Transversality for Holomorphic Supercurves

TL;DR

This paper establishes transversality results for holomorphic supercurves, showing that their moduli spaces are smooth manifolds under generic conditions, advancing the understanding of their geometric structure.

Contribution

It proves that perturbing the defining equations with a connection ensures the linearised operator is surjective, leading to smooth moduli spaces for holomorphic supercurves.

Findings

Moduli spaces are oriented finite-dimensional smooth manifolds.

Transversality holds under generic perturbations involving connections.

Dependence of moduli spaces on generic data is analyzed.

Abstract

We study holomorphic supercurves, which are motivated by supergeometry as a natural generalisation of holomorphic curves. We prove that, upon perturbing the defining equations by making them depend on a connection, the corresponding linearised operator is generically surjective. By this transversality result, we show that the resulting moduli spaces are oriented finite dimensional smooth manifolds. Finally, we examine how they depend on the choice of generic data.