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

TL;DR

This paper develops the foundational techniques for constructing virtual fundamental chains on spaces with Kuranishi structures, focusing on the de Rham approach and coefficients over real and rational numbers, independent of pseudo-holomorphic curve specifics.

Contribution

It provides a self-contained construction of the virtual fundamental chain on a single Kuranishi space, emphasizing the de Rham version and coefficient considerations, without relying on prior pseudo-holomorphic curve knowledge.

Findings

Constructed virtual fundamental chains over ℝ-coefficients.

Included a method for working over ℚ-coefficients in low-dimensional cases.

Established foundational techniques applicable to Kuranishi structures.

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).