On the topology of compact Stein surfaces

In this paper we obtain the following results: (1) Any compact Stein surface with boundary embeds naturally into a symplectic Lefschetz fibration over the 2-sphere. (2) There exists a minimal elliptic fibration over the 2-disk, which is not Stein. (3) The circle bundle over a genus n>1 surface with euler number e=-1 admits at least n+1 mutually non-homeomorphic simply-connected Stein fillings. (4) Any surface bundle over the circle, whose fiber is a closed surface of genus n>0 can be embedded into a closed symplectic 4-manifold, splitting the symplectic 4-manifold into two pieces both of which have positive b_2^+. (5) Every closed, oriented connected 3-manifold has a weakly symplectically fillable double cover, branched along a 2-component link.