Absolutely $k$-convex domains and holomorphic foliations on homogeneous manifolds

Abstract

We consider a holomorphic foliation $\mathcal{F}$ of codimension $k\geq 1$ on a homogeneous compact Kähler manifold $X$ of dimension $n \gt k$. Assuming that the singular set ${\rm Sing}(\mathcal{F})$ of $\mathcal{F}$ is contained in an absolutely $k$-convex domain $U\subset X$, we prove that the determinant of normal bundle $\det(N_{\mathcal{F}})$ of $\mathcal{F}$ cannot be an ample line bundle, provided $[n/k]\geq 2k+3$. Here $[n/k]$ denotes the largest integer $\leq n/k.$