Weil-Petersson volumes of moduli spaces of curves and the genus expansion in two dimensional gravity

On the Weil-Petersson Geometry of Calabi-Yau Moduli

In this paper we proved that the Weil-Petersson volume of the Chern class of any order over the moduli space of Calabi-Yau manifolds is a rational number. We also found the necessary and sufficient condition of the incompleteness of Weil-Petersson metric in several variables case.

Invertible Cohomological Field Theories and Weil-Petersson volumes

We show that the generating function for the higher Weil-Petersson volumes of the moduli spaces of stable curves with marked points can be obtained from Witten's free energy by a change of variables given by Schur polynomials. Since this generating function has a natural extension to the moduli space of invertible Cohomological Field Theories, this suggests the existence of a ``very large phase space'', correlation functions on which include Hodge integrals studied by C. Faber and R. Pandharipande. From this formula we derive an asymptotical expression for the Weil-Petersson volume as conjectu…

Weil-Petersson体积、稳定性条件与墙穿越

Weil-Petersson volumes, stability conditions and wall-crossing

本文研究锥形双曲曲面模空间的Weil-Petersson体积，通过Hassett稳定性条件空间的腔室分解，建立各腔室多项式与Mirzakhani多项式间的墙穿越关系，并揭示其体积性质。

Weil-Petersson Volumes of the Moduli Spaces of CY Manifolds

In this paper it is proved that the volumes of the moduli spaces of polarized CY manifolds with respect to the Weil-Petersson metrics are finite and they are rational numbers.

The asymptotic Weil-Petersson form and intersection theory on M_{g,n}

Moduli spaces of hyperbolic surfaces with geodesic boundary components of fixed lengths may be endowed with a symplectic structure via the Weil-Petersson form. We show that, as the boundary lengths are sent to infinity, the Weil-Petersson form converges to a piecewise linear form first defined by Kontsevich. The proof rests on the observation that a hyperbolic surface with large boundary lengths resembles a graph after appropriately scaling the hyperbolic metric. We also include some applications to intersection theory on moduli spaces of curves.

Higher Weil-Petersson Volumes of Moduli Spaces of Stable $n$-pointed Curves

Moduli spaces of compact stable $n$-pointed curves carry a hierarchy of cohomology classes of top dimension which generalize the Weil-Petersson volume forms and constitute a version of Mumford classes. We give various new formulas for the integrals of these forms and their generating functions. We also discuss their relation to the Kuenneth formula in quantum cohomology.

The Weil-Petersson current for moduli of vector bundles and applications to orbifolds

We investigate stable holomorphic vector bundles on a compact complex Kähler manifold and more generally on an orbifold that is equipped with a Kähler structure. We use the existence of Hermite-Einstein connections in this set-up and construct a generalized Weil-Petersson form on the moduli space of stable vector bundles with fixed determinant bundle. We show that the Weil-Petersson form extends as a (semi-)positive closed current for degenerating families that are restrictions of coherent sheaves. Such an extension will be called a Weil-Petersson current. When the orbifold is of Hodge type, t…

Moduli spaces of hyperbolic surfaces and their Weil-Petersson volumes

Moduli spaces of hyperbolic surfaces may be endowed with a symplectic structure via the Weil-Petersson form. Mirzakhani proved that Weil-Petersson volumes exhibit polynomial behaviour and that their coefficients store intersection numbers on moduli spaces of curves. In this survey article, we discuss these results as well as some consequences and applications.

模形式Weil-Petersson度量渐进行的模解释

Modular interpretation of the Weil-Petersson metric asymptotics for abelian varieties

本文研究了abelian varieties的Weil-Petersson度量渐近行为，通过多尺度塌缩极限与Odaka的分类联系，提供模形式的解释。

Weil-Petersson volumes and cone surfaces

The moduli spaces of hyperbolic surfaces of genus g with n geodesic boundary components are naturally symplectic manifolds. Mirzakhani proved that their volumes are polynomials in the lengths of the boundaries by computing the volumes recursively. In this paper we give new recursion relations between the volume polynomials.

