Holomorphic curves in exploded manifolds: regularity

Abstract

The category of exploded manifolds is an extension of the category of smooth manifolds related to tropical geometry in which some adiabatic limits appear as smooth families. This paper studies the dbar equation on variations of a given family of curves in an exploded manifold. Roughly, we prove that the dbar equation on variations of an exploded family of curves behaves as nicely as the dbar 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 used in a series of separate papers to construct Gromov Witten invariants for exploded manifolds.Comment: 52 pages. v2: The construction of Gromov Witten invariants has been removed to another paper. v3: rewritten introduction, improved exposition. v4, v5: improved exposition v6, v7: Minor improvements and some expanded explanations, (including weakened hypothesis for Proposition 3.11), as suggested by an anonymous referee of a different paper. Final version to appear in Geometry and Topolog