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 nonab...

We give a construction of the moduli space of stable maps to the classifying stack Bμr of a cyclic group by a sequence of r-th root constructions on M0,n. We prove a closed formula for the total Chern class of μreigenspaces of the Hodge bundle, and thus of the obstruction bundle of the genus zero Gromov-Witten theory of stacks of the form [C/μr ]. We deduce linear recursions for all genus-zero ...

We prove that any sequence of genus zero symplectic vortices with vanishing area has a subsequence that converges to a holomorphic map with zero average moment map. Conversely, any such map is the limit of such a sequence. We use these results to equate the gauged Gromov-Witten invariants for sufficiently small area with the invariant part of equivariant Gromov-Witten invariants, for convex tar...

We use analytic methods to define Family Gromov-Witten Invariants for Kähler surfaces. We prove that these are well-defined invariants of the deformation class of the Kähler structure. Gromov-Witten invariants are counts of holomorphic curves in a symplectic manifold X. To define them using the analytic approach one chooses an almost complex structure J compatible with the symplectic structure ...

We compute the local Gromov-Witten invariants of the “closed vertex”, that is, a configuration of three P’s meeting in a single triple point in a Calabi-Yau threefold. The method is to express the local invariants of the vertex in terms of ordinary Gromov-Witten invariants of a certain blowup of P and then to compute those invariants via the geometry of the Cremona transformation.

We show that certain hypergeometric series used to formulate mirror symmetry for Calabi-Yau hypersurfaces, in string theory and algebraic geometry, satisfy a number of interesting properties. Many of these properties are used in separate papers to verify the BCOV prediction for the genus one Gromov-Witten invariants of a quintic threefold and more generally to compute the genus one Gromov-Witte...

In this paper we study the geometry of the Severi varieties parametrizing curves on the rational ruled surface Fn. We compute the number of such curves through the appropriate number of fixed general points on Fn (Theorem 1.1), and the number of such curves which are irreducible (Theorem 1.3). These numbers are known as Severi degrees; they are the degrees of unions of components of the Hilbert...

Let X be a smooth projective variety. It was shown by A. Gathmann that in case X is a very ample hypersurface of some other smooth projective variety Y , the genus-0 (so-called “restricted” and “unrestricted” with a sufficiently low number of marked points) Gromov-Witten invariants of X can be computed in terms of genus-0 Gromov-Witten invariants of Y. The purpose of this article is to generali...