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Foliated affine and projective structures

Bertrand Deroin, Adolfo Guillot

Abstract

We formalize the concepts of holomorphic affine and projective structures along the leaves of holomorphic foliations by curves on complex manifolds. We show that many foliations admit such structures, we provide local normal forms for them at singular points of the foliation, and we prove some index formulae in the case where the ambient manifold is compact. As a consequence of these, we establish that a regular foliation of general type on a compact algebraic manifold of even dimension does not admit a foliated projective structure. Finally, we classify foliated affine and projective structures along regular foliations on compact complex surfaces.

Foliated affine and projective structures

Abstract

We formalize the concepts of holomorphic affine and projective structures along the leaves of holomorphic foliations by curves on complex manifolds. We show that many foliations admit such structures, we provide local normal forms for them at singular points of the foliation, and we prove some index formulae in the case where the ambient manifold is compact. As a consequence of these, we establish that a regular foliation of general type on a compact algebraic manifold of even dimension does not admit a foliated projective structure. Finally, we classify foliated affine and projective structures along regular foliations on compact complex surfaces.

Paper Structure

This paper contains 31 sections, 20 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Definitions and the problem of existence
  3. Foliated affine structures
  4. The foliated setting
  5. Foliated connections
  6. A cohomological obstruction
  7. More on foliated connections
  8. Foliated projective structures
  9. Local normal forms
  10. Normal forms for foliated affine structures
  11. Affine structures with singularities on curves
  12. The foliated case
  13. Normal forms for foliated projective structures
  14. Projective structures with singularities on curves
  15. The foliated case
  16. ...and 16 more sections

Key Result

Lemma 2.6

Let $M$ be a manifold, $\mathcal{F}$ a foliation on $M$, $p\in M\setminus\mathrm{Sing}(\mathcal{F})$. Let $X$ be a meromorphic vector field defined in a neighborhood of $p$ whose divisor of zeros and poles $D$ is invariant by $\mathcal{F}$ and which is tangent to $\mathcal{F}$ away from it. Then, in

Theorems & Definitions (62)

  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4: Inoue surfaces
  • Example 2.5: Hopf surfaces
  • Lemma 2.6: Extension Lemma guillot-rebelo
  • proof
  • Example 2.7: Elliptic fibrations
  • Lemma 2.8
  • proof
  • ...and 52 more