Note on Holomorphic Foliations with Turbulences on Elliptic Surfaces

Let M be a complex projective manifold. We say that M has a foliation by curves if there exists a line bundle L on M and a non-zero homomorphism i : L → T_M, where T_M is a tangent bundle of M. If the above homomorphism L → T_M is injective, then we say that a foliation is nonsingular. Let L_α be a 1-dimensional connected manifold. Then we say that L_α is a leaf of foliation i : L → TM if M = ∪_αL_α, L_α∩L_β =(0+/) for α≠β, and for x ∈L_α i(L)_x is a tangent bundle of L_α at x. In this paper we consider the case in which M is a projective surface. We use a notation S instead of M.
If S is a ruled surface, that is, there exists a surjective morphism with connected fibers π : S → C such that any general fiber of π is P^1, where C is a smooth projective curve, then the foliations by curves on S have been studied by Gomez-Mont ([G-M II]). Here we consider the case in which there exists a surjective morphism π : S → C with connected fibers such that any general fiber is an elliptic curve. We call this surface an elliptic surface over a smooth projective curve. Here we note that elliptic surfaces may have singular fibers and all types of singular fibers have been classified by Kodaira.
This paper consists of the following three parts;
(1) examples of special type of foliations on ellitpic surfaces,
(2) a family of foliations on elliptic surfaces,
(3) the existence of elliptic surfaces which have foliations.
In [B], Brunella obtained some interesting results for foliations without singularities on non-singular algebraic surfaces, and pointed out that a turbulent foliation can appear (un feuilletage tourbillonne). Here we mean by a foliation a holomorphic one, and discuss foliations on elliptic surfaces.