Weil-Petersson geometry on the space of Bridgeland stability conditions

Inspired by mirror symmetry, we investigate some differential geometric aspects of the space of Bridgeland stability conditions on a Calabi-Yau triangulated category. The aim is to develop theory of Weil-Petersson geometry on the stringy Kähler moduli space. A few basic examples are studied. In particular, we identify our Weil-Petersson metric with the Bergman metric on a Siegel modular variety in the case of the self-product of an elliptic curve.

Mirzakhani's recursion formula on Weil-Petersson volume and applications

We give an overview of the proof for Mirzakhani's volume recursion for the Weil-Petersson volumes of the moduli spaces of genus $g$ hyperbolic surfaces with $n$ labeled geodesic boundary components, and her application of this recursion to Witten's conjecture and the study of simple geodesic length spectrum growth rates.

Curvature of higher direct image sheaves and its application on negative-curvature criterion for the Weil-Petersson metric

We shall show that $q$-semipositivity of the vector bundle $E$ over a Kähler total space $\mathcal X$ implies the Griffiths-semipositivity of the $q$-th direct image of $\mathcal O(K_{\mathcal X/B}\otimes E)$. As an application, we shall give a negative-curvature criterion for the generalized Weil-Petersson metric on the base manifold.

Weil-Petersson Teichmüller theory of surfaces of infinite conformal type

Over the past two decades the theory of the Weil-Petersson metric has been extended to general Teichmüller spaces of infinite type, including for example the universal Teichmüller space. In this paper we give a survey of the main results in the Weil-Petersson geometry of infinite-dimensional Teichmüller spaces. This includes the rigorous definition of complex Hilbert manifold structures, Kähler geometry and global analysis, and generalizations of the period mapping. We also discuss the motivations of the theory in representation theory and physics beginning in the 1980s. Some examples of the a…

Weil-Petersson volume of moduli spaces, Mirzakhani's recursion and matrix models

We show that Mirzakhani's recursions for the volumes of moduli space of Riemann surfaces are a special case of random matrix loop equations, and therefore we confirm again that Kontsevitch's integral is a generating function for those volumes. As an application, we propose a formula for the Weil-Petersson volume Vol(M_{g,0}).

Mirzakhani 递推的 q-模拟用于 Weil-Petersson 体积

A q-analogue of Mirzakhani's recursion for Weil-Petersson volumes

本文定义了 Weil-Petersson 体积的 Mirzakhani 递推的 q-模拟，与 Okuyama 拟多项式最高次项一致，并推广到超设置。

Continuity of the Weil-Petersson potential

Let $\mathcal{M}_{\rm KSB}$ (resp. $\mathcal{M}_{\rm KSB}'$) be the the moduli space of $n$-dimensional Kähler-Einstein manifolds (resp. varieties) $X$ with $K_X$ ample. We prove that the Weil-Petersson metric on $\mathcal{M}_{\rm KSB}$ extends uniquely to the projective variety $\mathcal{M}_{\rm KSB}'$, as a closed positive current with continuous local potentials. This generalizes a theorem of Wolpert which treats the case $n=1$, and also confirms a conjecture of Berman-Guenancia. In addition, we derive uniform estimates for the volumes of sublevel sets of Kähler-Einstein potentials

从双标SYK相关子到Weil-Petersson体积

From double-scaled SYK correlators to Weil-Petersson volumes

本文通过严格证明，将双标SYK相关子与Weil-Petersson体积联系起来，揭示了量子几何与组合几何之间的关联。

模空间 $\overline{\cal M}_g$ 上 Weil-Petersson 度量平均标量曲率的估计

Estimates for the average scalar curvature of the Weil-Petersson metric on the Moduli space $\overline{\cal M}_g$

本文研究了模空间 $\overline{\cal M}_g$ 上 Weil-Petersson 度量的平均标量曲率，在 $g\to\infty$ 时给出了精确到 $1/g^2$ 阶的估计。

关于射影直线上加权点的模空间Weil-Petersson体积的连续性

On the continuity of Weil-Petersson volumes of the moduli space weighted points on the projective line

