Measures on differentiable stacks

TL;DR

This paper develops a theory of measures on differentiable stacks, proving invariance properties, formulas, and conjectures, with applications to Poisson manifolds and symplectic geometry.

Contribution

It introduces measures on differentiable stacks, proves Morita invariance, establishes a Stokes formula, and confirms Weinstein's volume conjecture in the symplectic case.

Findings

Morita invariance of measures on stacks

A Stokes formula for algebroid currents

Proof of Weinstein's volume conjecture in symplectic case

Abstract

We introduce and study measures and densities (= geometric measures) on differentiable stacks, using a rather straightforward generalization of Haefliger's approach to leaf spaces and to transverse measures for foliations. In general we prove Morita invariance, 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, we provide explicit (Weyl-type) formulas that shed light on Weinstein's notion of volumes of differentiable stacks, in particular, in the symplectic case, we prove the conjecture left open in Weinstein's "The volume of a differentiable stack". We also revisit the notion of Haar systems (and the existence of cut-off functions). Our original motivation comes from the study of Poisson manifolds of compact types, which provide two important examples of such measures: the affine and the Duistermaat-Heckman measures.