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Projectively flat foliations

Stéphane Druel

Abstract

We describe the structure of regular codimension $1$ foliations with numerically projectively flat tangent bundle on complex projective manifolds of dimension at least $4$. Along the way, we prove that either the normal bundle of a regular codimension $1$ foliation is pseudo-effective, or its conormal bundle is nef.

Projectively flat foliations

Abstract

We describe the structure of regular codimension foliations with numerically projectively flat tangent bundle on complex projective manifolds of dimension at least . Along the way, we prove that either the normal bundle of a regular codimension foliation is pseudo-effective, or its conormal bundle is nef.
Paper Structure (24 sections, 51 theorems, 115 equations)

This paper contains 24 sections, 51 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Notation, conventions, and used facts
  3. Global conventions
  4. Projective space bundles
  5. Stability
  6. Numerically flat vector bundles
  7. Projectively flat and numerically projectively flat vector bundles
  8. Foliation
  9. Ehresmann connection
  10. The standard setting
  11. Preparations for the proof of Theorem \ref{['thm_intro:main']}
  12. Bott (partial) connection and applications
  13. Abelian schemes
  14. Descent of line bundles
  15. Abundance
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $X$ be a complex projective manifold of dimension $n\geqslant 4$, and let $\mathscr{F} \subset T_X$ be a regular codimension $1$ foliation. Suppose that the normalized vector bundle $\textup{S}^{n-1}\mathscr{F}\otimes \det(\mathscr{F}^*)$ is numerically flat. Then one of the following holds.

Theorems & Definitions (132)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Lemma 2.5: demailly_peternell_schneider94
  • Definition 2.6
  • Lemma 2.7: jahnke_radloff_13
  • ...and 122 more