THE EVENTUALLY DISTANCE MINIMIZING RAYS IN MODULI SPACES

The eventually distance minimizing ray（EDM ray） in moduli spaces of the Riemann surfaces of analytic finite type with 3 g ＋ n-3 〉 0 is studied, which was introduced by Farb and Masur [5]. The asymptotic distance of EDM rays in a moduli space and the distance of end points of EDM rays in the boundary of the moduli space in the augmented moduli space are discussed in this article. A relation between the asymptotic distance of EDM rays and the distance of their end points is established. It is proved also that the distance of end points of two EDM rays is equal to that of end points of two Strebel rays in the Teichmu¨ller space of a covering Riemann surface which are leftings of some representatives of the EDM rays. Meanwhile, simpler proofs for some known results are given.

Moduli spaces of model real submanifolds:Two alternative approaches

Abstract Instead of the invariant theory approach employed by Beloshapka and Mamai for constructing the moduli spaces of Beloshapka＇s universal Cauchy-Riemann （CR） models, we consider two alternative approaches borrowed from the theories of equivalence problem and Lie symmetries, each of which having its own advan- tages. Also the moduli space M（1, 4） associated to the class of universal CR models of CR dimension 1 and codimension 4 is computed by means of the presented methods.

Spaces and moduli spaces of Riemannian metrics

These notes present and survey results about spaces and moduli spaces of complete Riemannian metrics with curvature bounds on open and closed manifolds, here focussing mainly on connectedness and disconnectedness properties. They also discuss several open problems and questions in the field.

Geometric aspects of the moduli space of Riemann surfaces

We describe some recent progress in the study of moduli space of Riemann surfaces in this survey paper. New complete Kahler metrics were introduced on the moduli space and Teichmuller space. Their curvature properties and asymptotic behavior were studied in details. These natural metrics served as bridges to connect all the known canonical metrics, especially the Kahler-Einstein metric. We showed that all the known complete metrics on the moduli space are equivalent and have Poincare type growth. Furthermore,the Kahler-Einstein metric has strongly bounded geometry. This also implied that the logarithm cotangent bundle of the moduli space is stable in the sense of Mumford.

New results on the geometry of the moduli space of Riemann surfaces

We briefly survey our recent results about the Mumford goodness of several canonical metrics on the moduli spaces of Riemann surfaces, including the Weil-Petersson metric, the Ricci metric, the Perturbed Ricci metric and the Kahler-Einstein metric. We prove the dual Nakano negativity of the Weil-Petersson metric. As applications of these results we deduce certain important results about the L 2-cohomology groups of the logarithmic tangent bundle over the compactified moduli spaces.

THE MODULI SPACE OF COMPLEX LAGRANGIAN SUBMANIFOLDS IN THE HYPER-KAEHLER MANIFOLD

The deformation of a compact complex Lagrangian submanifold in a hyper-Kaehler manifold and the moduli space are studied. It is proved that the moduli space Mc1 is a special Kaehler manifold, where special means that there is a real flat torsionfree symplectic connection satisfying dI = 0 (I is a complex structure of Mcl). Thus, following [4], one knows thatT*Mcl is a hyper-Kaehler manifold and then that Mcl is a complex Lagrangian submanifold in T* Mcl.

Cohomology of Moduli Space of Cubic Fourfolds Ⅰ

In this paper we compute the cohomology of moduli space of cubic fourfolds with ADE type singularities relying on Kirwan’s blowup and Laza’s GIT construction. More precisely, we obtain the Betti numbers of Kirwan’s resolution of the moduli space. Furthermore, by applying decomposition theorem we obtain the Betti numbers of the intersection cohomology of Baily-Borel compactification of the moduli space.

Moduli of Polarized Calabi–Yau Pairs

We prove that the irreducible components of the moduli space of polarized Calabi–Yau pairs are projective.

Partial Collapsing Degeneration of Floer Trajectories and Adiabatic Gluing

In the present paper,we study partial collapsing degeneration of Hamiltonian-perturbed Floer trajectories for an adiabatic ε-family and its reversal adiabatic gluing,as the prototype of the partial collapsing degeneration of 2-dimensional(perturbed)J-holomorphic maps to 1-dimensional gradient segments.We consider the case when the Floer equations are S^(1)-invariant on parts of their domains whose adiabatic limit has positive length as ε→0,which we call thimble-flow-thimble configurations.The main gluing theorem we prove also applies to the case with Lagrangian boundaries such as in the problem of recovering holomorphic disks out of pearly configuration.In particular,our gluing theorem gives rise to a new direct proof of the chain isomorphism property between the Morse-Bott version of Lagrangian intersection Floer complex of L by Fukaya-Oh-Ohta-Ono and the pearly complex of L Lalonde and Biran-Cornea.It also provides another proof of the present authors’earlier proof of the isomorphism property of the PSS map without involving the target rescaling and the scale-dependent gluing.

