{M}_{1,1} M 1 , 1 q m = q − m q_m=q_{-m} q m = q − m
Since the first nonzero cusp-form dimension is β 12 = 1 \beta_{12}=1 β 12 = 1 q d = 0 q_d=0 q d = 0 ∣ d ∣ < 11 |d|<11 ∣ d ∣ < 11
q 11 + q − 11 = 11 I 11 ( 2 ) Z Q ≈ 0.99996. q_{11}+q_{-11}=\frac{11I_{11}(2)}{Z_Q}\approx0.99996. q 11 + q − 11 = Z Q 11 I 11 ( 2 ) ≈ 0.99996. Tautological classes
Let R H 2 k ( M ‾ g , n ) ⊂ H 2 k ( M ‾ g , n ) RH^{2k}(\overline{\mathcal{M}}_{g,n})\subset H^{2k}(\overline{\mathcal{M}}_{g,n}) R H 2 k ( M g , n ) ⊂ H 2 k ( M g , n ) R k ( M ‾ g , n ) R^k(\overline{\mathcal{M}}_{g,n}) R k ( M g , n )
H g , n t a u t ( t ) = ∑ k dim Q R H 2 k ( M ‾ g , n ) t k . H^{\mathrm{taut}}_{g,n}(t)=\sum_k\dim_{\mathbb{Q}}RH^{2k}(\overline{\mathcal{M}}_{g,n})t^k. H g , n taut ( t ) = k ∑ dim Q R H 2 k ( M g , n ) t k . Let X g , n t a u t X^{\mathrm{taut}}_{g,n} X g , n taut H g , n t a u t ( t ) / H g , n t a u t ( 1 ) H^{\mathrm{taut}}_{g,n}(t)/H^{\mathrm{taut}}_{g,n}(1) H g , n taut ( t ) / H g , n taut ( 1 ) H 1 , n t a u t = H 1 , n H^{\mathrm{taut}}_{1,n}=H_{1,n} H 1 , n taut = H 1 , n [23 ] . In this section we show that, for every g ≥ 2 g\geq2 g ≥ 2 1 − Θ ( n − 1 ) 1-\Theta(n^{-1}) 1 − Θ ( n − 1 )
Theorem 7.1. Fix g ≥ 2 g\geq2 g ≥ 2 c g ′ , C g ′ > 0 c'_g,C'_g>0 c g ′ , C g ′ > 0 n n n
c g ′ n ≤ 1 − H g , n t a u t ( 1 ) H g , n ( 1 ) ≤ C g ′ n . (88) \frac{c'_g}{n}\leq1-\frac{H^{\mathrm{taut}}_{g,n}(1)}{H_{g,n}(1)}\leq\frac{C'_g}{n}. \tag*{(88)} n c g ′ ≤ 1 − H g , n ( 1 ) H g , n taut ( 1 ) ≤ n C g ′ . (88) Consequently,
1 2 ∑ k ∣ P ( X g , n t a u t = k ) − P ( X g , n H = k ) ∣ ≤ 1 − H g , n t a u t ( 1 ) H g , n ( 1 ) = O ( n − 1 ) , (89) \frac{1}{2}\sum_k\left|\mathbb{P}(X^{\mathrm{taut}}_{g,n}=k)-\mathbb{P}(X^H_{g,n}=k)\right|\leq1-\frac{H^{\mathrm{taut}}_{g,n}(1)}{H_{g,n}(1)}=O(n^{-1}), \tag*{(89)} 2 1 k ∑ P ( X g , n taut = k ) − P ( X g , n H = k ) ≤ 1 − H g , n ( 1 ) H g , n taut ( 1 ) = O ( n − 1 ) , (89) and
sup x ∈ R ∣ P ( X g , n t a u t − d n / 2 v n ≤ x ) − Φ ( x ) ∣ = O ( n − 1 / 2 ) . (90) \sup_{x\in\mathbb{R}}\left|\mathbb{P}\left(\frac{X^{\mathrm{taut}}_{g,n}-d_n/2}{\sqrt{vn}}\leq x\right)-\Phi(x)\right|=O(n^{-1/2}). \tag*{(90)} x ∈ R sup P ( v n X g , n taut − d n /2 ≤ x ) − Φ ( x ) = O ( n − 1/2 ) . (90) For the upper bound, we observe that the even cohomology classes vanishing on the preimage U n U_n U n M ‾ g \overline{\mathcal{M}}_g M g
