Holomorphic curves in exploded manifolds: regularity

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.

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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)

Brett Parker

Abstract