Codimension One Holomorphic Distributions on the Projective Three-space

Abstract

We study codimension one holomorphic distributions on the projective three-space, analyzing the properties of their singular schemes and tangent sheaves. In particular, we provide a classification of codimension one distributions of degree at most 2 with locally free tangent sheaves, and show that codimension one distributions of arbitrary degree with only isolated singularities have stable tangent sheaves. Furthermore, we describe the moduli space of distributions in terms of Grothendieck's Quot-scheme for the tangent bundle. In certain cases, we show that the moduli space of codimension one distributions on the projective space is an irreducible, nonsingular quasi-projective variety. Finally, we prove that every rational foliation, and certain logarithmic foliations have stable tangent sheaves.

On holomorphic distributions on Fano threefolds

“…In [5], O. Calvo-Andrade, M. Corrêa and M. Jardim showed a cohomological criterion for the connectedness of the singular scheme of codimension one distributions on P 3 [5, Theorem 3.7]. We extend this result for the others Fano threefolds with Picard number one.…”

“…Now, we will see when the singular locus of a codimension one distribution is connected. In [5], O. Calvo-Andrade, M. Corrêa and M. Jardim present a homological criterion for connectedness of the singular scheme of codimension 1 distributions on P 3 .…”

“…Proof: If ι X = 4, then the result follows from [5, Theorem 9.5]. If ι X = 3, it follows from Theorem 6.1.…”

“…In complex manifolds, holomorphic distributions and foliations have been much studied (see [1], [5], [8], [16]). An important problem is to analyse when the tangent and conormal sheaves split, as well as the properties of singular schemes of distributions.…”

“…In [5], O. Calvo-Andrade, M. Corrêa and M. Jardim showed that if F is a codimension one distribution on P 3 with T F is locally free and c 1 (T F ) = 0, then T F is either split or stable. We prove that this in fact holds for any Fano threefold with X Picard number ρ(X) = 1.…”