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Blown-up singular Riemannian foliations

Francisco C. Caramello, Laura Ribeiro dos Santos

TL;DR

The paper develops a blow-up desingularization framework for singular Riemannian foliations, linking the dynamics of the Molino sheaf to its blow-up and showing that blow-ups preserve or improve transverse structure, turning singular Killing foliations into regular Killing foliations. It proves that leaf closures are governed by the blown-up Molino sheaf, derives Euler-characteristic-based criteria for closed leaves, and demonstrates the leaf-space as a Gromov–Hausdorff limit of orbifolds. Additionally, it provides a Gysin-type decomposition for basic cohomology under blow-up, clarifying how H^i(Bl_Σ𝓕) splits into contributions from H^i(𝓕) and the exceptional divisor, with several natural special cases. Together, these results extend classical desingularization ideas to the foliated setting and connect dynamics, geometry of leaf spaces, and transverse cohomology, with potential applications to geometric analysis and orbifold theory.

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.

Blown-up singular Riemannian foliations

TL;DR

The paper develops a blow-up desingularization framework for singular Riemannian foliations, linking the dynamics of the Molino sheaf to its blow-up and showing that blow-ups preserve or improve transverse structure, turning singular Killing foliations into regular Killing foliations. It proves that leaf closures are governed by the blown-up Molino sheaf, derives Euler-characteristic-based criteria for closed leaves, and demonstrates the leaf-space as a Gromov–Hausdorff limit of orbifolds. Additionally, it provides a Gysin-type decomposition for basic cohomology under blow-up, clarifying how H^i(Bl_Σ𝓕) splits into contributions from H^i(𝓕) and the exceptional divisor, with several natural special cases. Together, these results extend classical desingularization ideas to the foliated setting and connect dynamics, geometry of leaf spaces, and transverse cohomology, with potential applications to geometric analysis and orbifold theory.

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.

Paper Structure

This paper contains 9 sections, 23 theorems, 55 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Singular Riemannian Foliations
  4. Homothetic Lemma and Stratification
  5. Molino's Theory for Riemannian foliations
  6. Blow-up of Singular Riemannian Foliations
  7. Dynamics of the blown-up foliation
  8. Blown-up Leaf Spaces
  9. Basic cohomology of the blown-up foliation

Key Result

Theorem 1

Let $(M,\mathcal{F})$ be a complete singular Riemannian foliation with smooth Molino sheaf $\mathscr{C}_\mathcal{F}$. Then In particular, if $\mathcal{F}$ is a singular Killing foliation, then $\mathrm{Bl}_{}{\mathcal{F}}$ is a (regular) Killing foliation.

Figures (1)

  • Figure 1: Blow-up of the foliation of $\mathbb{R}^2$ by orbits of $\mathrm{SO}(2)$.

Theorems & Definitions (47)

  • Theorem 1: Theorem \ref{['blup sheaf']}
  • Theorem 2: Theorem \ref{['closed leaves kill']}
  • Theorem 3: Theorem \ref{['chicos def']}
  • Theorem 4: Theorem \ref{['principal result']}
  • Definition 2.1: Singular Riemannian Foliations
  • Example 2.2: Homogeneous Singular Riemannian Foliations
  • Lemma 2.3: Molino's homothetic lemma molino
  • Proposition 2.4: molino
  • Definition 2.5: Transverse Killing Field
  • Definition 2.6: Killing Foliation
  • ...and 37 more