Kuranishi structure, Pseudo-holomorphic curve, and Virtual fundamental chain: Part 1

Abstract

This is the first part of the article we promised at the end of [FOOO13, Section 1]. We discuss the foundation of the virtual fundamental chain and cycle technique, especially its version appeared in [FOn] and also in [FOOO4, Section A1, Section 7.5], [FOOO7, Section 12], [Fu2]. In Part 1, we focus on the construction of the virtual fundamental chain on a single space with Kuranishi structure. We mainly discuss the de Rham version and so work over $\mathbb{R}$-coefficients, but we also include a self-contained account of the way how to work over $\mathbb{Q}$-coefficients in case the dimension of the space with Kuranishi structure is $\le 1$. Part 1 of this document is independent of our earlier writing [FOOO13]. We also do not assume the reader have any knowledge on the pseudo-holomorphic curve, in Part 1. Part 2 (resp. Part 3), which will appear in the near future, discusses the case of a system of Kuranishi structures and its simultaneous perturbations (resp. the way to implement the abstract story in the study of moduli spaces of pseudo-holomorphic curves).

arXiv comment203 pages, contentindex included

References in corpus (3)

• Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds
• Cyclic symmetry and adic convergence in LagrangianFloer theory
• Floer homology of Lagrangian submanifolds

Cited by in corpus (14)

• Contact homology and virtual fundamental cycles
• Smooth Kuranishi atlases with isotropy
• Family Floer program and non-archimedean SYZ mirror construction
• Involutions, obstructions and mirror symmetry
• Construction of Kuranishi structures on the moduli spaces of pseudo holomorphic disks: I
• Lectures on Polyfolds and Symplectic Field Theory
• $SO(3)$-Floer homology of 3-manifolds with boundary 1
• Lie groupoid, deformation of unstable curve, and construction of equivariant Kuranishi charts
• Virtual Neighborhood Technique for Holomorphic Curve Moduli Spaces
• The Künneth theorem for the Fukaya algebra of a product of Lagrangians
• A critical point analysis of Landau--Ginzburg potentials with bulk in Gelfand--Cetlin systems
• Shrinking good coordinate systems associated to Kuranishi structures
• The Calabi invariant and the least number of periodic solutions of locally Hamiltonian equations
• Monotone Lagrangian Floer theory in smooth divisor complements: I