Related papers: Singular symplectic flops and Ruan cohomology

For a symplectic manifold $(M,\om)$ with exact symplectic form we construct a 2-cocycle on the group of symplectomorphisms and indicate cases when this cocycle is not trivial.

A symplectic toric orbifold is a compact connected orbifold $M$, a symplectic form $\omega$ on $M$, and an effective Hamiltonian action of a torus $T$ on $M$, where the dimension of $T$ is half the dimension of $M$. We prove that there is a…

We study the space of pseudo-holomorphic spheres in compact symplectic manifolds with convex boundary. We show that the theory of Gromov-Witten invariants can be extended to the class of semi-positive manifolds with convex boundary. This…

Motivated by various possible generalizations of Taubes's \(SW=Gr\) theorem [T] to Floer-theoretic setting, we prove certain variants of Taubes's convergence theorem in \cite{T} (the first part of his proof of \(SW=Gr\)). In place of the…

We consider morphisms $\pi: X \to \mathbb{P}^1$ of smooth projective varieties over $\mathbb{C}$. We show that if $\pi$ has at most one singular fibre, then $X$ is uniruled and $\pi$ admits sections. We reach the same conclusions, but with…

Q-Gorenstein toric contact manifolds provide an interesting class of examples of contact manifolds with torsion first Chern class. They are completely determined by certain rational convex polytopes, called toric diagrams, and arise both as…

We study families of objects in Fukaya categories, specifically ones whose deformation behaviour is prescribed by the choice of an odd degree cohomology class. This leads to invariants of symplectic manifolds, which we apply to blowups…

For a monotone symplectic manifold and a smooth anticanonical divisor, there is a formal deformation of the symplectic cohomology of the divisor complement, defined by allowing Floer cylinders to intersect the divisor. We compute this…

The zeroth line bundle cohomology on Calabi-Yau three-folds encodes information about the existence of flop transitions and the genus zero Gromov-Witten invariants. We illustrate this claim by studying several Picard number 2 Calabi-Yau…

Let $X$ denote the `conifold smoothing', the symplectic Weinstein manifold which is the complement of a smooth conic in $T^*S^3$, or equivalently the plumbing of two copies of $T^*S^3$ along a Hopf link. Let $Y$ denote the `conifold…

We study the relation between the symplectomorphism group Symp M of a closed connected symplectic manifold M and the symplectomorphism and diffeomorphism groups Symp \TM and Diff \TM of its one point blow up \TM. There are three main…

Let either $GL(E)\times SO(F)$ or $GL(E)\times Sp(F)$ act naturally on the space of matrices $E\otimes F$. There are only finitely many orbits, and the orbit closures are orthogonal and symplectic generalizations of determinantal varieties,…

Let X be a Gorenstein orbifold and let Y be a crepant resolution of X. We state a conjecture relating the genus-zero Gromov--Witten invariants of X to those of Y, which differs in general from the Crepant Resolution Conjectures of Ruan and…

This text is a set of lecture notes for a series of four talks given at I.P.A.M., Los Angeles, on March 18-20, 2003. The first lecture provides a quick overview of symplectic topology and its main tools: symplectic manifolds, almost-complex…

In this note, genus zero Gromov-Witten invariants are reviewed and then applied in some examples of dimension four and six. It is also proved that the use of genus zero Gromov-Witten invariants in the class of embedded $J$-holomorphic…

We define relative Gromov-Witten invariants of a symplectic manifold relative to a codimension two symplectic submanifold. These invariants are the key ingredients in the symplectic sum formula of [IP4]. The main step is the construction of…

We give an expository account of a conjecture, developed by Coates--Corti--Iritani--Tseng and Ruan, which relates the quantum cohomology of a Gorenstein orbifold X to the quantum cohomology of a crepant resolution Y of X. We explore some…

In this paper we exploit the geometric approach to the virtual fundamental class, due to Fukaya-Ono and Li-Tian, to compare the virtual fundamental classes of stable maps to a symplectic manifold and a symplectic submanifold whenever all…

We prove that symplectic cohomology for open convex symplectic manifolds is invariant when the symplectic form undergoes deformations which may be non-exact and non-compactly supported, provided one uses the correct local system of…

This article describes the use of symplectic cut-and-paste methods to compute Gromov-Witten invariants. Our focus is on recent advances extending these methods to Kahler surfaces with geometric genus p_g>0, for which the usual GW invariants…