Symplectic four-manifolds and projective surfaces

Abstract

In this paper we consider symplectic versions of some results and constructions from the theory of complex projective surfaces with infinite fundamental groups. We introduce series of simple examples of symplectic fourfolds which are not Kähler. All of them have infinite fundamental groups which are fundamental groups of complex projective surfaces and contain symplectically embedded Riemann surfaces with positive self-intersection and a small image of their fundamental groups inside the fundamental group of the ambient symplectic fourfold. We have shown that there are no analogues of Zariski-Nori theorems for symplectic fourfolds. Our main results concern symplectic pencils of symplectically embedded Riemann surfaces. We give a universal construction of such pencils with rather arbitrary properties (any fundamental group in particular). We also give an obstruction for a symplectic Lefschetz pencil to be Kähler. Our construction suggests that the embedding of the local monodromy of the fiber of the above pencils in their global monodromy is an invariant of the symplectic structure.