Lemma 7.2. Fix g ≥ 2 g\geq2 g ≥ 2 B ≤ 1 ⊂ M ‾ g B_{\leq1}\subset\overline{\mathcal{M}}_g B ≤ 1 ⊂ M g M Γ \mathcal{M}_{\Gamma} M Γ π n : M ‾ g , n → M ‾ g \pi_n:\overline{\mathcal{M}}_{g,n}\to\overline{\mathcal{M}}_g π n : M g , n → M g U n = π n − 1 ( M ‾ g ∖ B ≤ 1 ) U_n=\pi_n^{-1}(\overline{\mathcal{M}}_g\setminus B_{\leq1}) U n = π n − 1 ( M g ∖ B ≤ 1 ) k k k
ker ( H 2 k ( M ‾ g , n ; Q ) ⟶ H 2 k ( U n ; Q ) ) ⊂ R H 2 k ( M ‾ g , n ) . \ker\left(H^{2k}(\overline{\mathcal{M}}_{g,n};\mathbb{Q})\longrightarrow H^{2k}(U_n;\mathbb{Q})\right)\subset RH^{2k}(\overline{\mathcal{M}}_{g,n}). ker ( H 2 k ( M g , n ; Q ) ⟶ H 2 k ( U n ; Q ) ) ⊂ R H 2 k ( M g , n ) . Consequently, writing b c ( Y ) = ∑ j dim H c j ( Y ; Q ) b_c(Y)=\sum_j\dim H^j_c(Y;\mathbb{Q}) b c ( Y ) = ∑ j dim H c j ( Y ; Q )
H g , n ( 1 ) − H g , n t a u t ( 1 ) ≤ b c ( U n ) . (91) H_{g,n}(1)-H^{\mathrm{taut}}_{g,n}(1)\leq b_c(U_n). \tag*{(91)} H g , n ( 1 ) − H g , n taut ( 1 ) ≤ b c ( U n ) . (91) Proof. The union B ≤ 1 B_{\leq1} B ≤ 1 (25) , a stratum M Γ \mathcal{M}_{\Gamma} M Γ B ≤ 1 B_{\leq1} B ≤ 1 Z n = π n − 1 ( B ≤ 1 ) Z_n=\pi_n^{-1}(B_{\leq1}) Z n = π n − 1 ( B ≤ 1 )
∏ v M ‾ g v , n v ⟶ Z n , \prod_v \overline{\mathcal{M}}_{g_v,n_v}\longrightarrow Z_n, v ∏ M g v , n v ⟶ Z n , one for each marked stable graph whose unmarked stabilization lies in B ≤ 1 B_{\leq1} B ≤ 1 n v n_v n v v v v
Although Z n Z_n Z n f : Y → Z f:Y\to Z f : Y → Z Y Y Y
im ( H m ( Y ; Q ) ⟶ H m ( Z ; Q ) ) = W − m H m ( Z ; Q ) . (92) \operatorname{im}\left(H^m(Y;\mathbb{Q})\longrightarrow H^m(Z;\mathbb{Q})\right)=W_{-m}H^m(Z;\mathbb{Q}). \tag*{(92)} im ( H m ( Y ; Q ) ⟶ H m ( Z ; Q ) ) = W − m H m ( Z ; Q ) . (92) This is the dual form of Deligne’s result [7 Proposition 8.2.5] . Indeed, cohomological descent for the Čech nerve Y ∙ Y_\bullet Y ∙ f f f E 1 p , q = H q ( Y p ; Q ) ⇒ H p + q ( Z ; Q ) E_1^{p,q}=H^q(Y_p;\mathbb{Q})\Rightarrow H^{p+q}(Z;\mathbb{Q}) E 1 p , q = H q ( Y p ; Q ) ⇒ H p + q ( Z ; Q ) [7 Section 8] . Each Y p Y_p Y p H q ( Y p ) H^q(Y_p) H q ( Y p ) q q q m m m p ≥ 1 p\geq1 p ≥ 1 m − 1 m-1 m − 1 H m ( Y ) H^m(Y) H m ( Y ) m m m H m ( Z ) → H m ( Y ) H^m(Z)\to H^m(Y) H m ( Z ) → H m ( Y ) W m − 1 H m ( Z ) W_{m-1}H^m(Z) W m − 1 H m ( Z ) (92) . The same argument applies to proper Deligne–Mumford stacks with rational coefficients.