本文证明了射影直线上加权点的模空间中，从Fano几何趋近Calabi-Yau几何时，Weil-Petersson体积连续，且局部化计算的CM体积收敛到McMullen的几何体积。

通往无穷没有捷径：Calabi-Yau复结构模空间中的Weil-Petersson距离

No Shortcuts to Infinity: Weil-Petersson Distance in Calabi-Yau Complex Structure Moduli Space

本文建立Calabi-Yau复结构模空间中边界层处于无穷Weil-Petersson距离的充要判据，证明无穷距离边界因子的交集仍为无穷距离，并验证Wang猜想。

大亏格下 Weil-Petersson 体积的渐近系数

Asymptotic coefficients of Weil-Petersson volumes in the large genus

本文针对 Mirzakhani-Zograf 关于 Weil-Petersson 体积大亏格渐近系数的猜想，证明这些系数仅属于 Q[π^{-2}]，而非更一般的 Q[π^{-2},π^2]。

超 Riemann 曲面模空间的高阶 Weil-Petersson 体积

Higher Weil-Petersson volumes of the moduli space of super Riemann surfaces

本文将 Stanford-Witten 关于超 Riemann 曲面模空间 Weil-Petersson 体积的递归公式推广到包含高次 κ 类的情形。

超Weil-Petersson体积的大亏格渐近性

Large genus asymptotics of super Weil-Petersson volumes

该研究得到超相交数的渐近展开并证明系数为多项式，给出显式计算系数的算法，进而证明超Weil-Petersson体积大亏格渐近展开的存在性，推广了Mirzakhani-Zograf的工作，还证实了两个相关猜想公式。

覆盖构造的局部刚性与曲线模空间的Weil–Petersson子簇

Local rigidity of covering constructions and Weil--Petersson subvarieties of the moduli space of curves

本文证明了曲线模空间$\u039c_{g,n}$的Weil–Petersson全测地子簇及覆盖构造是局部刚性的，该结论由一类轨形映射的一般刚性结果推导得到。

双模空间体积的故事：Weil-Petersson与Masur-Veech

A tale of two volumes of moduli spaces: Weil-Petersson and Masur-Veech

本文综述了Weil-Petersson与Masur-Veech体积的计算方法、关键结果及开放问题，探讨了两者在模空间研究中的联系与差异。

曲线模空间上的超Weil-Petersson测度

Super Weil-Petersson measures on the moduli space of curves

本文定义了光滑曲线模空间上由超Weil-Petersson度量导出的、按标记点处旋结构行为分类的有限测度，证明Ramond边界行为对应的体积是Neveu-Schwarz体积多项式的形变且满足Stanford-Witten递推关系的变体。

Towards large genus asymtotics of intersection numbers on moduli spaces of curves

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.

Potentials and Chern forms for Weil-Petersson and Takhtajan-Zograf metrics on moduli spaces

For the TZ metric on the moduli space $\mathscr{M}_{0,n}$ of $n$-pointed rational curves, we construct a Kähler potential in terms of the Fourier coefficients of the Klein's Hauptmodul. We define the space $\mathfrak{S}_{g,n}$ as holomorphic fibration $\mathfrak{S}_{g,n}\rightarrow\mathfrak{S}_{g}$ over the Schottky space $\mathfrak{S}_{g}$ of compact Riemann surfaces of genus $g$, where the fibers are configuration spaces of $n$ points. For the tautological line bundles $\mathscr{L}_{i}$ over $\mathfrak{S}_{g,n}$ we define Hermitian metrics $h_{i}$ in terms of Fourier coefficients of a coveri…

Geometry of moduli spaces of Higgs bundles

We construct a Petersson-Weil type Kähler form on the moduli spaces of Higgs bundles over a compact Kähler manifold. A fiber integral formula for this form is proved, from which it follows that the Petersson-Weil form is the curvature of a certain determinant line bundle, equipped with a Quillen metric, on the moduli space of Higgs bundles over a projective manifold. The curvature of the Petersson-Weil Kähler form is computed. We also show that, under certain assumptions, a moduli space of Higgs bundles supports of natural hyper-Kähler structure.

