The Abelian/Nonabelian correspondence and Frobenius manifolds

We propose an approach via Frobenius manifolds to the study (began in [BCK2] of the relation between rational Gromov–Witten invariants of nonabelian quotients X//G and those of the corresponding “abelianized” quotients X//T, for T a maximal torus in G. The ensuing conjecture expresses the Gromov–Witten potential of X//G in terms of the potential of X//T. We prove this conjecture when the nonabelian quotients are partial flag manifolds.     

Local Gromov–Witten invariants of cubic surfaces via nef toric degeneration

We compute local Gromov–Witten invariants of cubic surfaces at all genera. We use a deformation a of cubic surface to a nef toric surface and the deformation invariance of Gromov–Witten invariants.     

Counting curves on rational surfaces

In [CH3], Caporaso and Harris derive recursive formulas counting nodal plane curves of degree d and geometric genus g in the plane (through the appropriate number of fixed general points). We rephrase their arguments in the language of maps, and extend them to other rational surfaces, and other specified intersections with a divisor. As applications, (i) we count irreducible curves on Hirzebruch surfaces in a fixed divisor class and of fixed geometric genus, (ii) we compute the higher-genus Gromov–Witten invariants of (or equivalently, counting curves of any genus and divisor class on) del Pezzo surfaces of degree at least 3. In the case of the cubic surface in (ii), we first use a result of Graber to enumeratively interpret higher-genus Gromov–Witten invariants of certain K-nef surfaces, and then apply this to a degeneration of a cubic surface. Received: 30 June 1999 / Revised version: 1 January 2000     

Landau–Ginzburg/Calabi–Yau correspondence for quintic three-folds via symplectic transformations

We compute the recently introduced Fan–Jarvis–Ruan–Witten theory of W-curves in genus zero for quintic polynomials in five variables and we show that it matches the Gromov–Witten genus-zero theory of the quintic three-fold via a symplectic transformation. More specifically, we show that the J-function encoding the Fan–Jarvis–Ruan–Witten theory on the A-side equals via a mirror map the I-function embodying the period integrals at the Gepner point on the B-side. This identification inscribes the physical Landau–Ginzburg/Calabi–Yau correspondence within the enumerative geometry of moduli of curves, matches the genus-zero invariants computed by the physicists Huang, Klemm, and Quackenbush at the Gepner point, and yields via Givental’s quantization a prediction on the relation between the full higher genus potential of the quintic three-fold and that of Fan–Jarvis–Ruan–Witten theory.     

Gromov–Witten Invariants of Blow-ups Along Surfaces

In this paper, using the gluing formula of Gromov–Witten invariants for symplectic cutting developed by Li and Ruan, we established some relations between Gromov–Witten invariants of a semipositive symplectic manifold M and its blow-ups along a smooth surface.     

The quantum orbifold cohomology of weighted projective spaces

We calculate the small quantum orbifold cohomology of arbitrary weighted projective spaces. We generalize Givental’s heuristic argument, which relates small quantum cohomology to S ¹-equivariant Floer cohomology of loop space, to weighted projective spaces and use this to conjecture an explicit formula for the small J-function, a generating function for certain genus-zero Gromov–Witten invariants. We prove this conjecture using a method due to Bertram. This provides the first non-trivial example of a family of orbifolds of arbitrary dimension for which the small quantum orbifold cohomology is known. In addition we obtain formulas for the small J-functions of weighted projective complete intersections satisfying a combinatorial condition; this condition naturally singles out the class of orbifolds with terminal singularities.     

Geometry as seen by string theory

This is an introductory review of the topological string theory from physicist’s perspective. I start with the definition of the theory and describe its relation to the Gromov–Witten invariants. The BCOV holomorphic anomaly equations, which generalize the Quillen anomaly formula, can be used to compute higher genus partition functions of the theory. The open/closed string duality relates the closed topological string theory to the Chern–Simons gauge theory and the random matrix model. As an application of the topological string theory, I discuss the counting of bound states of D-branes.     

Quantum Cohomology of Minuscule Homogeneous Spaces

We study the quantum cohomology of (co)minuscule homogeneous varieties under a unified perspective. We show that three points Gromov–Witten invariants can always be interpreted as classical intersection numbers on auxiliary varieties. Our main combinatorial tools are certain quivers, in terms of which we obtain a quantum Chevalley formula and a higher quantum Poincaré duality. In particular, we compute the quantum cohomology of the two exceptional minuscule homogeneous varieties. DOI: .     

