Absolutely k-convex domains and holomorphic foliations on homogeneous manifolds

Abstract

We consider a holomorphic foliation ℱ of codimension k ≥ 1 on a homogeneous compact Kähler manifold X of dimension n > k. Assuming that the singular set Sing(ℱ) of ℱ is contained in an absolutely k-convex domain U ⊂ X, we prove that the determinant of normal bundle det(N_ℱ) of ℱ cannot be an ample line bundle, provided [n/k] ≥ 2k + 3. Here [n/k] denotes the largest integer ≤ n/k.