Related papers: Absolute vs. Relative Gromov-Witten Invariants

The mirror theorem is generalized to any smooth projective variety X. That is, a fundamental relation between the Gromov-Witten invariants of X and Gromov-Witten invariants of complete intersections Y in X is established.

We compute the quantum cohomology relative to a Lagrangian submanifold in some complete intersections. For quadric hypersurfaces, we also give a full computation of the genus zero open Gromov-Witten invariants.

We show that a holomorphic two-form $\theta$ on a smooth algebraic variety X localizes the virtual fundamental class of the moduli of stable maps $\mgn(X,\beta)$ to the locus where $\theta$ degenerates; it then enables us to define the…

Given a closed orientable Lagrangian surface L in a closed symplectic four-manifold X together with a relative homology class d in H_2 (X, L; Z) with vanishing boundary in H_1 (L; Z), we prove that the algebraic number of J-holomorphic…

In this article, we construct the reduced genus-two Gromov-Witten invariants for certain almost K\"{a}hler manifold $(X, \omega, J)$ such that $J$ is integrable and satisfies some regularity conditions. In particular, the standard…

In this paper, we study the all genus Gromov-Witten theory for any GKM orbifold $X$. We generalize the Givental formula which is studied in the smooth case in \cite{Giv2} \cite{Giv3} \cite{Giv4} to the orbifold case. Specifically, we…

The original proof of the Gromov's non-squeezing theorem [Gro85] is based on pseudo-holomorphic curves. The central ingredient is the compactness of the moduli space of pseudo-holomorphic spheres in the symplectic manifold…

We show that the standard generating functions for genus 0 two-point twisted Gromov-Witten invariants arising from concavex vector bundles over symplectic toric manifolds are explicit transforms of the corresponding one-point generating…

We discuss the GW/DT correspondence for 3-folds in both the absolute and relative cases. Descendents in Gromov-Witten theory are conjectured to be equivalent to Chern characters of the universal sheaf in Donaldson-Thomas theory. Relative…

We extend the definition of relative Gromov--Witten invariants with negative contact orders to all genera. Then we show that relative Gromov--Witten theory forms a partial CohFT. Some cycle relations on the moduli space of stable maps are…

We construct Ionel-Parker's proposed refinement of the standard relative Gromov-Witten invariants in terms of abelian covers of the symplectic divisor and discuss in what sense it gives rise to invariants. We use it to obtain some vanishing…

We compute, by two methods, the genus one degree zero orbifold Gromov-Witten invariants with non-stacky insertions which are exceptional cases of the dilaton and divisor equations. One method involves a detailed analysis of the relevant…

We extend the vanishing theorem for the Seiberg-Witten invariants of a manifold with positive scalar curvature to the case when the curvature is allowed to be negative on a set of small volume. (The precise curvature bounds are described in…

We show that for any minuscule or cominuscule homogeneous space X, the Gromov-Witten varieties of degree d curves passing through three general points of X are rational or empty for any d. Applying techniques of A. Buch and L. Mihalcea to…

Generalizing local Gromov-Witten theory, in this paper we define a local version of symplectic field theory. When the symplectic manifold with cylindrical ends is four-dimensional and the underlying simple curve is regular by automatic…

In the first part of the paper, we give an explicit algorithm to compute the (genus zero) Gromov-Witten invariants of blow-ups of an arbitrary convex projective variety in some points if one knows the Gromov-Witten invariants of the…

We describe a program for proving that the Gromov-Witten moduli spaces of compact symplectic manifolds carry a unique virtual fundamental class that satisfies certain naturality conditions. The virtual fundamental class is constructed using…

Let $(F,J,\omega)$ be an almost K\"ahler manifold, $\alpha$ a $J$-holomorphic action of a compact Lie group $\hat K$ on $F$, and $K$ a closed normal subgroup of $\hat K$ which leaves $\omega$ invariant. We introduce gauge theoretical…

We construct an invariant of closed ${\rm spin}^c$ 4-manifolds using families of Seiberg-Witten equations. This invariant is formulated as a cohomology class on a certain abstract simplicial complex consisting of embedded surfaces of a…

For any finite group $G$, the equivariant Gromov-Witten invariants of $[\mathbb{C}^r/G]$ can be viewed as a certain twisted Gromov-Witten invariants of the classifying stack $\mathcal{B} G$. In this paper, we use Tseng's orbifold quantum…