Deformations of geometric structures and Lie groupoids

Abstract

A central problem in geometry is that of understanding the behaviour of geometric structures under deformations. Typically, the final goal is to describe the moduli space of such structures up to an appropriate equivalence relation. Each class of geometric structures comes with its deformation theory, generally including a cohomology theory that controls such deformations. This project involves the study of the cohomology theories controlling deformations of several geometric structures, in a unified way. This unified approach is attained by studying the deformation theory of a Lie groupoid and the resulting deformation cohomology. Lie groupoids are geometric objects that can be used to encode many different geometric objects (Lie groups, principal bundles, Lie group actions, submersions, etc.) and so we can study the deformation theories of all these objects in equal footing. Lie groupoids are also used to do meaningful differential geometry on spaces which are singular (i.e. not smooth), by using classical techniques on Lie groupoids instead. Spaces arising in this way are called differentiable stacks. Part of the project deals with unravelling the algebraic structure of the deformation cohomology of a Lie groupoid. This will give us new information on the behaviour of the moduli space of Lie groupoids (and of the structures they model). In order to obtain a better understanding of the deformation theory of Lie groupoids, this project also studies the relation with deformations of other naturally associated objects: C*-algebras, Lie algebroids and differentiable stacks. We will also study deformations of groupoid morphisms (and their stability), and of geometric structures compatible with groupoids: symplectic, complex and Poisson structures. (AU)