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

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.

205 pages, 2 figures, to appear in Geometry and Topology

References in corpus (2)

• Smooth structures on Morse trajectory spaces, featuring finite ends and associative gluing
• The Hilbert--Smith conjecture for three-manifolds

Cited by in corpus (26)

• Covariantly functorial wrapped Floer theory on Liouville sectors
• Contact homology and virtual fundamental cycles
• Persistent homology and Floer-Novikov theory
• Bounds on spectral norms and barcodes
• Smooth Kuranishi atlases with isotropy
• Symplectic and contact differential graded algebras
• Non-contractible Periodic Orbits in Hamiltonian Dynamics on Closed Symplectic Manifolds
• A Log PSS morphism with applications to Lagrangian embeddings
• Mayer-Vietoris property for relative symplectic cohomology
• Higher symplectic capacities
• The topology of Kuranishi atlases
• Holomorphic curves in exploded manifolds: virtual fundamental class
• Quotient Theorems in Polyfold Theory and $S^1$-Equivariant Transversality
• Morse-Bott cohomology from homological perturbation theory
• Constructing the virtual fundamental class of a Kuranishi atlas
• On the generic behavior of the spectral norm
• On fillings of contact links of quotient singularities
• The Gromov width of generalized Bott manifolds
• Pseudoholomorphic Curves Relative to a Normal Crossings Symplectic Divisor: Compactification
• Equivariant Lagrangian Floer cohomology via semi-global Kuranishi structures
• Higher algebraic structures in Hamiltonian Floer theory
• Solomon-Tukachinsky's vs. Welschinger's Open Gromov-Witten Invariants of Symplectic Sixfolds
• Fundamental groups of rationally connected symplectic manifolds
• Homological invariants of codimension 2 contact submanifolds
• A Geometric Depiction of Solomon-Tukachinsky's Construction of Open GW-Invariants
• A landscape of contact manifolds via rational SFT