On The Problem of Splitting Deformations of Super Riemann Surfaces

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.