In theories of physics with supersymmetry, one will naturally happen upon the notion of a supermanifold and thereby motivations to study them. In this talk I will give an account of complex supermanifolds, with a view to study them through the lens of obstruction theory. The goal is to arrive at a classification and explore some of its limitations. The relevance of this classification to physics however is not immediately apparent. Nevertheless, some clues may be gleaned upon closer inspection of certain problems plaguing (perturbative) superstring theory. This motivates the study of superconformal structures.
In superstring theory, the supermanifolds of interest are defined over Riemann surfaces and come equipped with a certain structure, known as a superconformal structure. One of the main objectives in this talk will be on establishing a relation between: (1) the obstruction theory of these supermanifolds; and (2) certain deformations of their superconformal structures. If time permits, I will elaborate on some general constructions and comment on their relevance to universal aspects of superstring theory.