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

Abstract

We consider a holomorphic foliation $\mathcal{F}$ of codimension $k\ge 1$ on a homogeneous compact Kähler manifold $X$ of dimension $n>k$. Assuming that the singular set $Sing(\mathcal{F})$ 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 $[n/k]\ge 2k+3$. Here $[n/k]$ denotes the largest integer $\le n/k.$

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