THE FOLIATED LEFSCHETZ HYPERPLANE THEOREM

Abstract

A foliation$(M,{\mathcal{F}})$is said to be$2$-calibrated if it admits a closed 2-form$\unicode[STIX]{x1D714}$making each leaf symplectic. By using approximately holomorphic techniques, a sequence$W_{k}$of$2$-calibrated submanifolds of codimension-$2$can be found for$(M,{\mathcal{F}},\unicode[STIX]{x1D714})$. 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.