Witten's gauged linear sigma model [Wi1] is one of the universal frameworks or structures that lie behind stringy dualities. Its A-twisted moduli space at genus 0 case has been used in the Mirror Principle [LLY] that relates Gromov-Witten invariants and mirror symmetry computations. In this paper the A-twisted moduli stack for higher genus curves is defined and systematically studied. It is proved that such a moduli stack is an Artin stack. For genus 0, it has the A-twisted moduli space of [MP] as the coarse moduli space. The detailed proof of the regularity of the collapsing morphism by Jun Li in [LLY: I and II] can be viewed as a natural morphism from the moduli stack of genus 0 stable maps to the A-twisted moduli stack at genus 0.

Soit G un groupe algbrique complexe quasi-simple et G/P une varit de drapeaux partiels. La projection sur G/P des varits de Richardson (de la varit des drapeaux complets) forment une stratification de G/P. Nous montrons que les relations dadhrence des varits de Richardson projetes correspondent celles dun certain sous-ensemble de varits de Schubert sur la varit de drapeaux affine de G. Nous comparons aussi les classes de cohomologie quivariante et de K-thorie de ces deux stratifications. Notre travail gnralise celui de Knutson, Lam et Speyer pour la grassmannienne de type A.

We consider surfaces of geometric genus 3 with the property that their transcendental cohomology splits into 3 pieces, each piece coming from a K3 surface. For certain families of surfaces with this property, we can show there is a similar splitting on the level of Chow groups (and Chow motives).

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We study the limit of Calabi-Yau modular forms, and in particular, those resulting in classical modular forms. We then study two parameter families of elliptically fibred Calabi-Yau fourfolds and describe the modular forms arising from the degeneracy loci. In the case of elliptically fibred Calabi-Yau threefolds our approach gives a mathematical proof of many observations about modularity properties of topological string amplitudes starting with the work of Candelas, Font, Katz and Morrison. In the case of Calabi-Yau fourfolds we derive new identities not computed before.

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Let Y be the zero loci of a regular section of a convex vector bundle Y over Y . We provide a proof of a conjecture of Cox, Katz and Lee for the virtual class of the genus zero moduli of stable maps to Y . This in turn yields the expected relationship between Gromov-Witten theories of Y and Y which together with Mirror Theorems allows for the calculation of enumerative invariants of Y inside of Y .

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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.

We establish the asymptotic expansion of certain integrals of classes on moduli spaces of curves , when either the or goes to infinity. Our main tools are cut-join type recursion formulae from the WittenKontsevich theorem, as well as asymptotics of solutions to the first Painlev equation. We also raise a conjecture on large genus asymptotics for -point functions of classes and partially verify the positivity of coefficients in generalized Mirzakhanis formula of higher WeilPetersson volumes.

This is an expanded version of the third author's lecture in String-Math 2015 at Sanya. It summarizes some of our works in quantum cohomology. After reviewing the quantum Lefschetz and quantum Leray--Hirsch, we discuss their applications to the functoriality properties under special smooth flops, flips and blow-ups. Finally, for conifold transitions of Calabi--Yau 3-folds, formulations for small resolutions (blow-ups along Weil divisors) are sketched.

We give a short proof of Penner-Grushevsky-Schumacher-Trapanis large genus asymptotics of Weil-Petersson volumes of moduli spaces of curves. We also study asymptotic expansions for certain integrals of pure \psi classes and answer a question of Mirzakhani on the asymptotic behavior of one-point volume polynomials of moduli spaces of curves.

We introduce the notion of pseudo-N´eron model and give new examples of varieties admitting pseudo-N´eron models other than Abelian varieties. As an application of pseudo-N´eron models, given a scheme admitting a finite morphism to an Abelian scheme over a positive-dimensional base, we prove that for a very general genus-0, degree-d curve in the base with d sufficiently large, every section of the scheme over the curve is contained in a unique section over the entire base.

I will discuss results of three different types in geometry and topology.(1) General vanishing and rigidity theorems of elliptic genera proved by using modular forms, Kac-Moody algebras and vertex operator algebras.(2) The computations of intersection numbers of the moduli spaces of flat connections on a Riemann surface by using heat kernels.(3) The mirror principle about counting curves in Calabi-Yau and general projective manifolds by using hypergeometric series.

This is the second of a sequence of papers proving the quantum invariance for ordinary flops over an arbitrary smooth base. In this paper, we complete the proof of the invariance of the big quantum rings under ordinary flops of splitting type. To achieve that, several new ingredients are introduced. One is a quantum Leray--Hirsch theorem for the local model (a certain toric bundle) which extends the quantum D module of Dubrovin connection on the base by a Picard--Fuchs system of the toric fibers. Nonsplit flops as well as further applications of the quantum Leray--Hirsch theorem will be discussed in subsequent papers. In particular, a quantum splitting principle is developed in Part III which reduces the general ordinary flops to the split case solved here.

