On The Problem of Splitting Deformations of Super Riemann Surfaces

TL;DR

This paper investigates the obstruction theory of odd deformations of super Riemann surfaces, showing that second-order deformations with vanishing primary obstruction are split, advancing understanding of their structure.

Contribution

It proves that second-order odd deformations of super Riemann surfaces with genus greater than one are split if their primary obstruction vanishes, offering a step towards a full classification.

Findings

Primary obstruction class vanishes implies split deformation for second order.

Deformation theory of super Riemann surfaces is clarified.

Conjectural characterization of higher order odd deformations is proposed.

Abstract

An odd deformation of a super Riemann surface $\mathcal S$ is a deformation of $\mathcal S$ by variables of odd parity. In this article we study the obstruction theory of these odd deformations $\mathcal X$ of $\mathcal S$. We view $\mathcal X$ here as a complex supermanifold in its own right. Our objective in this article is to show, when $\mathcal X$ is a deformation of second order of $\mathcal S$ with genus $g>1$: if the primary obstruction class to splitting $\mathcal X$ vanishes, then $\mathcal X$ is in fact split. This result leads naturally to a conjectural characterisation of odd deformations of $\mathcal S$ of any order.