Let d = dim M ‾ g , n d=\dim\overline{\mathcal{M}}_{g,n} d = dim M g , n U n U_n U n 2 k 2k 2 k
H 2 d − 2 k ( Z n ; Q ) ( − d ) ⟶ H 2 k ( M ‾ g , n ; Q ) . H^{2d-2k}(Z_n;\mathbb{Q})(-d)\longrightarrow H^{2k}(\overline{\mathcal{M}}_{g,n};\mathbb{Q}). H 2 d − 2 k ( Z n ; Q ) ( − d ) ⟶ H 2 k ( M g , n ; Q ) . The target is pure of weight 2 k 2k 2 k (92) , applied to the disjoint union of the gluing sources, this image is spanned by Gysin pushforwards from the sources. These pushforwards shift degrees by even integers, so only the even cohomology of the sources is involved. That cohomology is tautological by Keel’s theorem in genus zero [18 ] , Petersen’s theorem in genus one [23 ] and the Künneth formula, because odd cohomology occurs in at most one factor. Since gluing pushforwards preserve tautological classes, the inclusion follows. The even part of the quotient by the tautological classes therefore has dimension at most ∑ k b 2 k ( U n ) ≤ b c ( U n ) \sum_k b_{2k}(U_n)\leq b_c(U_n) ∑ k b 2 k ( U n ) ≤ b c ( U n ) U n U_n U n
To bound b c ( U n ) b_c(U_n) b c ( U n ) [19 ] .
Lemma 7.3. Fix g ≥ 2 g\geq2 g ≥ 2 V ⊂ M ‾ g V\subset\overline{\mathcal{M}}_g V ⊂ M g r , s ≥ 0 r,s\geq0 r , s ≥ 0 M ‾ g , r ∣ s \overline{\mathcal{M}}_{g,r|s} M g , r ∣ s r r r 1 1 1 s s s ϵ \epsilon ϵ s ϵ < 1 s\epsilon<1 sϵ < 1
U r ∣ s = M ‾ g , r ∣ s × M ‾ g V , F r ( w ) = ∑ s ≥ 0 b c ( U r ∣ s ) w s s ! . U_{r|s}=\overline{\mathcal{M}}_{g,r|s}\times_{\overline{\mathcal{M}}_g}V,\qquad F_r(w)=\sum_{s\geq0}b_c(U_{r|s})\frac{w^s}{s!}. U r ∣ s = M g , r ∣ s × M g V , F r ( w ) = s ≥ 0 ∑ b c ( U r ∣ s ) s ! w s . Then, as formal power series,
∑ n ≥ 0 b c ( U n ∣ 0 ) z n n ! = F 0 ( T ( z , 1 ) ) . (93) \sum_{n\geq0}b_c(U_{n|0})\frac{z^n}{n!}=F_0(T(z,1)). \tag*{(93)} n ≥ 0 ∑ b c ( U n ∣0 ) n ! z n = F 0 ( T ( z , 1 )) . (93) Proof. By [19 Section 3] , for r ≥ 1 r\geq1 r ≥ 1 M ‾ g , r ∣ s → M ‾ g , r − 1 ∣ s + 1 \overline{\mathcal{M}}_{g,r|s}\to\overline{\mathcal{M}}_{g,r-1|s+1} M g , r ∣ s → M g , r − 1∣ s + 1 ( s j ) \binom{s}{j} ( j s ) M ‾ g , r ∣ s − j \overline{\mathcal{M}}_{g,r|s-j} M g , r ∣ s − j 2 ≤ j ≤ s 2\leq j\leq s 2 ≤ j ≤ s
