Complex Codimension One Singular Foliations and Godbillon–Vey Sequences

Abstract

Let F be a codimension one singular holomorphic foliation on a compact complex manifold M . Assume that there exists a meromorphic vector field X on M generically transversal to F . Then, we prove that F is the meromorphic pull-back of an algebraic foliation on an algebraic manifold N , or F is transversely projective outside a compact hypersurface, improving our previous work [7].Such a vector field insures the existence of a global meromorphic Godbillon-Vey sequence for the foliation F . We derive sufficient conditions on this sequence insuring such alternative. For instance, if there exists a finite Godbillon-Vey sequence or if the Godbillon-Vey invariant is zero, then either F is the pull-back of a foliation on a surface, or F is transversely projective. We illustrate these results with many examples.

Deformation of rational curves along foliations

“…Proof. The first two possibilities are described in [5,Lemma 2.20]. For describing the last possibility, assume that F is defined a rational 1-form ω 0 and observe that the existence of two non-equivalent virtually transversely Euclidean structures is equivalent to the existence of two linearly independent logarithmic 1-forms ω 1 , ω ′ 1 , both with periods commensurable to πi and satisfying…”

“…Consider a lift of f to a rational function on X (still denoted by f ) and set ω 0 = α 0 , ω 1 = α 1 , ω 2 = α 2 + f α 0 . According to [5,Corollary 2.4] there exists a unique ω 3 such that the system of equations (3.1) is satisfied. Moreover, the uniqueness of ω 3 implies that its restriction to the generic fiber of f must be zero.…”

Holonomy representation of quasi-projective leaves of codimension one foliations

“…Representations of fundamental groups of quasi-projective manifolds in Diff(C, 0) ⊂ Diff(C, 0) appear as holonomy representations of algebraic leaves of codimension one holomorphic foliations. There is a conjecture, formulated by Cerveau, Lins Neto and others [4], on the structure of codimension one foliations on projective manifolds of dimension at least three which predicts that they admit a singular transversely projective structure (see [10] for a precise definition) or contain a subfoliation of codimension two by algebraic leaves. Theorem A is in accordance with this conjecture, and is potentially useful to investigate it.…”

Isotrivial Unfoldings and Structural Theorems for Foliations on Projective Spaces

“…In the particular case when H 0 (X, T F) = H 1 (X, T F) = 0, which will be important to us later, we have the equality (2) Lie(Aut(F)) = H 0 (X, aut(F)) = H 0 (X, u(F)) = Υ(F).…”