Which algebraic surfaces admit algebraic foliations?

Question: Which projective algebraic surfaces admit algebraic foliations?

I realize that the question is a bit too general. There is a classification of minimal surfaces – not all the surfaces. And I do not think, one can say something about foliations after blow up. But I am looking for at least partial results.

More concrete question may be: is there a surface, admitting a foliation, in each birational class.

• What do mean by an algebraic foliation? A foliation with all leaves contained in algebraic curves? The foliations you want can be singular? – Alan Muniz Jan 21 '16 at 0:22
• In general you can twist the tangent bundle of the surface with a very ample line bundle and get a section that will define a foliation. – Alan Muniz Jan 21 '16 at 0:24
• I meant foliation without singularities. – quinque Jan 21 '16 at 7:48
• These are described in Brunella's paper numdam.org/numdam-bin/fitem?id=ASENS_1997_4_30_5_569_0 – Alan Muniz Jan 21 '16 at 12:11
• @Alan Muniz Sorry, I do not speak French. – quinque Jan 26 '16 at 18:31

Let me state Brunella's theorem that describes the regular foliated surfaces.

Thm: Let $X$ ba a complex projective surface and let $\mathcal{F}$ be a holomorphic foliation without singularities on $X$. Then we fall in one of the following cases:

1) $\mathcal{F}$ is a fibration;

2) $\mathcal{F}$ is trabnsverse to a fibration, hence it is the suspension of an automorphism group of an algebraic curve;

3) $\mathcal{F}$ in a turbulent foliation on an elliptic surface;

4) $\mathcal{F}$ is a linear foliation on a complex torus;

5) $\mathcal{F}$ a transversely hyperbolic foliation with dense leaves whose universal cover is a disk fibration over the disk.

A simple obstruction to the existence of regular foliations with tangent bundle $L$ on a surface $X$ is the vanishing of the Chern class $$c_2(TX-L)$$

This measures the number of singularities by Baum-Bott's Theorem.

Another way to see this is that $X$ admits a regular foliation if and only if $TX$ has a holomorphic sub-line-bundle.