b c ( U r ∣ s ) = b c ( U r − 1 ∣ s + 1 ) + ∑ j = 2 s ( s j ) ( j − 1 ) b c ( U r ∣ s − j ) , for r ≥ 1 b_c(U_{r|s})=b_c(U_{r-1|s+1})+\sum_{j=2}^{s}\binom{s}{j}(j-1)b_c(U_{r|s-j}),\qquad\text{for }r\geq1 b c ( U r ∣ s ) = b c ( U r − 1∣ s + 1 ) + j = 2 ∑ s ( j s ) ( j − 1 ) b c ( U r ∣ s − j ) , for r ≥ 1 that is,
F r ( w ) = F r − 1 ′ ( w ) + F r ( w ) ∑ j ≥ 2 ( j − 1 ) w j j ! , or F r ( w ) = F r − 1 ′ ( w ) R w ( w , 1 ) . F_r(w)=F'_{r-1}(w)+F_r(w)\sum_{j\ge2}(j-1)\frac{w^j}{j!},\qquad\text{or}\qquad F_r(w)=\frac{F'_{r-1}(w)}{R_w(w,1)}. F r ( w ) = F r − 1 ′ ( w ) + F r ( w ) j ≥ 2 ∑ ( j − 1 ) j ! w j , or F r ( w ) = R w ( w , 1 ) F r − 1 ′ ( w ) . since R w ( w , 1 ) = 1 − ∑ j ≥ 2 ( j − 1 ) w j / j ! R_w(w,1)=1-\sum_{j\ge2}(j-1)w^j/j! R w ( w , 1 ) = 1 − ∑ j ≥ 2 ( j − 1 ) w j / j ! (17) . Because R ( T ( z , 1 ) , 1 ) = z R(T(z,1),1)=z R ( T ( z , 1 ) , 1 ) = z F r ( T ( z , 1 ) ) = ∂ z F r − 1 ( T ( z , 1 ) ) F_r(T(z,1))=\partial_zF_{r-1}(T(z,1)) F r ( T ( z , 1 )) = ∂ z F r − 1 ( T ( z , 1 )) F r ( T ( z , 1 ) ) = ∂ z r F 0 ( T ( z , 1 ) ) F_r(T(z,1))=\partial_z^rF_0(T(z,1)) F r ( T ( z , 1 )) = ∂ z r F 0 ( T ( z , 1 )) T ( 0 , 1 ) = 0 T(0,1)=0 T ( 0 , 1 ) = 0 F r ( 0 ) = b c ( U r ∣ 0 ) F_r(0)=b_c(U_r|0) F r ( 0 ) = b c ( U r ∣0 ) z = 0 z=0 z = 0
∂ z r F 0 ( T ( z , 1 ) ) ∣ z = 0 = b c ( U r ∣ 0 ) ( r ≥ 0 ) . \left.\partial_z^rF_0(T(z,1))\right|_{z=0}=b_c(U_r|0)\qquad(r\ge0). ∂ z r F 0 ( T ( z , 1 )) ∣ z = 0 = b c ( U r ∣0 ) ( r ≥ 0 ) . Thus the formal Taylor expansion of F 0 ( T ( z , 1 ) ) F_0(T(z,1)) F 0 ( T ( z , 1 )) z = 0 z=0 z = 0
F 0 ( T ( z , 1 ) ) = ∑ r ≥ 0 b c ( U r ∣ 0 ) z r r ! , F_0(T(z,1))=\sum_{r\ge0}b_c(U_r|0)\frac{z^r}{r!}, F 0 ( T ( z , 1 )) = r ≥ 0 ∑ b c ( U r ∣0 ) r ! z r , which proves (7.6). □
Proof of Theorem 7.1. For the upper bound, we apply Lemma 7.3 with V = M ‾ g ∖ B ≤ 1 V=\overline{\mathcal{M}}_g\setminus B_{\le1} V = M g ∖ B ≤ 1 F 0 ( w ) F_0(w) F 0 ( w ) M Γ \mathcal{M}_\Gamma M Γ V V V (25) gives
