Holomorphic curves in exploded manifolds: virtual fundamental class

TL;DR

This paper develops a framework for defining Gromov--Witten invariants of exploded manifolds using a virtual fundamental class, enabling integration and pushforward of differential forms to produce numerical invariants.

Contribution

It introduces a construction of a virtual fundamental class for Kuranishi categories in exploded manifolds, generalizing previous methods and ensuring invariants are well-defined and compatible with various operations.

Findings

Constructed a virtual fundamental class for Kuranishi categories.

Defined integration of differential forms over the virtual class.

Provided an alternative approach to Gromov--Witten invariants for symplectic manifolds.

Abstract

We define Gromov--Witten invariants of exploded manifolds. The technical heart of this paper is a construction of a virtual fundamental class $[\mathcal K]$ of any Kuranishi category $\mathcal K$ (which is a simplified, more general version of an embedded Kuranishi structure.) We also show how to integrate differential forms over $[\mathcal K]$ to obtain numerical invariants, and push forward differential forms from $\mathcal K$ over suitable evaluation maps. We show that such invariants are independent of any choices, and are compatible with pullbacks, products, and tropical completion of Kuranishi categories. In the case of a compact symplectic manifold, this gives an alternative construction of Gromov--Witten invariants, including gravitational descendants.