Holomorphic curves in exploded manifolds: virtual fundamental class

Abstract

We define Gromov–Witten invariants of exploded manifolds. The technical heart of this paper is a construction of a virtual fundamental class $\left[\mathcal{K}\right]$ 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 $\left[\mathcal{K}\right]$ to obtain numerical invariants, and push forward such differential forms over suitable 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.