The foliated Lefschetz hyperplane theorem

TL;DR

This paper extends the Lefschetz hyperplane theorem to foliations with symplectic leaves, using approximately holomorphic techniques to analyze the topology of submanifolds within foliated structures.

Contribution

It establishes a Lefschetz hyperplane theorem for 2-calibrated foliations, linking leaf topology with symplectic geometry and transverse structure.

Findings

Lefschetz theorem applies to pairs (F, F ∩ W_k) in foliations

Constructs sequences of 2-calibrated submanifolds using holomorphic techniques

Derives consequences for the transverse geometry of foliations

Abstract

A foliation $(M,\mathcal{F})$ is said to be $2$--calibrated if it admits a closed 2-form $\omega$ making each leaf symplectic. By using approximately holomorphic techniques, a sequence $W_k$ of $2$--calibrated submanifolds of codimension--$2$ can be found for $(M, \mathcal{F}, \omega)$. Our main result says that the Lefschetz hyperplane theorem holds for the pairs $(F, F \cap W_k)$, with $F$ any leaf of $\mathcal{F}$. This is applied to draw important consequences on the transverse geometry of such foliations.