New recursions for genus-zero Gromov-Witten invariants

New relations among the genus-zero Gromov-Witten invariants of a complex projective manifold X are exhibited. When the cohomology of X is generated by divisor classes and classes “with vanishing one-point invariants,” the relations determine many-point invariants in terms of one-point invariants.     

Hamiltonian Gromov-Witten invariants

In this paper we introduce invariants of semi-free Hamiltonian actions of S¹ on compact symplectic manifolds using the space of solutions to certain gauge theoretical equations. These equations generalise both the vortex equations and the holomorphicity equation used in Gromov-Witten theory. In the definition of the invariants we combine ideas coming from gauge theory and the ideas underlying the construction of Gromov-Witten invariants.     

Gromov-Witten invariants and quantum cohomology

This article is an elaboration of a talk given at an international conference on Operator Theory, Quantum Probability, and Noncommutative Geometry held during December 20–23, 2004, at the Indian Statistical Institute, Kolkata. The lecture was meant for a general audience, and also prospective research students, the idea of the quantum cohomology based on the Gromov-Witten invariants. Of course there are many important aspects that are not discussed here. Dedicated to Professor K B Sinha on the occasion of his 60th birthday     

Gromov-Witten invariants on Grassmannians

We prove that any three-point genus zero Gromov-Witten invariant on a type Grassmannian is equal to a classical intersection number on a two-step flag variety. We also give symplectic and orthogonal analogues of this result; in these cases the two-step flag variety is replaced by a sub-maximal isotropic Grassmannian. Our theorems are applied, in type , to formulate a conjectural quantum Littlewood-Richardson rule, and in the other classical Lie types, to obtain new proofs of the main structure theorems for the quantum cohomology of Lagrangian and orthogonal Grassmannians.

     

Gromov-Witten invariants of jumping curves

Given a vector bundle on a smooth projective variety , we can define subschemes of the Kontsevich moduli space of genus-zero stable maps parameterizing maps such that the Grothendieck decomposition of has a specified splitting type. In this paper, using a ``compactification' of this locus, we define Gromov-Witten invariants of jumping curves associated to the bundle . We compute these invariants for the tautological bundle of Grassmannians and the Horrocks-Mumford bundle on . Our construction is a generalization of jumping lines for vector bundles on . Since for the tautological bundle of the Grassmannians the invariants are enumerative, we resolve the classical problem of computing the characteristic numbers of unbalanced scrolls.

     

Gromov-Witten invariants in algebraic geometry

Gromov-Witten invariants for arbitrary non-singular projective varieties and arbitrary genus are constructed using the techniques from [K. Behrend, B. Fantechi. The Intrinsic Normal Cone.] Oblatum 26-II-1996 & 27-VI-1996     

Reconstruction theorems for Gromov-Witten invariants

We prove a Reconstruction Theorem for (ordinary) Gromov-Witten invariants which improves the First Reconstruction Theorem of Kontsevich and Manin for manifolds whose Picard number is not one. In some cases our Reconstruction Theorem gives 1-point reconstruction.We discuss some interesting examples in detail, and finally we describe four applications: rational surfaces, Fano threefolds, the blow-up of the projective space along a linear subspace, and the non-Fano moduli space of curves .     

Gromov-Witten invariants and pseudo symplectic capacities

We introduce the concept of pseudo symplectic capacities which is a mild generalization of that of symplectic capacities. As a generalization of the Hofer-Zehnder capacity we construct a Hofer-Zehnder type pseudo symplectic capacity and estimate it in terms of Gromov-Witten invariants. The (pseudo) symplectic capacities of Grassmannians and some product symplectic manifolds are computed. As applications we first derive some general nonsqueezing theorems that generalize and unite many previous versions then prove the Weinstein conjecture for cotangent bundles over a large class of symplectic uniruled manifolds (including the uniruled manifolds in algebraic geometry) and also show that any closed symplectic submanifold of codimension two in any symplectic manifold has a small neighborhood whose Hofer-Zehnder capacity is less than a given positive number. Finally, we give two results on symplectic packings in Grassmannians and on Seshadri constants. Partially supported by the NNSF 10371007 of China and the Program for New Century Excellent Talents of the Education Ministry of China.     

