# Families of Supermanifolds: Splitting Types and Obstruction Maps

## TL;DR

This paper investigates the classification and deformation of supermanifolds, focusing on splitting types and obstruction maps, and introduces a secondary obstruction theory for specific families over non-reduced superspaces.

## Contribution

It provides a detailed analysis of supermanifold families, especially splitting types and the development of a secondary obstruction theory for gtm-families.

## Key findings

- Classification of supermanifold families based on splitting types
- Construction of variations of splitting types
- Introduction of secondary obstruction theory for gtm-families

## Abstract

In this article we study the notion of supermanifolds families, starting from Green's general classification of supermanifolds. The topics studied divide this article into two distinct parts, labelled I and II respectively. Part I concerns the splitting type of a supermanifold and our attempt to construct variations thereof. In Part II we look at a particular class of families, referred to as `$gtm$-families'. We are concerned with the notion of a `secondary obstruction theory', which is apparent in any $gtm$-family of supermanifolds over a (connected and Stein) non-reduced, superspace base.