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

John Pardon

Geometry & Topology 20 (2016) 779–1034

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 for general symplectic manifolds. Our framework generalizes to the S1–equivariant setting, and we use S1–localization to calculate Hamiltonian Floer homology. The Arnold conjecture (as treated by Floer, by Hofer and Salamon, by Ono, by Liu and Tian, by Ruan, and by Fukaya and Ono) is a well-known corollary of this calculation.

virtual fundamental cycles, pseudo-holomorphic curves, implicit atlases, Gromov–Witten invariants, Floer homology, Hamiltonian Floer homology, Arnold conjecture, $S^1$–localization, transversality, gluing
Mathematical Subject Classification 2010
Primary: 37J10, 53D35, 53D40, 53D45, 57R17
Secondary: 53D37, 53D42, 54B40
Received: 26 May 2014
Revised: 20 May 2015
Accepted: 1 July 2015
Published: 28 April 2016
Proposed: Leonid Polterovich
Seconded: Richard Thomas, Ronald Stern
John Pardon
Department of Mathematics
Stanford University
450 Serra Mall
Building 380
Stanford, CA 94305