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Principalization on logarithmically foliated orbifolds

Dan Abramovich, André Belotto da Silva, Michael Temkin, Jarosław Włodarczyk

TL;DR

This work develops a characteristic-zero theory of principalization of ideals on smooth orbifolds endowed with a normal crossings divisor and a foliation. By embedding the problem into the language of foliated logarithmic varieties and employing orbifold (stack) weighted/cobordant blow-ups, the authors construct a foliation-aware invariant ${\rm inv}_{\mathcal{F}}$ and a notion of ${\mathcal{F}}$-aligned centers to drive a functorial desingularization algorithm. The main results include a principalization theorem for foliated manifolds, embedded desingularization of subvarieties, and preservation of foliation properties under aligned blow-ups, with two key applications: resolution of Darboux totally integrable foliations and a method to render generically transverse sections into transverse sections while preserving the transverse locus. They also develop a thick-class framework, showing that log-smooth and ${\mathcal{K}}$-monomial foliations admit functorial resolution and connecting to Darboux first integrals. The methods unify Rees-algebra techniques, rectification theory, and stack-theoretic blow-ups to achieve canonical, functorial, and compatible resolutions in both algebraic and analytic settings, with broad implications for foliations and differential equations in birational geometry.

Abstract

In characteristic zero, we construct principalization of ideals on smooth orbifolds endowed with a normal crossings divisor and a foliation. We then illustrate how the method can be used in the general study of foliations via two applications. First, we provide a resolution of singularities of Darboux totally integrable foliations in arbitrary dimensions -- including rational and meromorphic Darboux foliations. Second, we show how to transform a generically transverse section into a transverse section.

Principalization on logarithmically foliated orbifolds

TL;DR

This work develops a characteristic-zero theory of principalization of ideals on smooth orbifolds endowed with a normal crossings divisor and a foliation. By embedding the problem into the language of foliated logarithmic varieties and employing orbifold (stack) weighted/cobordant blow-ups, the authors construct a foliation-aware invariant and a notion of -aligned centers to drive a functorial desingularization algorithm. The main results include a principalization theorem for foliated manifolds, embedded desingularization of subvarieties, and preservation of foliation properties under aligned blow-ups, with two key applications: resolution of Darboux totally integrable foliations and a method to render generically transverse sections into transverse sections while preserving the transverse locus. They also develop a thick-class framework, showing that log-smooth and -monomial foliations admit functorial resolution and connecting to Darboux first integrals. The methods unify Rees-algebra techniques, rectification theory, and stack-theoretic blow-ups to achieve canonical, functorial, and compatible resolutions in both algebraic and analytic settings, with broad implications for foliations and differential equations in birational geometry.

Abstract

In characteristic zero, we construct principalization of ideals on smooth orbifolds endowed with a normal crossings divisor and a foliation. We then illustrate how the method can be used in the general study of foliations via two applications. First, we provide a resolution of singularities of Darboux totally integrable foliations in arbitrary dimensions -- including rational and meromorphic Darboux foliations. Second, we show how to transform a generically transverse section into a transverse section.

Paper Structure

This paper contains 70 sections, 58 theorems, 242 equations.

Table of Contents

  1. Introduction
  2. Principalization in the presence of a foliation
  3. Main results: principalization and embedded resolution
  4. Relation to resolution of morphisms.
  5. Resolution of classes of foliations
  6. Resolution preserving transverse locus
  7. Main methods of the proof
  8. Orbifolds and weighted (or cobordant) blow-ups
  9. Rees algebras, lifting and the foliated Replacement Lemma
  10. The invariant ${\rm inv}_{{\mathcal{F}}}$ and the desingularization algorithm.
  11. Nested-regular Rectification Theorem and splitting along transverse sections.
  12. An example
  13. Preserving properties of foliations
  14. Algebraic and analytic categories
  15. Structure of the paper
  16. ...and 55 more sections

Key Result

Theorem 1.2.1

Let $(X,{\mathcal{F}},E)$ be a smooth foliated logarithmic variety and ${\mathcal{I}}$ be a coherent ideal sheaf over $X$. If $X$ is algebraic, let $X_0=X$; otherwise, let $X_0 \subset X$ be a relatively compact open set, and let $(X_0,{\mathcal{F}}_0,E_0)$ be the restricted triple. There exists a s where ${\mathcal{F}}_{i+1}$ is the controlled (resp., strict) transform of ${\mathcal{F}}_i$ by $\s

Theorems & Definitions (173)

  • Theorem 1.2.1: Principalization over foliated manifolds
  • Theorem 1.2.2: Embedded desingularization of a subvariety
  • Remark 1.2.3: On more general foliations
  • Definition 1.3.1: Thick Classes
  • Theorem 1.3.2
  • Theorem 1.3.3: Thick Reduction of singularities
  • Definition 1.3.4: Globally totally integrable foliations
  • Theorem 1.3.5: Resolution of rationally totally integrable foliations
  • Definition 1.3.6: Globally $\mathcal{K}$-Darboux totally integrable foliations
  • Theorem 1.3.7: Resolution of Darboux totally integral foliations
  • ...and 163 more