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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 ̄ equation on variations of a given family of curves in an exploded manifold. Roughly, we prove that the ̄ equation on variations of an exploded family of curves behaves as nicely as the ̄ 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.

Keywords
holomorphic curves, exploded manifolds, regularity of dbar equation, gluing analysis
Mathematical Subject Classification 2010
Primary: 58J99
References
Publication
Received: 11 February 2011
Revised: 6 August 2018
Accepted: 22 October 2018
Published: 17 June 2019
Proposed: Tomasz Mrowka
Seconded: Peter Ozsváth, Dan Abramovich
Authors
Brett Parker
School of Mathematics
Monash University
Melbourne, VIC
Australia