Due to the orbifold singularities, the intersection numbers on the moduli space of curves Mg, n are in general rational numbers rather than integers. We study the properties of the denominators of these intersection numbers and their relationship with the orders of automorphism groups of stable curves. We also present a conjecture about a multinomial type numerical property for a general class of Hodge integrals.

Let X be a compact Khler manifold and X a subvariety of X with higher co-dimension. The aim is to study complete constant scalar curvature Khler metrics on non-compact Khler manifold X with Poincar--Mok--Yau asymptotic property (see Definition\ref {def}). In this paper, the methods of Calabi's ansatz and the moment construction are used to provide some special examples of such metrics.

Spin polynomial invariants, as denned by Pidstrigatch and Tyurin, are used to define differentiable invariants for Dolgachev surfaces. In order to define and compute these invariants, some machinery is developed to study the wall structure for Spin polynomials, to deal with the fact that the moduli spaces have the wrong dimension and to handle nonreduced structures. The invariants are then computed.

In this note, we give a list of Calabi-Yau hypersurfaces in weighted projective 4-spaces with the property that a hypersurface contains naturally a pencil of K3 variety. For completeness we also obtain a similar list in the case K3 hypersurfaces in weighted projective 3-spaces. The first list significantly enlarges the list of K3-fibrations of\KlemmLercheMayr~ which has been obtained on some assumptions on the weights. Our lists are expected to correspond to examples of the so-called heterotic-type II duality\KachruVafa\KachruSilverstein.

Let X be a compact K\" ahler manifold and X be a simple normal crossing divisor. If X is the support of some effective X -ample divisor, we show <div class="gsh_dspfr"> X

By a case-free approach we give a precise description of the closure of a Steinberg fiber within a twisted wonderful compactification of a simple linear algebraic group. In the nontwisted case this description was earlier obtained by the first author.

Let X be a smooth complex projective variety. Using a construction devised by Gathmann, we present a recursive formula for some of the Gromov–Witten invariants of X. We prove that, when X is homogeneous, this formula gives the number of osculating rational curves at a general point of a general hypersurface of X. This generalizes the classical well known pairs of inflection (asymptotic) lines for surfaces in P^3 of Salmon, as well as Darboux’s 27 osculating conics.

We survey the theory of complex manifolds that are related to Riemann surface, Hodge theory, Chern class, Kodaira embedding and HirzebruchRiemannRoch, and some modern development of uniformization theorems, KhlerEinstein metric and the theory of DonaldsonUhlenbeckYau on Hermitian YangMills connections. We emphasize mathematical ideas related to physics. At the end, we identify possible future research directions and raise some important open questions.

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We use the Atiyah-Bott-Segal-Singer Lefschetz fixed point formula to calculate the Hirzebruch ?_y genus X_y(S^[n]), where S^[n] is the Hilbert scheme of points of length n of a surface S. Combinatorial interpretation of the weights of the fixed points of the natural torus action on (²)^[n] is used. This is the first step to prove a conjectural formula about the elliptic genus of the Hilbert schemes.

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This paper is a contribution to the general problem of giving an explicit description of the basic locus in the reduction modulo p of Shimura varieties. Motivated by\cite {Vollaard-Wedhorn} and\cite {Rapoport-Terstiege-Wilson}, we classify the cases where the basic locus is (in a natural way) the union of classical Deligne-Lusztig sets associated to Coxeter elements. We show that if this is satisfied, then the Newton strata and Ekedahl-Oort strata have many nice properties.

Let V be a convex vector bundle over a smooth projective manifold X, and let Y be the subset of X which is the zero locus of a regular section of V. This mostly expository paper discusses a conjecture which relates the virtual fundamental classes of X and Y. Using an argument due to Gathmann, we prove a special case of the conjecture. The paper concludes with a discussion of how our conjecture relates to the mirror theorems in the literature.

In this paper, we study the zero loci of local systems of the form \delta\Pi , where \delta\Pi is the period sheaf of the universal family of CY hypersurfaces in a suitable ambient space \delta\Pi , and \delta\Pi is a given differential operator on the space of sections \delta\Pi . Using earlier results of three of the authors and their collaborators, we give several different descriptions of the zero locus of \delta\Pi . As applications, we prove that the locus is algebraic and in some cases, non-empty. We also give an explicit way to compute the polynomial defining equations of the locus in some cases. This description gives rise to a natural stratification to the zero locus.

