Blown-up singular Riemannian foliations

Abstract

We investigate new properties and applications of the blow-up desingularization method in the context of singular Riemannian foliations. First, we relate the dynamics of such a foliation, which is governed by the so-called Molino sheaf, with that of its blow-up. In the particular case of singular Killing foliations, this leads to a strong constraint: when the Euler characteristic of the ambient manifold is non-vanishing and the singular strata are all odd-codimensional, the leaves of such foliations are all closed. Next, we show that the space of leaf closures of a singular Killing foliation is the Gromov--Hausdorff limit of a sequence of orbifolds, whose dimensions are the codimension of the foliation. Finally, we relate the basic cohomology of a singular Riemannian foliation with that of its blow-up, generalizing well-known, classical analogous results in algebraic and complex geometry.