3 g − 3 − ∣ E ( Γ ) ∣ = dim M Γ ≥ 2. 3g-3-\lvert E(\Gamma)\rvert=\dim\mathcal{M}_\Gamma\ge2. 3 g − 3 − ∣ E ( Γ )∣ = dim M Γ ≥ 2. Thus every such graph has at most 3 g − 5 3g-5 3 g − 5
Fix one of these graphs Γ \Gamma Γ Proposition 3.4 , we stratify the space of light curves with core graph Γ \Gamma Γ r r r k e k_e k e e e e
X Γ , r × ∏ e ∈ E ( Γ ) L M k e , r + ∑ e k e = n . X_{\Gamma,r}\times\prod_{e\in E(\Gamma)}\mathrm{LM}_{k_e},\qquad r+\sum_e k_e=n. X Γ , r × e ∈ E ( Γ ) ∏ LM k e , r + e ∑ k e = n . The exponential Betti bound (30) gives b c ( X Γ , r ) ≤ K Γ B Γ r b_c(X_{\Gamma,r})\le K_\Gamma B_\Gamma^r b c ( X Γ , r ) ≤ K Γ B Γ r r r r Lemma 3.6 at t = 1 t=1 t = 1 b c ( L M k ) = k ! b_c(\mathrm{LM}_k)=k! b c ( LM k ) = k ! k = 0 k=0 k = 0
∑ r ≥ 0 b c ( X Γ , r ) w r r ! ≤ K Γ e B Γ w , ∑ k ≥ 0 b c ( L M k ) w k k ! = 1 1 − w , \sum_{r\ge0}b_c(X_{\Gamma,r})\frac{w^r}{r!}\le K_\Gamma e^{B_\Gamma w},\qquad
\sum_{k\ge0}b_c(\mathrm{LM}_k)\frac{w^k}{k!}=\frac{1}{1-w}, r ≥ 0 ∑ b c ( X Γ , r ) r ! w r ≤ K Γ e B Γ w , k ≥ 0 ∑ b c ( LM k ) k ! w k = 1 − w 1 , where ≼ \preccurlyeq ≼ n ! / ( r ! ∏ e k e ! ) n!/(r!\prod_e k_e!) n ! / ( r ! ∏ e k e !) r r r k e k_e k e 1 / ( 1 − w ) 1/(1-w) 1/ ( 1 − w ) V V V K g , B g > 0 K_g,B_g>0 K g , B g > 0
F 0 ( w ) ≼ ∑ Γ K Γ e B Γ w ( 1 − w ) ∣ E ( Γ ) ∣ ≼ K g e B g w ( 1 − w ) 3 g − 5 . (94) F_0(w)\preccurlyeq\sum_\Gamma\frac{K_\Gamma e^{B_\Gamma w}}{(1-w)^{\lvert E(\Gamma)\rvert}}
\preccurlyeq\frac{K_g e^{B_gw}}{(1-w)^{3g-5}}. \tag*{(94)} F 0 ( w ) ≼ Γ ∑ ( 1 − w ) ∣ E ( Γ )∣ K Γ e B Γ w ≼ ( 1 − w ) 3 g − 5 K g e B g w . (94) The coefficients of T ( z , 1 ) T(z,1) T ( z , 1 ) 7.3 give
H g , n ( 1 ) − H g , n t a u t ( 1 ) ≤ K g n ! [ z n ] e B g T ( z , 1 ) ( 1 − T ( z , 1 ) ) 3 g − 5 . H_{g,n}(1)-H^{\mathrm{taut}}_{g,n}(1)\le K_gn![z^n]\frac{e^{B_gT(z,1)}}{(1-T(z,1))^{3g-5}}. H g , n ( 1 ) − H g , n taut ( 1 ) ≤ K g n ! [ z n ] ( 1 − T ( z , 1 ) ) 3 g − 5 e B g T ( z , 1 ) . By (21) at t = 1 t=1 t = 1 z → e − 2 z\to e-2 z → e − 2 Δ \Delta Δ