Counting maximal subbundles via Gromov-Witten invariants

In this article we explicitly compute the number of maximal subbundles of rank k of a general stable bundle of rank r and degree d over a smooth projective curve C of genus g2 over , when the dimension of the quot scheme of maximal subbundles is zero. Our method is to describe these numbers purely in terms of the Gromov-Witten invariants of the Grassmannian and use the formula of Vafa and Intriligator to compute them.     

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.     

Gromov-Witten invariants of blow-ups along points and curves

Abstract. In this paper, using the gluing formula of Gromov-Witten invariants under symplectic cutting, due to Li and Ruan, we studied the Gromov-Witten invariants of blow-ups at a smooth point or along a smooth curve. We established some relations between Gromov-Witten invariants of M and its blow-ups at a smooth point or along a smooth curve. Received February 4, 1999     

Local and relative Gromov-Witten invariants of the projective plane

Let B be a line or a smooth conic in ². We give a recursive formula for certain linear combinations of the genus 0 relative Gromov-Witten invariants of (²,B). After change of sign, this is equivalent to the WDVV equation for the equivariant local Gromov-Witten invariants of ² in ᵊA(–B). We show that these invariants coincide up to sign. Mathematics Subject Classification (2000):14N35, 14N10     

Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties

In this paper, we define the virtual moduli cycle of moduli spaces with perfect tangent-obstruction theory. The two interesting moduli spaces of this type are moduli spaces of vector bundles over surfaces and moduli spaces of stable morphisms from curves to projective varieties. As an application, we define the Gromov-Witten invariants of smooth projective varieties and prove all its basic properties.

     

Local Gromov-Witten invariants and tautological sheaves on Hilbert schemes

Abstract We study the local Gromov-Witten invariants of O(k)⊕O(-k-2) → P1 by localization techniques and the Marino-Vafa formula, using suitable circle actions. They are identified with the equivariant Riemann-Roch indices of some power of the determinant of the tautological sheaves on the Hilbert schemes of points on the affine plane. We also compute the corresponding Gopakumar-Vafa invariants and make some conjectures about them.     

Finite group action and Gromov-Witten invariants in symplectic four-manifolds

Let a finite group act semi-freely on a closed symplectic four-manifold with a 2-dimensional fixed point set. Then we show that the relative Gromov-Witten invariants are the same as the invariants on the quotient set-up with respect to the fixed point set.     

Orbifold Gromov–Witten Invariants of Weighted Blow-up at Smooth Points

In this paper, one considers the change of orbifold Gromov–Witten invariants under weighted blow-up at smooth points. Some blow-up formula for Gromov–Witten invariants of symplectic orbifolds is proved. These results extend the results of manifolds case to orbifold case.     

Hamiltonian / Gauge theoretical Gromov-Witten invariants of toric varieties

We present a purely algebraic approach to the Hamiltonian / Gauge theoretical invariants associated to torus actions on affine spaces. Secondly, we address the issue of computing the invariants: a localization and a genus recursion formula are deduced. Partially supported by: EAGER - European Algebraic Geometry Research Training Network, contract No. HPRN-CT-2000-00099 (BBW).     

Genus-decreasing relation of Gromov-Witten invariants for surfaces under blow-up

Using the degeneration formula, we study the change of Gromov-Witten invariants under blow-up for symplectic 4-manifolds and obtain a genus-decreasing relation of Gromov-Witten invariant of symplectic four manifold under blow-up.     

A genus-4 topological recursion relation for Gromov-Witten invariants

In this paper,we give a new genus-4 topological recursion relation for Gromov-Witten invariants of compact symplectic manifolds via Pixton’s relations on the moduli space of curves.As an application,we prove that Pixton’s relations imply a known topological recursion relation on Mg,1 for genus g≤4.