Exploded Manifolds

TL;DR

This paper introduces exploded manifolds, a mathematical framework extending smooth manifolds with tropical and holomorphic curve theories, facilitating Gromov-Witten invariants computation relative to divisors.

Contribution

It presents the concept of exploded manifolds, integrating tropical geometry with holomorphic curve theory to advance Gromov-Witten invariants calculations.

Findings

Exploded manifolds extend smooth manifolds with tropical structures.

They enable computation of Gromov-Witten invariants relative to divisors.

The framework connects tropical curve counts with holomorphic curve theory.

Abstract

This paper provides an introduction to exploded manifolds. The category of exploded manifolds is an extension of the category of smooth manifolds with an excellent holomorphic curve theory. Each exploded manifold has a tropical part which is a union of convex polytopes glued along faces. Exploded manifolds are useful for defining and computing Gromov-Witten invariants relative to normal crossing divisors, and using tropical curve counts to compute Gromov-Witten invariants.