Holomorphic curves in exploded manifolds: regularity

Abstract

The category of exploded manifolds is an extension of the category of smooth manifolds; for exploded manifolds, some adiabatic limits appear as smooth families. This paper studies the $\bar{\partial }$ equation on variations of a given family of curves in an exploded manifold. Roughly, we prove that the $\bar{\partial }$ equation on variations of an exploded family of curves behaves as nicely as the $\bar{\partial }$ equation on variations of a smooth family of smooth curves, even though exploded families of curves allow the development of normal-crossing or log-smooth singularities. The resulting regularity results are foundational to the author’s construction of Gromov–Witten invariants for exploded manifolds.