Holomorphic curves in exploded manifolds: regularity

Parker, Brett

doi:10.2140/gt.2019.23.1621

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Holomorphic curves in exploded manifolds: regularity

Brett Parker

Geometry & Topology 23 (2019) 1621–1690

DOI: 10.2140/gt.2019.23.1621

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.

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

Volume 23, issue 4 (2019)