Quantum cohomology of <Emphasis Type="Italic">G/P</Emphasis> and homology of affine Grassmannian

Let G be a simple and simply-connected complex algebraic group, P ⊂ G a parabolic subgroup. We prove an unpublished result of D. Peterson which states that the quantum cohomology QH ^*(G/P) of a flag variety is, up to localization, a quotient of the homology H _*(Gr _G ) of the affine Grassmannian Gr _G of G. As a consequence, all three-point genus-zero Gromov–Witten invariants of G/P are identified with homology Schubert structure constants of H _*(Gr _G ), establishing the equivalence of the quantum and homology affine Schubert calculi.     

Relative Gromov-Witten invariants and the mirror formula

Let X be a smooth complex projective variety, and let be a smooth very ample hypersurface such that is nef. Using the technique of relative Gromov-Witten invariants, we give a new short and geometric proof of (a version of) the “mirror formula”, i.e. we show that the generating function of the genus zero 1-point Gromov-Witten invariants of Y can be obtained from that of X by a certain change of variables (the so-called “mirror transformation”). Moreover, we use the same techniques to give a similar expression for the (virtual) numbers of degree-d plane rational curves meeting a smooth cubic at one point with multiplicity 3d, which play a role in local mirror symmetry. Received: 11 July 2001 / Published online: 4 February 2003 Funded by the DFG scholarships Ga 636/1–1 and Ga 636/1–2.     

Variations of mixed Hodge structure, Higgs fields, and quantum cohomology

Following C. Simpson, we show that every variation of graded-polarized mixed Hodge structure defined over ℚ carries a natural Higgs bundle structure which is invariant under the ℂ^* action studied in [20]. We then specialize our construction to the context of [6], and show that the resulting Higgs field θ determines (and is determined by) the Gromov–Witten potential of the underlying family of Calabi–Yau threefolds. Received: 14 February 2000     

A General Fredholm Theory II: Implicit Function Theorems

This is the second paper in a series introducing a generalized Fredholm theory in a new class of smooth spaces called polyfolds. In general, these spaces are not locally homeomorphic to open sets in Banach spaces. The current paper develops the Fredholm theory in M-polyfold bundles. It consists of a transversality and a perturbation theory. In upcoming papers the generalized Fredholm theory will be applied to the Floer Theory, the Gromov–Witten Theory and the Symplectic Field Theory H.H.’s research partially supported by NSF grant DMS-0603957. K.W.’s research partially supported by NSF grant DMS-0606588.     

Moduli Spaces of Higher Spin Curves and Integrable Hierarchies

We prove the genus zero part of the generalized Witten conjecture, relating moduli spaces of higher spin curves to Gelfand–Dickey hierarchies. That is, we show that intersection numbers on the moduli space of stable r-spin curves assemble into a generating function which yields a solution of the semiclassical limit of the KdV _r equations. We formulate axioms for a cohomology class on this moduli space which allow one to construct a cohomological field theory of rank r–1 in all genera. In genus zero it produces a Frobenius manifold which is isomorphic to the Frobenius manifold structure on the base of the versal deformation of the singularity A _r–1. We prove analogs of the puncture, dilaton, and topological recursion relations by drawing an analogy with the construction of Gromov–Witten invariants and quantum cohomology.     

Matroids and quotients of spheres

For any linear quotient of a sphere, where is an elementary abelian p–group, there is a corresponding representable matroid which only depends on the isometry class of X. When p is 2 or 3 this correspondence induces a bijection between isometry classes of linear quotients of spheres by elementary abelian p–groups, and matroids representable over Not only do the matroids give a great deal of information about the geometry and topology of the quotient spaces, but the topology of the quotient spaces point to new insights into some familiar matroid invariants. These include a generalization of the Crapo–Rota critical problem inequality and an unexpected relationship between and whether or not the matroid is affine. Received: 7 February 2001; in final form: 30 October 2001/ Published online: 29 April 2002     

New symplectic 4–manifolds with <Emphasis Type="Italic">b</Emphasis><Subscript>+</Subscript>=1

