Holomorphic curves in exploded manifolds Regularity

摘要

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 (partial derivative) over bar equation on variations of a given family of curves in an exploded manifold. Roughly, we prove that the (partial derivative) over bar equation on variations of an exploded family of curves behaves as nicely as the (partial derivative) over bar 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.