Virtual neighborhoods and pseudo-holomorphic curves

TL;DR

This paper develops a comprehensive framework using virtual neighborhood techniques to define and compute various invariants in symplectic geometry, leading to proofs of longstanding conjectures like Arnold's.

Contribution

It introduces a unified approach to establish GW-invariants, quantum cohomology, and Floer cohomology for general symplectic manifolds, including families.

Findings

Established GW-invariants and quantum cohomology for general symplectic manifolds

Proved Arnold conjecture for nondegenerate Hamiltonian symplectomorphisms

Extended invariants to families of symplectic manifolds

Abstract

We use virtual neighborhood technique to establish GW-invariants, Quantum cohomology, equivariant GW-invariants, equivariant quantum cohomology and Floer cohomology for general symplectic manifold. We also establish GW-invariants for a family of symplectic manifolds. As a consequence, we prove Arnold conjecture for nondegenerate Hamiltonian symplectomorphisms.