We present a series of new results we obtained recently about the intersection numbers of tautological classes on moduli spaces of curves, including a simple formula of the n-point functions for Witten's $τ$ classes, an effective recursion formula to compute higher Weil-Petersson volumes, several new recursion formulae of intersection numbers and our proof of a conjecture of Itzykson and Zuber concerning denominators of intersection numbers. We also present Virasoro and KdV properties of generating functions of general mixed $κ$ and $ψ$ intersections.
We study intersection cohomology of moduli spaces of semistable vector bundles on a complex algebraic surface. Our main result relates intersection Poincare polynomials of the moduli spaces to Donaldson-Thomas invariants of the surface. In support of this result, we compute explicitly intersection Poincare polynomials for sheaves with rank two and three on ruled surfaces.
We provide a general method for computing rational Chow rings of moduli of smooth complete intersections. We specialize this result in different ways: to compute the integral Picard group of the associated stack ; to obtain an explicit presentation of rational Chow rings of moduli of smooth complete intersections of codimension two; to prove old and new results on moduli of smooth curves of genus $\leq 5$ and polarized K3 surfaces of degree $\leq 8$.
This is a note on calculating intersection numbers on moduli spaces of curves. A codimension 3 relation among tautological classes on the moduli space of genus 4 curves is given.
Generalized Hypergeometric Functions for Degree k Hypersurface in CP^{N-1} and Intersection Numbers of Moduli Space of Quasimaps from CP^{1} with Two Marked Points to CP^{N-1}
本文将 CP^{N-1} 中 k 次超曲面的镜像计算所用广义超几何函数实现为带两个标记点的拟映射模空间相交数的生成函数，建立了超曲面镜像对称与拟映射计数之间的具体联系。
We establish the Airy curve case of a conjecture of Gukov and Sułkowski by reducing to Dijkgraaf-Verlinde-Verlinde Virasoro constraints satisfied by the intersection numbers on moduli spaces of algebraic curves.
We compute Betti numbers of both the components of the moduli space of rank 2 semi-stable torsion-free sheaves with fixed determinant over a reducible nodal curve with two smooth components intersecting at a node. We also compute the intersection Betti numbers of the moduli space.
Period Integrals (Givental's I-function) of Calabi-Yau Hypersurface in $CP^{N-1}$ and Intersection Numbers of Moduli Space of Quasimaps from $CP^1$ with Two Marked Points to $CP^{N-1}$
We explain how to compute top-dimensional intersections of psi-classes on moduli spaces of m-stable curves. On the moduli spaces of Deligne-Mumford stable pointed curves of genus one, these intersection numbers are determined by two recursions, namely the string equation and the dilaton equation. We establish, for each integer m>0, an analogous pair of recursions which determine these intersection numbers on the moduli spaces of m-stable pointed curves of genus one.
We study the moduli space of smooth complete intersections of two quadrics in $\mathbb{P}^n$ by relating it to the geometry of the singular members of the corresponding pencils. Giving an alternative presentation for the moduli space of complete intersections, we compute the Picard group for all $n\geq 3$.
We introduce conjectures relating the Chow ring of a smooth Artin stack $\mathcal{X}$ to the Chow groups of its possibly singular good moduli space $X$. In particular, we conjecture the existence of an intersection product on a subgroup of the full Chow group $A^*(X)$ coming from strong cycles on $\mathcal{X}$.
Hausel and Rodriguez-Villegas conjectured that the intersection form on the moduli space of stable PGL_n-Higgs bundles on a curve vanishes if the degree is coprime to n. In this note we prove this conjecture. Along the way we show that moduli spaces of stable chains are irreducible for stability parameters larger than the stability condition induced form stability of Higgs bundles.
This paper proves the existence of homeomorphic (diffeomorphic) complex 6-dimensional (7-dim) complete intersections that belong to components of the moduli space of different dimensions. These results are given as a supplement to earlier result on 5-dimensional complete intersections.
Using the celebrated Witten-Kontsevich theorem, we prove a recursive formula of the $n$-point functions for intersection numbers on moduli spaces of curves. It has been used to prove the Faber intersection number conjecture and motivated us to find some conjectural vanishing identities for Gromov-Witten invariants. The latter has been proved recently by X. Liu and R. Pandharipande. We also give a combinatorial interpretation of $n$-point functions in terms of summation over binary trees.
Stabilization of intersection Betti numbers for moduli spaces of one-dimensional sheaves on surfaces
The School of Mathematics and Statistics, Xi’an Jiaotong University（西安交通大学数学与统计学院）；Shanghai Center for Mathematical Sciences, Fudan University（复旦大学上海数学中心）
We explicitly compute the diverging factor in the large genus asymptotics of the Weil-Petersson volumes of the moduli spaces of $n$-pointed complex algebraic curves. Modulo a universal multiplicative constant we prove the existence of a complete asymptotic expansion of the Weil-Petersson volumes in the inverse powers of the genus with coefficients that are polynomials in $n$. This is done by analyzing various recursions for the more general intersection numbers of tautological classes, whose large genus asymptotic behavior is also extensively studied.
This paper is to give some concrete examples of the general fibers of the evaluation map of some Kontsevich mapping spaces parametrize low degree rational curves on low degree complete intersection varieties. We prove these examples are Fano or Calabi-Yau vareities of low dimension, e.g, surface and threefold. In usual, the "moduli" fibers are constructed in an abstract way, but here we give a down-to-earth interpretation to some of these fibers from the perspective of projective geometry.