A Weak∞-Functor in Morse Theory

In the spirit of Morse homology initiated by Witten and Floer,we construct two∞-categories A and B.The weak one A comes out of the Morse-Smale pairs and their higher homotopies,and the strict one B concerns the chain complexes of the Morse functions.Based on the boundary structures of the compactified moduli space of gradient flow lines of Morse functions with parameters,we build up a weak∞-functor F:A→B.Higher algebraic structures behind Morse homology are revealed with the perspective of defects in topological quantum field theory.

Modular Invariants and Singularity Indices of Hyperelliptic Fibrations

The modular invariants of a family of curves are the degrees of the pullback of the corresponding divisors by the moduli map. The singularity indices were introduced by Xiao(1991) to classify singular fibers of hyperelliptic fibrations and to compute global invariants locally. In semistable case, the author shows that the modular invariants corresponding to the boundary divisor classes are just the singularity indices. As an application,the author shows that the formula of Xiao for relative Chern numbers is the same as that of Cornalba-Harris in semistable case.

Remarks on lines and minimal rational curves

We determine all of lines in the moduli space M of stable bundles for arbitrary rank and degree. A further application of minimal rational curves is also given in last section.

Families of hyperelliptic curves with maximal slopes

For each integer g≥2, we construct a family of hyperelliptic curves of genus g whose slope reaches the upper bound obtained by Xiao.

Factorization of Generalized Theta Functions Revisited

This survey is based on my lectures given in the last few years. As a reference, constructions of moduli spaces of parabolic sheaves and generalized parabolic sheaves are provided. By a refinement of the proof of vanishing theorems, we show, without using vanishing theorems, a new observation that dim H0（Uc, θuc） is independent of all of the choices for any smooth curves. The estimate of various codimensions and computation of canonical line bundle of moduli space of generalized parabolic sheaves on a reducible curve are provided in Section 6, which is completely new.

Hodge integrals with one λ-class

Hodge integrals over moduli space of stable curves play an important roles in understanding the topological properties of moduli space. ELSV formula connects the Hodge integrals with Hurwitz numbers, and the generating function of Hurwitz numbers satisfies the cut-and-join equation. Therefore, it is natural to consider how to use the cut-and-join equation for Hurwitz numbers to compute Hodge integrals which appear in ELSV formula. In this paper, at first, we will review the method introduced in Goulden et al.＇s paper to get the λg conjecture for Hodge integral. Through some variables transformation, the generating function of Hurwitz number becomes a symmetric polynomial which satisfies a symmetrized cut-and-join equation. By comparing the coefficients of the lowest degree term of both sides in this equation, we can get the ,λg conjecture. Then, in a similar way, we obtain our main result in this paper： a recursive formula for Hodge integral of type contains only one ,λg-l-class. We also point out that our results are closely related to the degree 0 Virasoro conjecture for a curve.

Correlation functions of gauged linear σ-model

This is the second paper in a series following Tian and Xu(2015), on the construction of a mathematical theory of the gauged linear σ-model(GLSM). In this paper, assuming the existence of virtual moduli cycles and their certain properties, we define the correlation function of GLSM for a fixed smooth rigidified r-spin curve.

The morphism induced by Frobenius push-forward

Let X be a smooth projective curve of genus g 2 over an algebraically closed field k of characteristic p>0,and F:X→X(1)the relative Frobenius morphism.Let M s X(r,d)(resp.M ss X(r,d))be the moduli space of(resp.semi-)stable vector bundles of rank r and degree d on X.We show that the set-theoretic map S ss Frob:M ss X(r,d)→M ss X(1)(rp,d+r(p-1)(g-1))induced by[E]→[F(E)]is a proper morphism.Moreover,the induced morphism S s Frob:M s X(r,d)→M s X(1)(rp,d+r(p-1)(g-1))is a closed immersion.As an application,we obtain that the locus of moduli space M s X(1)(p,d)consisting of stable vector bundles whose Frobenius pull backs have maximal Harder-Narasimhan polygons is isomorphic to the Jacobian variety Jac X of X.

On the Equivalence of Multiplicative Structures in Floer Homology and Quantum Homology

In this paper,we will prove that Floer homology equipped with either the intrinsic or exterior product is isomorphic to quantum homology as a ring.We will also prove that GW-invariants in Floer homology and quantum homology are equivalent.

Landau-Ginzburg/Calabi-Yau Correspondence via Wall-Crossing

Landau-Ginzburg/Calabi-Yau correspondence claims the equivalence between the Gromov-Witten theory and the Fan-Jarvis-Ruan-Witten theory. The authors survey recently developed wall-crossing approach to the Landau-Ginzburg/Calabi-Yau correspondence for a quintic threefold.

Boundedness of the period maps and global Torelli theorem

Recently, we proved the Griffiths conjecture on the boundedness of the period maps, and then used it to prove global Torelli theorem on the Torelli space under certain natural conditions. This paper serves as a review of these results and an introduction to the main ideas behind the proofs.