Minimal invariant varieties and first integrals for algebraic foliations

Abstract

Let X be an irreducible algebraic variety over C, endowed with an algebraic foliation F. In this paper, we introduce the notion of minimal invariant variety

Parallelisms & Lie Connections

“…Around each point of the domain of ∇ there is a parallelism, by possibly transcendental vector fields, such that ∇ is its associated connection. Then, Lemma 3.3 states (1)⇒ (2). Taking into account that (∇ rec ) rec = ∇ we have the desired equivalence.…”

Deformation of rational curves along foliations

“…We denote by F tang the foliation on M with general leaf given by the Zariski closure of a leaf of F tang . The existence of a foliation with these properties follows from [1]. Theorem 4.6 (Theorem B of the Introduction).…”

“…Transversely projective foliations. Similarly, a foliation F on X is called transversely projective if for any rational 1-form ω 0 defining F there exists rational 1-forms ω 1 and ω 2 such that…”

“…Let F be a codimension one foliation of degree three on P 3 . The canonical bundle of F is K F = O P 3 (1). Therefore, if f : P 1 → P 3 is the parametrization of a general line then h 0 (P 1 , f * T F ) = 0.…”

“…If the general leaf of F tang is not algebraic then Theorem C guarantees that F is defined by a closed rational 1-form without divisorial components in its zero set. Thus F is described by item (1).…”