In 1995 Dusa McDuff and Dietmar Salamon conjectured the existence of symplectic 4–manifolds (X,ω) which satisfy b₊=1, K²=0, K·ω>0, and which fail to be of Lefschetz type. This is equivalent to finding a symplectic, homology T²×S² manifold with nontorsion canonical class and a cohomology ring which is not isomorphic to the cohomology ring of T²×S². They needed such examples to complete a list of possible symplectic 4–manifolds with b₊=1. In that same year Tian-Jun Li and Ai-ko Liu, working from a different point of view, questioned whether there existed symplectic 4–manifolds with b₊=1 with Seiberg- Witten invariants that did not depend on the chamber structure of the moduli space. The purpose of this paper is to construct an infinite number of examples which satisfy both requirements. The author was partially supported by NSF grant DMS-0406021.     

A comparison theorem for Gromov–Witten invariants in the symplectic category

We exploit the geometric approach to the virtual fundamental class, due to Fukaya–Ono and Li–Tian, to compare Gromov–Witten invariants of a symplectic manifold and a symplectic submanifold whenever all constrained stable maps to the former are contained in the latter to first order. Various special cases of the comparison theorem in this paper have long been used in the algebraic category; some of them have also appeared in the symplectic setting. Combined with the inherent flexibility of the symplectic category, the main theorem leads to a confirmation of Pandharipande?s Gopakumar–Vafa prediction for GW-invariants of Fano classes in 6-dimensional symplectic manifolds. The proof of the main theorem uses deformations of the Cauchy–Riemann equation that respect the submanifold and Carleman Similarity Principle for solutions of perturbed Cauchy–Riemann equations. In a forthcoming paper, we apply a similar approach to relative Gromov–Witten invariants and the absolute/relative correspondence in genus 0.     

Euler–Poincaré formula in equal characteristic¶under ordinariness assumptions

Let X be an irreducible smooth projective curve over an algebraically closed field of characteristic p>0. Let ? be either a finite field of characteristic p or a local field of residue characteristic p. Let F be a constructible étale sheaf of $\BF$-vector spaces on X. Suppose that there exists a finite Galois covering π:Y→X such that the generic monodromy of π^* F is pro-p and Y is ordinary. Under these assumptions we derive an explicit formula for the Euler–Poincaré characteristic χ(X,F) in terms of easy local and global numerical invariants, much like the formula of Grothendieck–Ogg–Shafarevich in the case of different characteristic. Although the ordinariness assumption imposes severe restrictions on the local ramification of the covering π, it is satisfied in interesting cases such as Drinfeld modular curves. Received: 7 December 1999 / Revised version: 28 January 2000     

Microlocal branes are constructible sheaves

Let X be a compact real analytic manifold, and let T* X be its cotangent bundle. In a recent paper with Zaslow (J Am Math Soc 22:233–286, 2009), we showed that the dg category Sh _c (X) of constructible sheaves on X quasi-embeds into the triangulated envelope F(T* X) of the Fukaya category of T* X. We prove here that the quasi-embedding is in fact a quasi-equivalence. When X is a complex manifold, one may interpret this as a topological analogue of the identification of Lagrangian branes in T* X and regular holonomic D_X{{\mathcal D}_X} -modules developed by Kapustin (A-branes and noncommutative geometry, arXiv:hep-th/0502212) and Kapustin and Witten (Commun Number Theory Phys 1(1):1–236, 2007) from a physical perspective. As a concrete application, we show that compact connected exact Lagrangians in T* X (with some modest homological assumptions) are equivalent in the Fukaya category to the zero section. In particular, this determines their (complex) cohomology ring and homology class in T* X, and provides a homological bound on their number of intersection points. An independent characterization of compact branes in T* X has recently been obtained by Fukaya et al. (Invent Math 172(1):1–27, 2008).     

A generalization of the Razmyslov-Procesi theorem

It is proved that all relations between the invariants of several n x n-matrices over an infinite field of arbitrary characteristic follow from σ_n+1,σ_n+2,... where σ_i is the ith coefficient of a characteristic polynomial extended to matrices of any order ≥i. Similarly, all relations between the concomitants are implied by X_n+1, X_n+2, …, where X_i is a characteristic polynomial in the general n x n-matrix, also extended to matrices of any order. Supported by RFFR grant No. 95-01-00513. Translated fromAlgebra i Logika, Vol. 35, No. 4, pp. 433–457, July–August, 1996.