Technical details on Kuranishi structure and virtual fundamental chain

Abstract

This is an expository article on the theory of Kuranishi structure and is based on a series of pdf files we uploaded for the discussion of the google group named `Kuranishi' (with its administrator H. Hofer). There we replied to several questions concerning Kuranishi structure raised by K. Wehrheim. At this stage we submit this article to the e-print arXiv, all the questions or objections asked in that google group were answered, supplemented or confuted by us. We first discuss the abstract theory of Kuranishi structure and virtual fundamental chain/cycle. This part can be read independently from other parts. We then describe the construction of Kuranishi structure on the moduli space of pseudoholomorphic curves, including the complete analytic detail of the gluing construction as well as the smoothness of the resulting Kuranishi structure. The case of S^1 equivariant Kuranishi structure which appears in the study of time independent Hamiltonian and the moduli space of Floer's equation is included.

arXiv comment257 pages. arXiv admin note: substantial text overlap with arXiv:1208.1340 by other authors

References in corpus (2)

• Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds
• Lagrangian Floer theory over integers: spherically positive symplectic manifolds

Cited by in corpus (27)

• Contact homology and virtual fundamental cycles
• Smooth Kuranishi atlases with isotropy
• Virtual neighborhood technique for pseudo-holomorphic spheres
• Functors of wrapped Fukaya categories from Lagrangian correspondences
• A new definition of Kuranishi space
• Exponential decay estimates and smoothness of the moduli space of pseudoholomorphic curves
• Seidel elements and potential functions of holomorphic disc counting
• Wrapped Floer cohomology and Lagrangian correspondences
• Derived manifolds and Kuranishi models
• Lectures on Polyfolds and Symplectic Field Theory
• Construction of Kuranishi structures on the moduli spaces of pseudo holomorphic disks: I
• $SO(3)$-Floer homology of 3-manifolds with boundary 1
• Lie groupoid, deformation of unstable curve, and construction of equivariant Kuranishi charts
• Smoothness of Kuranishi atlases on Gromov-Witten moduli spaces
• Finite group actions on Lagrangian Floer theory
• Virtual Neighborhood Technique for Holomorphic Curve Moduli Spaces
• Some new homology and cohomology theories of manifolds and orbifolds
• The Künneth theorem for the Fukaya algebra of a product of Lagrangians
• The Gromov-Witten axioms for symplectic manifolds via polyfold theory
• On the Existence of a Global Neighbourhood
• Shrinking good coordinate systems associated to Kuranishi structures
• Automatic transversality in contact homology I: Regularity
• A Note on Disk Counting in Toric Orbifolds
• Monotone Lagrangian Floer theory in smooth divisor complements: I
• Lagrangian correspondences and the generalized Viterbo restriction functor
• A symplectic prolegomenon
• Gromov-Witten theory of a locally conformally symplectic manifold