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.