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Ueda foliation problem for complex tori

Laurent Stolovitch, Xiaojun Wu

Abstract

We consider an embedded general complex torus $C_n$ into a complex manifold $M_{n+d}$ with a unitary flat normal bundle $N_C$. We show the existence of (non-singular) holomorphic foliation in a neighborhood of $C$ in $M$ having $C$ as leaf under some conditions.

Ueda foliation problem for complex tori

Abstract

We consider an embedded general complex torus into a complex manifold with a unitary flat normal bundle . We show the existence of (non-singular) holomorphic foliation in a neighborhood of in having as leaf under some conditions.
Paper Structure (3 sections, 9 theorems, 138 equations, 1 figure)

This paper contains 3 sections, 9 theorems, 138 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Setting
  3. Proof of the main result

Key Result

Theorem 1.1

Let $C$ be an $n$-dimensional complex torus, holomorphically embedded into a complex manifold $M_{n+d}$. Assume that $T_M|_C$ splits. Assume the normal bundle $N_C$ has (locally constant) unitary transition functions. Assume that $N_C$ is vertically strongly Diophantine (see Definition (dioph)). The

Figures (1)

  • Figure 1: Domains and their translates by the $\tilde{T}_j$'s for $\epsilon=0$ and their convex hull. When $\epsilon>0$, $\tilde{T}_j^k \omega'^*_{\epsilon}$ overlaps both $\tilde{T}_j^{k-1} \omega'^*_{\epsilon}$ and $\tilde{T}_j^{k+1} \omega'^*_{\epsilon}$.

Theorems & Definitions (24)

  • Theorem 1.1
  • Lemma 2.1
  • Proposition 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Definition 2.6
  • Lemma 2.7
  • proof
  • Remark 2.8
  • ...and 14 more