Compactness for Holomorphic Supercurves

TL;DR

This paper extends Gromov compactness to moduli spaces of holomorphic supercurves, providing explicit limits, a suitable energy bound, and a topology making these spaces compact and metrizable.

Contribution

It introduces a compactness theorem for holomorphic supercurves, including explicit limiting objects and a topology on the enlarged moduli space.

Findings

Sequences of holomorphic supercurves have convergent subsequences under energy bounds

A new topology makes the moduli space of supercurves compact and metrizable

Explicit construction of limiting objects for supercurves

Abstract

We study the compactness problem for moduli spaces of holomorphic supercurves which, being motivated by supergeometry, are perturbed such as to allow for transversality. We give an explicit construction of limiting objects for sequences of holomorphic supercurves and prove that, in important cases, every such sequence has a convergent subsequence provided that a suitable extension of the classical energy is uniformly bounded. This is a version of Gromov compactness. Finally, we introduce a topology on the moduli spaces enlarged by the limiting objects which makes these spaces compact and metrisable.