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Distributions with locally free tangent sheaf

J. V. Pereira, J. P. dos Santos

Abstract

In the paper Stability of Holomorphic Foliations with Split Tangent Sheaf one finds a study of the locus $\mathrm{Dec}$ where the tangent sheaf of a {\it family} of foliations in $\mathbb{P}_{\mathbb C}^n$ is {\it decomposable}, i.e. a sum of line bundles. A prime conclusion is an ``openness'' result: once the singular locus has sufficiently large codimension, $\mathrm{Dec}$ turns out to be open. In the present paper, we study the locus $\mathrm{LF}$ of points of a family of distributions where the tangent sheaf is {\it locally free}. Through general Commutative Algebra, we show that $\mathrm{LF}$ is open provided that singularities have codimension at least three. When dealing with foliations rather than distributions, the condition on the lower bound of the singular set can be weakened by the introduction of ``Kupka'' points. We apply the available ``openness'' results to families in $\mathbb{P}_{\mathbb C}^n$ and in $\mathcal B$, the variety of Borel subgroups of a simple group. By establishing a theorem putting in bijection irreducible components of the space of two-dimensional subalgebras of a given semi-simple Lie algebra and its nilpotent orbits, we conclude that the space of foliations of {\it rank two} on $\mathbb{P}_{\mathbb C}^n$ and $\mathcal B$, may have quite many irreducible components as $n$ and $\dim \mathcal B$ grow. We also set in place several algebro-geometric foundations for the theory of families of distributions in two appendices.

Distributions with locally free tangent sheaf

Abstract

In the paper Stability of Holomorphic Foliations with Split Tangent Sheaf one finds a study of the locus where the tangent sheaf of a {\it family} of foliations in is {\it decomposable}, i.e. a sum of line bundles. A prime conclusion is an ``openness'' result: once the singular locus has sufficiently large codimension, turns out to be open. In the present paper, we study the locus of points of a family of distributions where the tangent sheaf is {\it locally free}. Through general Commutative Algebra, we show that is open provided that singularities have codimension at least three. When dealing with foliations rather than distributions, the condition on the lower bound of the singular set can be weakened by the introduction of ``Kupka'' points. We apply the available ``openness'' results to families in and in , the variety of Borel subgroups of a simple group. By establishing a theorem putting in bijection irreducible components of the space of two-dimensional subalgebras of a given semi-simple Lie algebra and its nilpotent orbits, we conclude that the space of foliations of {\it rank two} on and , may have quite many irreducible components as and grow. We also set in place several algebro-geometric foundations for the theory of families of distributions in two appendices.
Paper Structure (38 sections, 68 theorems, 175 equations)

This paper contains 38 sections, 68 theorems, 175 equations.

Table of Contents

  1. Introduction
  2. "Openess" results
  3. The space of Lie subalgebras
  4. Irreducible components of the spaces of foliations
  5. Generalities
  6. The scheme of Lie subalgebras
  7. The scheme $\mathbf{SLie}$ parametrising subalgebras richardson67
  8. Commuting varieties
  9. Two dimensional non-abelian subalgebras and nilpotent orbits
  10. Irreducible components of $\mathrm{Fol}$ for a projective space of dimension at least 4
  11. Partitions and nilpotent classes
  12. Vector fields in projective space associated to certain partitions
  13. Irreducible components of spaces of foliations
  14. Distributions with a locally free tangent sheaf
  15. Constructibility results for set defined by the tangent sheaf
  16. ...and 23 more sections

Key Result

Theorem 1

Let $f: X \to S$ be a proper and smooth morphism of Noetherian schemes. Let $\mathcal{V}$ be a relative distribution on $X$, i.e. a strongly saturated coherent submodule of the relative tangent sheaf $T_f$. For each $s\in S$, let $X_s$ denote the (schematic) fibre of $X$ above $s$ and denote by $\ Then for all $s$ in a certain neighbourhood of $o$, the tangent sheaf $T_{\mathcal{V}_s}$ is also l

Theorems & Definitions (144)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 4
  • Theorem 6
  • Theorem 2.1: Richardson
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • proof
  • ...and 134 more