We study various geometrical quantities for CalabiYau varieties realized as cones over Gorenstein Fano varieties, obtained as toric varieties from reflexive polytopes in various dimensions. Focus is made on reflexive polytopes up to dimension 4 and the minimized volumes of the SasakiEinstein base of the corresponding CalabiYau cone are calculated. By doing so, we conjecture new bounds for the SasakiEinstein volume with respect to various topological quantities of the corresponding toric varieties. We give interpretations about these volume bounds in the context of associated field theories via the AdS/CFT correspondence.

We describe the proof that the period map from the Torelli space of Calabi-Yau manifolds to the classifying space of polarized Hodge structures is an embedding. The proof is based on the constructions of holomorphic affine structure on the Teichmller space and Hodge metric completion of the Torelli space. A canonical global holomorphic section of the holomorphic (n, 0) class on the Teichmller space is constructed.

In this note we derive some consequences from the Mario-Vafa formula and the cut-and-join equation, these include unified simple proofs of the \lambda_g conjecture and some other Hodge integral identities. We also describe a proof of the ELSV formula relating Hurwitz numbers and Hodge integrals by using the cut-and-join equation, following our proof of the Mario-Vafa formula.

Tautological systems are Picard-Fuchs type systems arising from varieties with large symmetry. In this survey, we discuss recent progress on the study of tautological systems. This includes tautological systems for vector bundles, a new construction of Jacobian rings for homogenous vector bundles, and relations between period integrals and zeta functions.

This is the first of a sequence of papers proving the quantum invariance under ordinary flops over an arbitrary smooth base. In this first part, we determine the defect of the cup product under the canonical correspondence and show that it is corrected by the small quantum product attached to the extremal ray. We then perform various reductions to reduce the problem to the local models. In Part II, we develop a quantum Leray--Hirsch theorem and use it to show that the big quantum cohomology ring is invariant under analytic continuations in the Kähler moduli space for ordinary flops of splitting type. In Part III, together with F. Qu, we remove the splitting condition by developing a quantum splitting principle, and hence solve the problem completely.

In this paper, we study effective recursion formulae for computing intersection numbers of mixed \psi and \psi classes on moduli spaces of curves. By using the celebrated WittenKontsevich theorem, we generalize MulaseSafnuk form of Mirzakhani's recursion and prove a recursion formula of higher WeilPetersson volumes. We also present recursion formulae to compute intersection pairings in the tautological rings of moduli spaces of curves.

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 K\" ahler 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.

Let Λ be a numerical semigroup, C⊆An the monomial curve singularity associated to Λ, and T its tangent cone. In this paper we provide a sharp upper bound for the least positive integer in Λ in terms of the codimension and the maximum degree of the equations of T, when T is not a complete intersection. A special case of this result settles a question of J. Herzog and D. Stamate.

Motivated by mirror symmetry, we study certain integral representations of solutions to the Gelfand-Kapranov-Zelevinsky (GKZ) hypergeometric system. Some of these solutions arise as period integrals for Calabi-Yau manifolds in mirror symmetry. We prove that for a suitable compactification of the parameter space, there exist certain special boundary points, which we called maximal degeneracy points, at which all solutions but one become singular.

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.

In [LL-Y1, III: Sec. 5.4] on mirror principle, a method was developed to compute the integral \int_ {X} ^{st} e^{H\cdot t}\cap {\mathbf 1} _d for a flag manifold $ X=\Fl_ {r_1,..., r_I}({\Bbb C}^ n) $ via an extended mirror principle diagram. This method turns the required localization computation on the augmented moduli stack $\bar {\cal M} _ {0, 0}(\CP^ 1\times X) $ of stable maps to a localization computation on a hyper-Quot-scheme $\HQuot ({\cal E}^ n) $. In this article, the detail of this localization computation on $\HQuot ({\cal E}^ n) $ is carried out. The necessary ingredients in the computation, notably, the \int_ {X} ^{st} e^{H\cdot t}\cap {\mathbf 1} _d -fixed-point components and the distinguished ones \int_ {X} ^{st} e^{H\cdot t}\cap {\mathbf 1} _d in $\HQuot ({\cal E}^ n) $, the \int_ {X} ^{st} e^{H\cdot t}\cap {\mathbf 1} _d -equivariant Euler class of \int_ {X} ^{st} e^{H\cdot t}\cap {\mathbf 1} _d in $\HQuot ({\cal E}^ n) $, and a push-forward formula of cohomology classes involved in the problem from the total space of a restrictive flag manifold bundle to its base manifold are given. With these, an exact expression of \int_ {X} ^{st} e^{H\cdot t}\cap {\mathbf 1} _d is obtained. Comments on the Hori-Vafa conjecture are given in the end.