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Holomorphic foliations with no transversely projective structure

Indranil Biswas, Sorin Dumitrescu

Abstract

We prove that on the product of two elliptic curves a generic nonsingular turbulent foliation does not admit any transversely projective structure.

Holomorphic foliations with no transversely projective structure

Abstract

We prove that on the product of two elliptic curves a generic nonsingular turbulent foliation does not admit any transversely projective structure.
Paper Structure (11 sections, 7 theorems, 98 equations)

This paper contains 11 sections, 7 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. A turbulent foliation
  3. Projective structure on a Riemann surface
  4. Definition of a projective structure
  5. A flat projective bundle
  6. Transversal projective structures
  7. Foliation and projective structure
  8. Another description of $\mathcal{F}$-projective structures
  9. $\mathcal{F}$--projective structure on the torus
  10. Properties of unique $\mathcal{F}$--projective structure
  11. Nonexistence of $\mathcal{F}$--projective structure

Key Result

Lemma 2.1

Take $\underline{x},\,\, \underline{y}\,\in\, {\rm Sym}^d_0(C)$ as above such that $\{x_1,\, \cdots,\, x_d\}\cap\{y_1,\, \cdots,\, y_d\}\, =\, \emptyset$. If there is a meromorphic function on $C$ whose zero divisor is $\underline{y}$ and the pole divisor is $\underline{x}$, then the two elements $\

Theorems & Definitions (16)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 4.1
  • proof
  • Remark 4.2
  • Lemma 4.3
  • proof
  • Proposition 5.1
  • ...and 6 more