THE FOLIATED LEFSCHETZ HYPERPLANE THEOREM

Abstract

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