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2016 An algebraic approach to virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves
John Pardon
Geom. Topol. 20(2): 779-1034 (2016). DOI: 10.2140/gt.2016.20.779


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.


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John Pardon. "An algebraic approach to virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves." Geom. Topol. 20 (2) 779 - 1034, 2016.


Received: 26 May 2014; Revised: 20 May 2015; Accepted: 1 July 2015; Published: 2016
First available in Project Euclid: 16 November 2017

zbMATH: 1342.53109
MathSciNet: MR3493097
Digital Object Identifier: 10.2140/gt.2016.20.779

Primary: 37J10 , 53D35 , 53D40 , 53D45 , 57R17
Secondary: 53D37 , 53D42 , 54B40

Keywords: $S^1$–localization , Arnold conjecture , Floer homology , gluing , Gromov–Witten Invariants , Hamiltonian Floer homology , implicit atlases , pseudo-holomorphic curves , transversality , virtual fundamental cycles

Rights: Copyright © 2016 Mathematical Sciences Publishers


Vol.20 • No. 2 • 2016
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