#### Volume 23, issue 4 (2019)

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Holomorphic curves in exploded manifolds: regularity

### Brett Parker

Geometry & Topology 23 (2019) 1621–1690
##### 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 $\stackrel{̄}{\partial }$ equation on variations of a given family of curves in an exploded manifold. Roughly, we prove that the $\stackrel{̄}{\partial }$ equation on variations of an exploded family of curves behaves as nicely as the $\stackrel{̄}{\partial }$ 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.

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##### Keywords
holomorphic curves, exploded manifolds, regularity of dbar equation, gluing analysis
Primary: 58J99