Related papers: Relative Gromov-Witten Invariants

We construct invariants under deformation of real symplectic 4-manifolds. These invariants are obtained by counting three different kinds of real rational J-holomorphic curves which realize a given homology class and pass through a given…

In this paper, we propose a conjecture that clarifies the relationship between the number of degree d elliptic curves in complex four-dimensional projective Fano hypersurfaces and their degree d elliptic Gromov-Witten (GW) invariants. The…

Here we review background in differential topology related to the calculation of an euler characteristic, and background on localization in equivariant cohomology. We then outline Gromov-Witten invariants in algebraic geometry and give…

We outline the construction of invariants of Hamiltonian group actions on symplectic manifolds. These invariants can be viewed as an equivariant version of Gromov-Witten invariants. They are derived from solutions of a PDE involving the…

The quantum cohomology algebra of the (full) flag manifold is a fundamental example in quantum cohomology theory, with connections to combinatorics, algebraic geometry, and integrable systems. Using a differential geometric approach, we…

We establish a product formula for Gromov-Witten invariants for closed, connected, relatively semi-positive Hamiltonian fibrations over any symplectic base. Furthermore, we show that the fibration projection induces a locally trivial…

We give explicit formulas for the multipoint series of Gromov-Witten invariants of CP^1. We use the recursions of the Toda hierarchy to calculate these in degree 0, and the Toda equation and the degree 0 formulas to extend the calculation…

Let $\ix$ be a smooth Deligne-Mumford stack over the complex numbers. One can define twisted orbifold Gromov-Witten invariants of $\ix$ by considering multiplicative invertible characteristic classes of various bundles on the moduli spaces…

The Gopakumar-Vafa conjecture predicts that the Gromov-Witten invariants of a Calabi-Yau 3-fold can be canonically expressed in terms of integer invariants called BPS numbers. Using the methods of symplectic Gromov-Witten theory, we prove…

A projected Gromov-Witten variety is the union of all rational curves of fixed degree that meet two opposite Schubert varieties in a homogeneous space X = G/P. When X is cominuscule we prove that the map from a related Gromov-Witten variety…

We introduce the process of symplectic reduction along a submanifold as a uniform approach to taking quotients in symplectic geometry. This construction holds in the categories of smooth manifolds, complex analytic spaces, and complex…

We construct the Gromov-Witten invariants of moduli of stable morphisms to $\Pf$ with fields. This is the all genus mathematical theory of the Guffin-Sharpe-Witten model, and is a modified twisted Gromov-Witten invariants of $\Pf$. These…

We study Gromov-Witten invariants on the blow-up of P^n at a point, which is probably the simplest example of a variety whose moduli spaces of stable maps do not have the expected dimension. It is shown that many of these invariants can be…

We give a summation over graphs type formula for the permutation-equivariant K-theoretic Gromov-Witten total potential of a projective manifold X in terms of cohomological Gromov-Witten (GW) invariants of X. We achieve this by describing…

Let $(m_1, m_2)$ be a pair of positive integers. Denote by $\mathbb{P}^1$ the complex projective line, and by $\mathbb{P}^1_{m_1,m_2}$ the orbifold complex projective line obtained from $\mathbb{P}^1$ by adding $\mathbb{Z}_{m_1}$ and…

In this paper, we explicitly prove that statistical manifolds, related to exponential families and with flat structure connection have a Frobenius manifold structure. This latter object, at the interplay of beautiful interactions between…

These are notes of lectures given at the NATO Summer School, Montreal 1995. Taubes's recent spectacular work setting up a correspondence between $J$-holomorphic curves in symplectic 4-manifolds and solutions of the Seiberg-Witten equations…

We prove a splitting formula that reconstructs the logarithmic Gromov- Witten invariants of simple normal crossing varieties from the punctured Gromov- Witten invariants of their irreducible components, under the assumption of the gluing…

Using reduced Gromov-Witten theory, we define new invariants which capture the enumerative geometry of curves on holomorphic symplectic 4-folds. The invariants are analogous to the BPS counts of Gopakumar and Vafa for Calabi-Yau 3-folds,…

After reviewing recent results on symplectic Lefschetz pencils and symplectic branched covers of CP^2, we describe a new construction of maps from symplectic manifolds of any dimension to CP^2 and the associated monodromy invariants. We…