Transformation groups of holomorphic foliations

Abstract

We prove that the self-bimeromorphisms group of a foliation of general type on a projective surface is finite. Along the proof we study the structure of arbitrary codimension foliations on projective varieties invariant by an infinite linear algebraic group.

The Noether–Fano inequalities for codimension one singular holomorphic foliations

“…This is a weaker version of a result of [2], [16] and [3], asserting that the group of birational symmetries of a reduced foliation with maximal foliated Kodaira dimension κ = 2 (cf. [12], [1], [11]) coincides with its group of automorphisms and is a finite group.…”

On transcendental automorphisms of algebraic foliations

“…Denote by Bim(F) the group of bimeromorphic maps Φ : M M which preserve F in M (these maps take leaves of F onto leaves of F wherever defined). Combining the extended Lemma 1 with Theorem 1 of [20] we immediately obtain: Corollary 1. Let F be a codimension one singular holomorphic foliation on CP (2) and Γ ⊂ CP (2) a non-invariant algebraic curve.…”

“…In this paper we study the group of (self) isomorphisms of a foliation. Given a codimension one holomorphic foliation F with singularities on a complex manifold M we denote by Aut(F) the maximal subgroup of Aut(M ) whose elements preserve the foliation F. This object has been studied in [20] where it is proven that Aut(F) is finite for F of general type on a (compact) projective surface. We recall (cf.…”

“…If in addition F is of general type (cf. [20]) then Aut(F ) is finite. For a foliation with hyperbolic singularities at infinity, if there is a transcendental automorphism then the foliation is either linear logarithmic, Riccati or chaotic (cf.…”

“…We recall (cf. [20]) that a foliation F on a projective surface M 2 is of general type if its Kodaira dimension (cf. [17]) is equal to 2.…”