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Turbulent holomorphic foliations on compact complex tori and transversely holomorphic Cartan geometry

Indranil Biswas, Sorin Dumitrescu

TL;DR

The paper addresses the classification of transversely holomorphic Cartan geometries on nonsingular foliations of compact complex tori. It develops generating distributions with associated Bott connections and introduces transversely branched Cartan geometries, proving that every smooth turbulent foliation admits only flat transverse $G/H$-structures and that such structures are unique when the underlying flat partial connection is fixed. It also establishes a rigidity result: any transversely branched Cartan geometry on a torus is flat and unique up to the underlying data. These results impose strong restrictions on transverse geometric structures on torus foliations and point toward a complete higher-codimension classification.

Abstract

We define a class of nonsingular holomorphic foliations on compact complex tori which generalizes (in higher codimension) the turbulent foliations of codimension one constructed by Ghys. For those smooth turbulent foliations we prove that all transversely holomorphic Cartan geometries are flat. We also establish a uniqueness result for the transversely holomorphic Cartan geometries.

Turbulent holomorphic foliations on compact complex tori and transversely holomorphic Cartan geometry

TL;DR

The paper addresses the classification of transversely holomorphic Cartan geometries on nonsingular foliations of compact complex tori. It develops generating distributions with associated Bott connections and introduces transversely branched Cartan geometries, proving that every smooth turbulent foliation admits only flat transverse -structures and that such structures are unique when the underlying flat partial connection is fixed. It also establishes a rigidity result: any transversely branched Cartan geometry on a torus is flat and unique up to the underlying data. These results impose strong restrictions on transverse geometric structures on torus foliations and point toward a complete higher-codimension classification.

Abstract

We define a class of nonsingular holomorphic foliations on compact complex tori which generalizes (in higher codimension) the turbulent foliations of codimension one constructed by Ghys. For those smooth turbulent foliations we prove that all transversely holomorphic Cartan geometries are flat. We also establish a uniqueness result for the transversely holomorphic Cartan geometries.

Paper Structure

This paper contains 7 sections, 4 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Generating distributions on a torus
  3. Turbulent foliation and transversely branched Cartan geometry
  4. Partial connection along a foliation
  5. Transversely branched Cartan geometry
  6. Connection on $E_G$
  7. Rigidity of branched Cartan geometries on torus

Key Result

Lemma 2.1

Let $\mathcal{F}$ as in e2 be a generating subbundle. Let $V$ be a holomorphic vector bundle on $A$ with $\mu_{\rm max}(V)\, \leq\, 0$ (see e5). Then for all $j\, \geq\, 1$, where $\mathcal{N}$ is constructed in e4.

Theorems & Definitions (6)

  • Lemma 2.1
  • proof
  • Lemma 3.1: BD
  • Lemma 3.2: BD
  • Theorem 4.1
  • proof