1 − T ( z , 1 ) = 2 ( e − 2 ) e ( 1 − z e − 2 ) 1 / 2 + O ( 1 − z e − 2 ) . (95) 1-T(z,1)=\sqrt{\frac{2(e-2)}{e}}\left(1-\frac{z}{e-2}\right)^{1/2}+O\left(1-\frac{z}{e-2}\right). \tag*{(95)} 1 − T ( z , 1 ) = e 2 ( e − 2 ) ( 1 − e − 2 z ) 1/2 + O ( 1 − e − 2 z ) . (95) Consequently,
e B g T ( z , 1 ) ( 1 − T ( z , 1 ) ) 3 g − 5 = e B g ( e 2 ( e − 2 ) ) ( 3 g − 5 ) / 2 ( 1 − z e − 2 ) − ( 3 g − 5 ) / 2 ( 1 + O ( ( 1 − z e − 2 ) 1 / 2 ) ) . \frac{e^{B_gT(z,1)}}{(1-T(z,1))^{3g-5}}=e^{B_g}\left(\frac{e}{2(e-2)}\right)^{(3g-5)/2}\left(1-\frac{z}{e-2}\right)^{-(3g-5)/2}\left(1+O\left(\left(1-\frac{z}{e-2}\right)^{1/2}\right)\right). ( 1 − T ( z , 1 ) ) 3 g − 5 e B g T ( z , 1 ) = e B g ( 2 ( e − 2 ) e ) ( 3 g − 5 ) /2 ( 1 − e − 2 z ) − ( 3 g − 5 ) /2 ( 1 + O ( ( 1 − e − 2 z ) 1/2 ) ) . Applying the transfer theorem gives
[ z n ] e B g T ( z , 1 ) ( 1 − T ( z , 1 ) ) 3 g − 5 = O ( ( e − 2 ) − n n ( 3 g − 5 ) / 2 − 1 ) . [z^n]\frac{e^{B_gT(z,1)}}{(1-T(z,1))^{3g-5}}=O\left((e-2)^{-n}n^{(3g-5)/2-1}\right). [ z n ] ( 1 − T ( z , 1 ) ) 3 g − 5 e B g T ( z , 1 ) = O ( ( e − 2 ) − n n ( 3 g − 5 ) /2 − 1 ) . Combining this estimate with the preceding coefficientwise bound yields
H g , n ( 1 ) − H g , n t a u t ( 1 ) = O ( n ! ( e − 2 ) − n n ( 3 g − 7 ) / 2 ) . (96) H_{g,n}(1)-H^{\mathrm{taut}}_{g,n}(1)=O\left(n!(e-2)^{-n}n^{(3g-7)/2}\right). \tag*{(96)} H g , n ( 1 ) − H g , n taut ( 1 ) = O ( n ! ( e − 2 ) − n n ( 3 g − 7 ) /2 ) . (96) On the other hand, (34) at t = 1 t=1 t = 1
H g , n ( 1 ) = C g ( 1 ) n ! ( e − 2 ) − n n ( 3 g − 5 ) / 2 ( 1 + O ( n − 1 / 2 ) ) , C g ( 1 ) > 0. H_{g,n}(1)=C_g(1)n!(e-2)^{-n}n^{(3g-5)/2}\left(1+O(n^{-1/2})\right),\qquad C_g(1)>0. H g , n ( 1 ) = C g ( 1 ) n ! ( e − 2 ) − n n ( 3 g − 5 ) /2 ( 1 + O ( n − 1/2 ) ) , C g ( 1 ) > 0. Dividing the two estimates, we obtain