Weil-Petersson随机曲面的多项式速率谱间隙

Spectral gap with polynomial rate for Weil-Petersson random surfaces

该研究证明Weil-Petersson随机采样的亏格$g$闭双曲曲面拉普拉斯谱间隙至少为$\frac{1}{4}-O(\frac{1}{g^c})$（$g\to\infty$时概率趋近1），拓展并新证Anantharaman和Monk的结果，适配随机矩阵强收敛多项式方法至该场景。

Asymptotic curvature of moduli spaces for Calabi-Yau threefolds

Motivated by the classical statements of Mirror Symmetry, we study certain Kahler metrics on the complexified Kahler cone of a Calabi-Yau threefold, conjecturally corresponding to approximations to the Weil-Petersson metric near large complex structure limit for the mirror. In particular, the naturally defined Riemannian metric (defined via cup-product) on a level set of the Kahler cone is seen to be analogous to a slice of the Weil-Petersson metric near large complex limit. This enables us to give counterexamples to a conjecture of Ooguri and Vafa that the Weil-Petersson metric has non-positi…

$\mathscr{N}=2$ 超Weil-Petersson体积的离散类似物

A discrete analogue of $\mathscr{N}=2$ super Weil-Petersson volumes

本研究从双标度SYK引入离散谱曲线，利用拓扑递归定义整数边界长度下的拟多项式体积，聚焦局域N=2理论计算有限q体积，经弦方程验证后在连续极限下重现超Weil-Petersson体积，还给出其全亏格相交理论解释。

具有紧边界的双曲曲面 Weil-Petersson 体积的拓扑递归

Topological recursion of the Weil-Petersson volumes of hyperbolic surfaces with tight boundaries

本文将 Mirzakhani 的 Weil-Petersson 体积递归推广到具有紧特殊测地边界和锥形缺陷的双曲曲面，证明体积及生成函数仍为多项式并满足广义拓扑递归，进而用紧 Weil-Petersson 生成函数表示带缺陷的 JT 引力多边界关联函数。

带边双曲曲面模空间Weil-Petersson体积的大$n$渐近

Large-$n$ asymptotics for Weil-Petersson volumes of moduli spaces of bordered hyperbolic surfaces

研究大$n$极限下Weil-Petersson随机曲面的谱理论，证明小Laplace特征值个数与$n$线性相关且最优，并给出模空间Weil-Petersson体积的新渐近公式。

Weil-Petersson geometry and determinant bundles on inductive limits of moduli spaces

In an earlier paper [Acta Mathematica, v. 176, 1996, 145-169, alg-geom/9505024 ] the present authors and Dennis Sullivan constructed the universal direct system of the classical Teichmüller spaces of Riemann surfaces of varying genus. The direct limit, which we called the universal commensurability Teichmüller space, $T_{infty}$, was shown to carry on it a natural action of the universal commensurability mapping class group, $MC_{\infty}$. In this paper we identify an interesting cofinal sub-system corresponding to the tower of finite-sheeted characteristic coverings over any fixed base surfac…

$N=1$ 超对称、Weil-Petersson 体积递归和谱曲线

$N=1$ Supersymmetry, Weil-Petersson Volume Recursion, and a Spectral Curve

证明 Stanford-Witten-Norbury 体积递归可直接从谱曲线导出，并推导出另一种体积递归，区别在于 Ramond 信息位于初始数据或递归核中。

Weil-Petersson体积的边界加法恒等式及其超对称推广

A boundary-addition identity for Weil-Petersson volumes and their supersymmetric generalizations

本文推导出Weil-Petersson体积及其超对称推广的边界加法恒等式，将其与KdV层级关联，统一并推广了已知递推关系，并揭示了N=2和小N=4情形的相交理论描述。

用于更高 Weil-Petersson 体积的 cut-and-join 算子

Cut-and-join operators for higher Weil-Petersson volumes

本文构造了刻画 $\overline{\mathcal M}_{g,n}$ 上 $ψ$、$κ$、$Θ$ 类相交数生成函数的 cut-and-join 算子，并建立代数拓扑递归以递归计算这些相交数及相关高 Weil-Petersson 体积。

