Holomorphic curves in exploded manifolds: regularity

Parker, Brett¹

¹ School of Mathematics, Monash University, Melbourne, VIC, Australia

Geometry & topology, Tome 23 (2019) no. 4, pp. 1621-1690.

Résumé

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.

Détail

DOI : 10.2140/gt.2019.23.1621

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

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