A relatively fast algorithm for evaluating Weil-Petersson volumes of moduli spaces of complex algebraic curves is proposed. On the basis of numerical data, a conjectural large genus asymptotics of the Weil-Petersson volumes is computed. Asymptotic formulas for the intersection numbers involving $ψ$-classes are conjectured as well. The accuracy of the formulas is high enough to believe that they are exact.
We study the asymptotics of the Weil-Petersson volumes of the moduli spaces of compact Riemann surfaces of genus $g$ with $n$ punctures, for fixed $n$ as $g \to \infty$.
An explicit upper bound for the Weil-Petersson volumes of the moduli spaces of punctured Riemann surfaces is obtained, using Penner's combinatorial integration scheme with embedded trivalent graphs. It is shown that for a fixed number of punctures n and for genus g going to infinity, the Weil-Petersson volume of M_{g,n} has an upper bound c^g g^{2g}. Here c is an independent of n constant, which is given explicitly.
A formula for the generating function of the Weil-Petersson volumes of moduli spaces of pointed curves that is identical to the genus expansion of the free energy in two dimensional gravity is obtained. The contribution of arbitrary genus is expressed in terms of the Bessel function $J_0$.
We give a Hodge-theoretic criterion for a Calabi--Yau variety to have finite Weil--Petersson distance on higher dimensional bases up to a set of codimension $\geq 2$. The main tool is variation of Hodge structures and variation of mixed Hodge structures. We also give a description on the codimension 2 locus for the moduli space of Calabi--Yau threefolds. We prove that the points lying on exactly one finite and one infinite divisor have infinite Weil--Petersson distance along angular slices. Finally, by giving a classification of the dominant term of the candidates of the Weil--Petersson potent…
In this paper we study effective recursion formulae for computing intersection numbers of mixed $ψ$ and $κ$ classes on moduli spaces of curves. By using the celebrated Witten-Kontsevich theorem, we generalize Mulase-Safnuk form of Mirzakhani's recursion and prove a recursion formula of higher Weil-Petersson volumes. We also present recursion formulae to compute intersection pairings in the tautological rings of moduli spaces of curves.
The Weil-Petersson form on a cluster variety is a 2-form on a certain open smooth subvariety; the union of the cluster tori. We show that for acyclic cluster varieties, the Weil-Petersson 2-form extends to a regular Kähler 2-form on the entire cluster variety.
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.
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…
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.
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.
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.
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 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.
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.
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.
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.
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.
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…
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}).
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
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.
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…
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.
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…
A discrete analogue of $\mathscr{N}=2$ super Weil-Petersson volumes
University of California, Santa Barbara（加州大学圣塔芭芭拉分校）；Beijing Institute of Mathematical Sciences and Applications（北京数学科学与应用研究所）；Kavli Institute for Theoretical Physics（卡弗里理论物理研究所）
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…
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…
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.
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.
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…
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.