1 − H g , n t a u t ( 1 ) H g , n ( 1 ) = H g , n ( 1 ) − H g , n t a u t ( 1 ) H g , n ( 1 ) = O ( n − 1 ) , 1-\frac{H^{\mathrm{taut}}_{g,n}(1)}{H_{g,n}(1)}=\frac{H_{g,n}(1)-H^{\mathrm{taut}}_{g,n}(1)}{H_{g,n}(1)}=O(n^{-1}), 1 − H g , n ( 1 ) H g , n taut ( 1 ) = H g , n ( 1 ) H g , n ( 1 ) − H g , n taut ( 1 ) = O ( n − 1 ) , which proves the upper bound in (88) .
For the lower bound, substitute w = T ( z , 1 ) w=T(z,1) w = T ( z , 1 ) (76) . Using (95) and applying the transfer theorem as above, we obtain
∑ p − q = 22 h p , q ( M ‾ g , n ) = a g Γ ( ( 3 g − 5 ) / 2 ) ( e 2 ( e − 2 ) ) ( 3 g − 5 ) / 2 n ! ( e − 2 ) − n n ( 3 g − 7 ) / 2 ( 1 + O ( n − 1 / 2 ) ) . (97) \sum_{p-q=22}h^{p,q}(\overline{\mathcal{M}}_{g,n})=\frac{a_g}{\Gamma((3g-5)/2)}\left(\frac{e}{2(e-2)}\right)^{(3g-5)/2}n!(e-2)^{-n}n^{(3g-7)/2}\left(1+O(n^{-1/2})\right). \tag*{(97)} p − q = 22 ∑ h p , q ( M g , n ) = Γ (( 3 g − 5 ) /2 ) a g ( 2 ( e − 2 ) e ) ( 3 g − 5 ) /2 n ! ( e − 2 ) − n n ( 3 g − 7 ) /2 ( 1 + O ( n − 1/2 ) ) . (97) Every class counted in (97) has even degree and lies off the Hodge diagonal, whereas tautological classes are algebraic and hence of type ( k , k ) (k,k) ( k , k )
H g , n ( 1 ) − H g , n t a u t ( 1 ) ≥ ∑ p − q = 22 h p , q ( M ‾ g , n ) . H_{g,n}(1)-H^{\mathrm{taut}}_{g,n}(1)\geq\sum_{p-q=22}h^{p,q}(\overline{\mathcal{M}}_{g,n}). H g , n ( 1 ) − H g , n taut ( 1 ) ≥ p − q = 22 ∑ h p , q ( M g , n ) . Since a g > 0 a_g>0 a g > 0 (34) at t = 1 t=1 t = 1 (88) . The total variation bound (89) follows immediately from the degreewise inclusion of tautological cohomology and (88) . Combining this bound with the CLT for X g , n H X^H_{g,n}\! X g , n H Theorem 4.3 gives (90) . □
By (45) , the tautological classes also account for a proportion 1 − O ( n − 1 / 2 ) 1-O(n^{-1/2}) 1 − O ( n − 1/2 ) M ‾ g , n \overline{\mathcal{M}}_{g,n} M g , n
Department of Mathematics and Research Institute of Natural Sciences, Sookmyung Women’s University, Seoul 04310, Korea
Email address: jwchoi@sookmyung.ac.kr
School of Mathematics, Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 02455, Korea
Email address: kiem@kias.re.kr
