The foliated Lefschetz hyperplane theorem

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.

Title and abstract modified. Section 2 on Lie groupoids and essential equivalence greatly reduced. bibliography updated. DOI added (to appear in Nagoya Math. J.)

References in corpus (1)

• Regularization and minimization of Haefliger structures of codimension one

Cited by in corpus (1)

• Conformal Symplectic structures, Foliations and Contact Structures