Blown-up singular Riemannian foliations

TL;DR

This paper explores the blow-up desingularization method in singular Riemannian foliations, revealing how it constrains leaf properties, relates leaf closure spaces to orbifold limits, and connects basic cohomology between original and blown-up foliations.

Contribution

It introduces new applications of blow-up techniques to analyze the dynamics, leaf closures, and cohomology of singular Riemannian foliations, especially singular Killing foliations.

Findings

Singular Killing foliations have all leaves closed under certain conditions.

The space of leaf closures is a Gromov--Hausdorff limit of orbifolds.

Basic cohomology relates between original and blown-up foliations.

Abstract

In this paper we investigate new 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: the leaves of such foliations are all closed, provided the Euler characteristic of the ambient manifold is non-vanishing and its singular strata are all odd-codimensional. 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.