Geometry & Topology

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

John Pardon

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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.

Article information

Geom. Topol., Volume 20, Number 2 (2016), 779-1034.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37J10: Symplectic mappings, fixed points 53D35: Global theory of symplectic and contact manifolds [See also 57Rxx] 53D40: Floer homology and cohomology, symplectic aspects 53D45: Gromov-Witten invariants, quantum cohomology, Frobenius manifolds [See also 14N35] 57R17: Symplectic and contact topology
Secondary: 53D37: Mirror symmetry, symplectic aspects; homological mirror symmetry; Fukaya category [See also 14J33] 53D42: Symplectic field theory; contact homology 54B40: Presheaves and sheaves [See also 18F20]

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


Pardon, John. An algebraic approach to virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves. Geom. Topol. 20 (2016), no. 2, 779--1034. doi:10.2140/gt.2016.20.779.

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