An algebraic approach to virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves

TL;DR

This paper introduces an algebraic framework using implicit atlases for defining virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves, facilitating the construction of symplectic invariants without geometric perturbations.

Contribution

It develops an algebraic, sheaf-theoretic approach with implicit atlases for virtual fundamental cycles, avoiding geometric perturbations and enabling applications to Gromov--Witten invariants and Floer homology.

Findings

Defined Gromov--Witten invariants over  for general symplectic manifolds.

Constructed Hamiltonian Floer homology over .

Applied S^1-localization to compute Floer homology and verify the Arnold conjecture.

Abstract

We develop techniques for defining and working with virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves which are not necessarily cut out transversally. Such techniques have the potential for applications as foundations for invariants in symplectic topology arising from "counting" pseudo-holomorphic curves. We introduce the notion of an *implicit atlas* on a moduli space, which is (roughly) a convenient system of local finite-dimensional reductions. We present a general intrinsic strategy for constructing a canonical implicit atlas on any moduli space of pseudo-holomorphic curves. The main technical step in applying this strategy in any particular setting is to prove appropriate gluing theorems. We require only topological gluing theorems, that is, smoothness of the transition maps between gluing charts need not be addressed. Our approach to virtual fundamental cycles is algebraic rather than geometric (in particular, we do not use perturbation). Sheaf-theoretic tools play an important role in setting up our functorial algebraic "VFC package". We illustrate the methods we introduce by giving definitions of Gromov--Witten invariants and Hamiltonian Floer homology over $\mathbb Q$ for general symplectic manifolds. Our framework generalizes to the $S^1$-equivariant setting, and we use $S^1$-localization to calculate Hamiltonian Floer homology. The Arnold conjecture (as treated by Floer, Hofer--Salamon, Ono, Liu--Tian, Ruan, and Fukaya--Ono) is a well-known corollary of this calculation.