Kuranishi structures

Kuranishi structures
Name	Kuranishi structures
Field	Symplectic topology; Differential geometry; Algebraic geometry
Introduced	1985
Introduced by	Masatake Kuranishi
Related	Gromov–Witten theory; Floer homology; Moduli spaces; Virtual fundamental class

Kuranishi structures are a toolkit introduced by Masatake Kuranishi for describing local models of moduli spaces that fail to be smooth manifolds. Developed to treat deformation problems with obstruction spaces, they provide a way to patch local finite-dimensional reductions into a global object used in enumeration problems in Gromov–Witten theory, Floer homology, and related areas. Kuranishi structures connect ideas from Masatake Kuranishi's work to later developments by researchers associated with Fukaya–Oh–Ohta–Ono, Kontsevich, and Ruan, enabling intersection-theoretic constructions such as the virtual fundamental class.

Introduction

Kuranishi structures originated in the study of moduli of complex structures by Masatake Kuranishi and were adapted to symplectic topology in constructions by Kenji Fukaya, Kaoru Ono, Yong-Geun Oh, and Hiroshi Ohta to handle moduli spaces of pseudoholomorphic curves encountered in Gromov's work. They appear alongside other technical devices in foundational programs associated with Kontsevich's homological mirror symmetry proposals and Ruan's virtual cycle program. Subsequent expositions and critiques involved mathematicians such as McDuff, Wehrheim, Pardon, and Joyce, reflecting the cross-disciplinary influence of ideas stemming from the Kuranishi approach on modern enumerative geometry.

Definitions and basic properties

A Kuranishi structure on a compact topological space of solutions is a collection of local charts indexed by marked points, each chart consisting of a finite-dimensional smooth manifold (the domain), a finite-dimensional vector space (the obstruction space), a smooth section, and an isotropy group action drawn from symmetry groups like those studied by Deligne–Mumford in the context of moduli of curves. Compatibility data include coordinate changes between charts satisfying cocycle-like conditions analogous to those considered by Cheeger and Gromov in geometric analysis. Important properties involve notions of orientation, tangent-obstruction complexes reminiscent of constructions by Atiyah and Singer, and virtual dimension computations paralleling index theory developed by Atiyah–Singer.

Construction and examples

Constructions typically begin with a Fredholm setup such as the Cauchy–Riemann operator in the analytic framework employed by Gromov for pseudoholomorphic curve theory, with local finite-dimensional reductions provided by choices akin to Kuranishi maps used in deformation theory for complex structures studied by Kodaira and Spencer. Classic examples include moduli spaces of stable maps in Gromov–Witten theory investigated by groups around Kontsevich and Manin, as well as moduli of solutions to gauge-theoretic equations considered by researchers influenced by Donaldson and Uhlenbeck. Other instances arise in enumerative problems related to Seiberg–Witten theory and counts appearing in programs initiated by Taubes.

Virtual fundamental class and applications

Using Kuranishi structures one defines a virtual fundamental class (VFC) via perturbation of multi-valued sections or multisections, constructions influenced by techniques from Thom and transversality methods used in classical intersection theory by Lefschetz. The resulting VFC underpins invariants such as Gromov–Witten invariants central to the work of Kontsevich–Manin and calculations used in mirror symmetry conjectures advanced by Candelas and collaborators. Applications include computations in quantum cohomology appearing in studies by Witten, structural results in symplectic field theory linked to Eliashberg and Hofer, and invariants in low-dimensional topology inspired by programs of Floer and Donaldson.

Comparisons and alternative frameworks

Alternative approaches to handling transversality and virtual classes include polyfold theory developed by Hofer–Wysocki–Zehnder, algebraic virtual cycle machinery by Behrend–Fantechi in the setting of Deligne–Mumford stacks, and sheaf-theoretic categorical methods advocated by Joyce and others. Pardon's algebraic-topological formulation offers another route to VFC without explicit multisections, whereas McDuff–Wehrheim developed a Kuranishi atlas perspective emphasizing topological coherence. Each framework addresses similar examples from Gromov–Witten theory and Floer homology but trades off between analytic choices, categorical structures like those in Derived Algebraic Geometry proponents such as Lurie, and the level of explicit perturbations needed.

Technical issues and developments

Technical issues historically centered on gluing analysis, choices of perturbations, and coherence of coordinate changes; these were topics in exchanges among teams led by Fukaya, Ono, McDuff, Wehrheim, and commentators like Pardon and Joyce. Subsequent developments clarified orientations, equivariant structures involving actions studied in representation-theoretic contexts by Weyl and obstruction bundle techniques reminiscent of methods used by Thomason in algebraic K-theory. Ongoing work explores comparisons between Kuranishi-based VFCs and those from polyfolds or algebraic stacks, with contributions from researchers such as Biran, Cornea, Seidel, and Smith influencing computations in homological mirror symmetry and symplectic topology.

Category:Symplectic topology