Irreducible metric maps and Weil-Petersson volumes

We consider maps on a surface of genus $g$ with all vertices of degree at least three and positive real lengths assigned to the edges. In particular, we study the family of such metric maps with fixed genus $g$ and fixed number $n$ of faces with circumferences $α_1,\ldots,α_n$ and a $β$-irreducibility constraint, which roughly requires that all contractible cycles have length at least $β$. Using recent results on the enumeration of discrete maps with an irreducibility constraint, we compute the volume $V_{g,n}^{(β)}(α_1,\ldots,α_n)$ of this family of maps that arises naturally from the Lebesgu…

从随机中导出拉蒙德：带NS边界和R尖点的超黎曼曲面的Weil-Petersson体积

Ramond from Random: Weil-Petersson Volumes for Super-Riemann surfaces with NS Boundaries and R Punctures

本文通过随机矩阵模型统一处理NS边界和R尖点，计算了超黎曼曲面模空间的Weil-Petersson体积，并首次给出了包含R尖点的谱曲线，利用拓扑递归推导了闭式公式。

A proof of the Hasse-Weil inequality for genus 2 à la Manin

We prove the Hasse-Weil inequality for genus 2 curves given by an equation of the form y^2 = f(x) with f a polynomial of degree 5, using arguments that mimic the elementary proof of the genus 1 case obtained by Yu. I. Manin in 1956.

Completion of the moduli space for polarized Calabi-Yau manifolds

In this paper, we construct a completion of the moduli space for polarized Calabi-Yau manifolds by using Ricci-flat Kähler-Einstein metrics and the Gromov-Hausdorff topology, which parameterizes certain Calabi-Yau varieties. We then study the algebro-geometric perperties and the Weil-Petersson geometry of such completion. We show that the completion can be exhausted by sequences of quasi-projective varieties, and new points added have finite Weil-Petersson distance to the interior.

Omega Admissible Theory II: New metrics on determinant of cohomology And Their applications to moduli spaces of punctured Riemann surfaces

For singular metrics, there is no Quillen metric formalism on cohomology determinant. In this paper, we develop an admissible theory, with which the arithmetic Deligne-Riemann-Roch isometry can be established for singular metrics. As an application, we first study Weil-Petersson metrics and Takhtajan-Zograf metrics on moduli spaces of punctured Riemann surfaces, and then give a more geometric interpretation of our determinant metrics in terms of Selberg zeta functions. We end this paper by proposing an arithmetic factorization for Weil-Petersson metrics, cuspidal metrics and Selberg zeta funct…

关于某些Fano K-模空间体积的研究

On the volume of some Fano K-moduli spaces

本文计算了任意维数下四次del Pezzo及低维对数Fano超平面排列的K-模空间的CM体积，并推广Weil-Petersson度量以建立其与Weil-Petersson体积的联系。

Quasi-projectivity of the moduli space of smooth Kahler-Einstein Fano manifolds

In this note, we prove that there is a canonical continuous Hermitian metric on the CM line bundle over the proper moduli space $\bar{\mathcal{M}}$ of smoothable Kahler-Einstein Fano varieties. The curvature of this metric is the Weil-Petersson current, which exists as a positive (1,1)-current on $\bar{\mathcal{M}}$ and extends the canonical Weil-Petersson current on the moduli space parametrizing smooth Kahler-Einstein Fano manifolds $\mathcal{M}$. As a consequence, we show that the CM line bundle is nef and big on $\bar{\mathcal{M}}$ and its restriction on $\mathcal{M}$ is ample.

A genus two arithmetic Siegel-Weil formula on X_0(N)

We define a family of arithmetic zero cycles in the arithmetic Chow group of a modular curve X_0(N), for N>3 odd and squarefree, and identify the arithmetic degrees of these cycles as q-coefficients of the central derivative of a Siegel Eisenstein series of genus two. This parallels work of Kudla-Rapoport-Yang for Shimura curves.

超体积与KdV tau函数

Super volumes and KdV tau functions

本文通过引入Ramond穿孔，证明超曲线模空间的超Weil-Petersson体积与广义BGW KdV tau函数相关，并满足Stanford-Witten递归。