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

Abstract

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.

Residues for flags of holomorphic foliations

“…Brunella and Perrone in [8] using integration current show a formula for residues for codimension one foliations. Recently Corrêa and Fernandez in [11] generalize this result for higher codimensional holomorphic foliations. Here using these tools and the Godbillon-Vey classes for flags of Dominguez [13], we present a formula for residue of flags.…”

“…Here using these tools and the Godbillon-Vey classes for flags of Dominguez [13], we present a formula for residue of flags. For a basic reference, see [11,22,13,8].…”

Analytic varieties invariant by foliations and Pfaff systems

“…An explicit calculation of the residues is difficult in general, see [5,29,9,19,32,38,10,39]. If the foliation F has dimension one with isolated singularities, Baum and Bott in [6] show that residues can be expressed in terms of a Grothendieck residue, i.e, for each x ∈ Sing(F ) we have…”

Dynamical aspects of foliations with ample normal bundle

“…In [6], Brunella gave an affirmative answer to his conjecture when X is a complex torus or, more generally, a compact homogeneous manifold (cf. [9] for higher codimensional foliations). Also, under the assumption that Pic(X) = Z, Brunella and Perrone confirmed the conjecture in [7].…”

“…[7, §2]), although we use it in a different way from the strategy of Brunella sketched in [5, §4]. In previous approaches [6,7,9], Baum-Bott's formula was used to localize the square of the first Chern class c 2 1 (N F ) to Sing(F ). Instead, we use the vanishing of the Baum-Bott class over a neighborhood of M to construct a holomorphic connection…”