Differentiable stacks: stratifications, measures and deformations

Abstract

The core of this thesis arises as two studies on the differential geometry of Lie groupoids and of the singular (i .e., not smooth) spaces modelled by them: one about measures on orbit spaces and another about deformations of Lie groupoids and deformation cohomology. These are complemented by some more expository material (surveying various approaches to "orbit spaces" and stratifications on them). We introduce the notions of transverse measure and transverse density for any Lie groupoid, generalizing Haefliger's approach for étale Lie groupoids. We then prove some fundamental results on transverse measures and densities: For general Lie groupoids we prove Morita invariance, which lets us interpret transverse measures as objects associated to stacks, a Stokes formula which provides reinterpretations in terms of (Ruelle-Sullivan type) algebroid currents and a Van Est isomorphism. In the proper case we reduce the theory to classical (Radon) measures on the underlying space and we provide explicit (Weyl-type) formulas that shed light on Weinstein's notion of volumes of differentiable stacks. We also revisit the notions of modular class and of Haar systems. We then study deformations of Lie groupoids. We introduce the deformation cohomology of a Lie groupoid, which provides an intrinsic model for the cohomology of a Lie groupoid with values in its adjoint representation. We then prove several fundamental properties of the deformation cohomology including Morita invariance, a van Est theorem, and a vanishing result in the proper case. Finally, we use the deformation cohomology in order to prove rigidity and normal form results.