Flags of holomorphic foliations

Abstract

A flag of holomorphic foliations on a complex manifold M is an object consisting of a finite number of singular holomorphic foliations on M of growing dimensions such that the tangent sheaf of a fixed foliation is a subsheaf of the tangent sheaf of any of the foliations of higher dimension. We study some basic properties of these objects and, in P n C , n ≥ 3, we establish some necessary conditions for a foliation of lower dimension to leave invariant foliations of codimension one. Finally, still in P n C , we find bounds involving the degrees of polar classes of foliations in a flag.

Codimension One Holomorphic Distributions on the Projective Three-space

“…Let F be a codimension one distribution of degree 2 on P 3 with singular scheme Z; let U be the maximal 0-dimensional subsheaf of O Z , and let C ⊂ Z be the corresponding subscheme of pure dimension 1. The inequality (33) becomes deg(C) ≤ 7; using Theorem 3.1, we obtain (46) c 2 (T F ) = 6 − deg(C) and c 3 (T F ) = 18 − 4 deg(C) + 2p a (C).…”

“…If T F splits as a sum of line bundles, then only the two possibilities listed in the statement of the theorem occur. The degree and genus of the singular set are easily computed via formula (46).…”

Inequalities for Characteristic Numbers of Flags of Distributions and Foliations

“…This gives K 2 = 3 T P 3 (t 3 ) and Bott's formulae tells us that H 1 (P 3 , 3 T P 3 (t 3 )) = 0. In this case (7) and (8) coincide and Theorem 1.1 holds.…”

“…To prove the surjectivity of ζ it's enough, by (7), to show that H 1 (P n , K 2 ) = 0. This is done in the Lemma 3.1.…”

Residues for flags of holomorphic foliations

“…Some authors studied flags of singular holomorphic foliations: in [12] Corrêa and Soares proved an inequality involving the degrees of two distributions (foliations) which form a flag on projective spaces. Mol in [22] studied the polar classes of flags of holomorphic foliations.…”