Blown-up singular Riemannian foliations
arXiv:2512.07069
The paper studies how blowing up singular Riemannian foliations affects their dynamics, leaf closure spaces, and basic cohomology, obtaining constraints on closed leaves for singular Killing foliations and describing leaf closure spaces as Gromov–Hausdorff limits of orbifolds.
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.
Added Proposition 3.2. Minor structural changes. Numbering changed