We study the arithmetic of complete intersections in projective space over number fields. Our main results include arithmetic Torelli theorems and versions of the Shafarevich conjecture, as proved for curves and abelian varieties by Faltings. For example, we prove an analogue of the Shafarevich conjecture for cubic and quartic threefolds and intersections of two quadrics.
A curve on a projective variety is called movable if it belongs to an algebraic family of curves covering the variety. We consider when the cone of movable curves can be characterized without existence statements of covering families by studying the complete intersection cone on a family of blow-ups of complex projective space, including the moduli space of stable six-pointed rational curves, $\bar{M}_{0,6}$, and the permutohedral or Losev-Manin moduli space of four-pointed rational curves. Our main result is that the movable and complete intersection cones coincide for the toric members of th…
In [GH], Griffiths and Harris asked whether a projective complex submanifold of codimension two is determined by the moduli of its second fundamental forms. More precisely, given a nonsingular subvariety $X^n \subset {\mathbb P}^{n+2}$, the second fundamental form $II_{X,x}$ at a point $x \in X$ is a pencil of quadrics on $T_x(X)$, defining a rational map $μ^X$ from $X$ to a suitable moduli space of pencils of quadrics on a complex vector space of dimension $n$. The question raised by Griffiths and Harris was whether the image of $μ^X$ determines $X$. We study this question when $X^n \subset {…
The main result of this paper is the statement that the Hodge theoretic Donaldson-Thomas invariant for a quiver with zero potential and a generic stability condition agrees with the compactly supported intersection cohomology of the closure of the stable locus inside the associated coarse moduli space of semistable quiver representations. In fact, we prove an even stronger result relating the Donaldson-Thomas "function" to the intersection complex. The proof of our main result relies on a relative version of the integrality conjecture in Donaldson-Thomas theory. This will be the topic of the s…
Given a collection of boundary divisors in the moduli space of stable genus-zero n-pointed curves, Giansiracusa proved that their intersection is nonempty if and only if all pairwise intersections are nonempty. We give a complete classification of the pairs (g,n) for which the analogous statement holds in the moduli space of n-pointed curves of genus g.
This article explores to which extent the algebro-geometric theory of rational descendant Gromov-Witten invariants can be carried over to the tropical world. Despite the fact that the tropical moduli-spaces we work with are non-compact, the answer is surprisingly positive. We discuss the string, divisor and dilaton equations, we prove a splitting lemma describing the intersection with a "boundary" divisor and we prove general tropical versions of the WDVV resp. topological recursion equations (under some assumptions). As a direct application, we prove that the toric varieties $\mathbb{P}^1$, $…
We study the ring generated by the Chern classes of tautological line bundles on the moduli space of parabolic bundles of arbitrary rank on a Riemann surface. We show the Poincaré duals to these Chern classes have simple geometric representatives. We use this construction to show that the ring generated by these Chern classes vanishes below the dimension of the moduli space, in analogy with the Newstead-Ramanan conjecture for stable bundles.
The intersection form in the cohomology of the moduli space of genus 0 $n$-pointed curves $H^*(\overline{M}_{0,n})$ and the explicit Künneth formula in quantum cohomology
The goal of this paper is to study a $p$-adic analog of the joint of the conjectures of André--Oort and André--Pink. More precisely, on a product of ordinary Siegel formal moduli schemes, we study the distribution of points whose components are either CM points or points in Hecke orbits. We use linearity of formal subschemes of the product as the $p$-adic analog of geodesicness over complex numbers. Moreover, we relax the usual incidence relations by using $p$-adic distance. We also study a $p$-adic formal scheme theoretic analog of the Ax--Lindemann theorem.
For a split reductive group G over a finite field, we show that the intersection (cohomology) motive of the moduli stack of iterated G-shtukas with bounded modification and level structure is defined independently of the standard conjectures on motivic t-structures on triangulated categories of motives. This is in accordance with general expectations on the independence of l in the Langlands correspondence for function fields.
We determine the integral Chow and cohomology rings of the moduli stack $\mathcal{B}_{r,d}$ of rank $r$, degree $d$ vector bundles on $\mathbb{P}^1$ bundles. We first show that the rational Chow ring $A_{\mathbb{Q}}^*(\mathcal{B}_{r,d})$ is a free $\mathbb{Q}$-algebra on $2r+1$ generators. The isomorphism class of this ring happens to be independent of $d$. Then, we prove that the integral Chow ring $A^*(\mathcal{B}_{r,d})$ is torsion-free and provide multiplicative generators for $A^*(\mathcal{B}_{r,d})$ as a subring of $A_{\mathbb{Q}}^*(\mathcal{B}_{r,d})$. From this description, we see that…
We compute the Picard group of the moduli stack of smooth curves of genus $g$ for $3\leq g\leq 5$, using methods of equivariant intersection theory. We base our proof on the computation of some relations in the integral Chow ring of certain moduli stacks of smooth complete intersections. As a byproduct, we compute the cycle classes of some divisors on $\mathcal{M}_g$.
Let $C$ be a smooth projective curve of genus $2$. Following a method by O' Grady, we construct a semismall desingularization $\tilde{\mathcal{M}}_{Dol}^G$ of the moduli space $\mathcal{M}_{Dol}^G$ of semistable $G$-Higgs bundles of degree 0 for $G=GL(2,\mathbb{C}), SL(2,\mathbb{C})$. By the decomposition theorem by Beilinson, Bernstein, Deligne one can write the cohomology of $\tilde{\mathcal{M}}_{Dol}^G$ as a direct sum of the intersection cohomology of $\mathcal{M}_{Dol}^G$ plus other summands supported on the singular locus. We use this splitting to compute the intersection cohomology of $…