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The paper is Part III of our ongoing project to study a case of Crepant Transformation Conjecture: K-equivalence Conjecture for ordinary flops. In this paper we prove the invariance of quantum rings for general ordinary flops, whose local models are certain non-split toric bundles over arbitrary smooth base. An essential ingredient in the proof is a quantum splitting principle, which reduces a statement in Gromov--Witten theory on non-split bundles to the case of split bundles.

Let X be an equivariant embedding of a connected reductive group X over an algebraically closed field X of positive characteristic. Let X denote a Borel subgroup of X . A X -Schubert variety in X is a subvariety of the form $\diag (G)\cdot V $, where X is a X -orbit closure in X . In the case where X is the wonderful compactification of a group of adjoint type, the X -Schubert varieties are the closures of Lusztig's X -stable pieces. We prove that X admits a Frobenius splitting which is compatible with all X -Schubert varieties. Moreover, when X is smooth, projective and toroidal, then any X -Schubert variety in X admits a stable Frobenius splitting along an ample divisors. Although this indicates that X -Schubert varieties have nice singularities we present an example of a non-normal X -Schubert variety in the wonderful compactification of a group of type X . Finally we also extend the Frobenius splitting results to the more general class of X -Schubert varieties.

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In two very detailed, technical, and fundamental works, Jun Li constructed a theory of Gromov-Witten invariants for a singular scheme of the gluing form Y_1\cup_D Y_2 that arises from a degeneration Y_1\cup_D Y_2 and a theory of relative Gromov-Witten invariants for a codimension-1 relative pair Y_1\cup_D Y_2 . As a summit, he derived a degeneration formula that relates a finite summation of the usual Gromov-Witten invariants of a general smooth fiber Y_1\cup_D Y_2 of Y_1\cup_D Y_2 to the Gromov-Witten invariants of the singular fiber Y_1\cup_D Y_2 via gluing the relative pairs Y_1\cup_D Y_2 and Y_1\cup_D Y_2 . The finite sum mentioned above depends on a relative ample line bundle Y_1\cup_D Y_2 on Y_1\cup_D Y_2 . His theory has already applications to string theory and mathematics alike. For other new applications of Jun Li's theory, one needs a refined degeneration formula that depends on a curve class Y_1\cup_D Y_2 in Y_1\cup_D Y_2 or Y_1\cup_D Y_2 , rather than on the line bundle Y_1\cup_D Y_2 . Some monodromy effect has to be taken care of to deal with this. For the simple but useful case of a degeneration Y_1\cup_D Y_2 that arises from blowing up a trivial family Y_1\cup_D Y_2 , we explain how the details of Jun Li's work can be employed to reach such a desired degeneration formula. The related set Y_1\cup_D Y_2 of admissible triples adapted to Y_1\cup_D Y_2 that appears in the formula can be obtained via an analysis on the intersection numbers of relevant cycles and a study of Mori cones that appear in the problem. This set is intrinsically determined by Y_1\cup_D Y_2 and the normal bundle Y_1\cup_D Y_2 of the smooth subscheme Y_1\cup_D Y_2 in Y_1\cup_D Y_2 to be blown up.

For projective conifold transitions between Calabi-Yau threefolds X and Y, with X close to Y in the moduli, we show that the combined information provided by the A model (Gromov--Witten theory in all genera) and B model (variation of Hodge structures) on X, linked along the vanishing cycles, determines the corresponding combined information on Y. Similar result holds in the reverse direction when linked with the exceptional curves.

We realize the moduli spaces of cubic fourfolds with specified group actions as arithmetic quotients of complex hyperbolic balls or type IV symmetric domains, and study their compactifications. We prove the geometric (GIT) compactifications are naturally isomorphic to the Hodge theoretic (Looijenga, in many cases Baily–Borel) compactifications. The key ingredients of the proof are the global Torelli theorem by Voisin, the characterization of the image of the period map given by Looijenga and Laza independently, and the functoriality of Looijenga compactifications proved in the Appendix.

We present a series of results we obtained recently about the intersection numbers of tautological classes on moduli spaces of curves, including a simple formula of the <i>n</i>-point functions for Witten9s classes, an effective recursion formula to compute higher WeilPetersson 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 and intersections.

We define tautological relations for the moduli space of stable maps to a target variety. Using the double ramification cycle formula for target varieties of Janda–Pandharipande–Pixton–Zvonkine, we construct non-trivial tautological relations parallel to Pixton’s double ramification cycle relations for the moduli of curves. Examples and applications are discussed.