Obstructed Thickenings and Supermanifolds

TL;DR

This paper investigates the structure and classification of thickenings associated with supermanifolds in complex geometry, addressing embedding questions, moduli problems, and obstructions, with a focus on complex projective plane examples.

Contribution

It provides a detailed classification of supermanifold thickenings, explores their embedding conditions, and discusses moduli and obstructions in the complex-analytic setting.

Findings

Classification of thickenings of a given order

Conditions for embedding thickenings in supermanifolds

Examples of obstructed thickenings of the complex projective plane

Abstract

Associated to any supermanifold is a filtration by spaces, referred to as thickenings. It is the objective of this article to study them up to a certain equivalence and then up to isomorphism in the complex-analytic setting. We study them from two points of view: (1) as structures embedded in supermanifolds and (2) abstractly. Throughout, we will be guided by the goal to clarify and address the question: when does a given thickening embed in a supermanifold? Such a question was, in essence, first studied by Eastwood and LeBrun. In this article we begin with a pedagogical account of their study, after which we further study thickenings in supergeometry and present a classification of thickenings of a given order. As a complement to our study, we comment on the moduli problem for complex supermanifolds and consider the analogous problem for thickenings. Finally, to illustrate the ideas in this article, we conclude by describing some obstructed thickenings of the complex projective plane.