On symplectic coverings of the projective plane

Abstract

We prove that a resolution of singularities of any finite covering of the projective complex plane branched along a Hurwitz curve , and possibly along the line “at infinity”, can be embedded as a symplectic submanifold in some projective algebraic manifold equipped with an integer Kähler symplectic form. (If has negative nodes, then the covering is assumed to be non-singular over them.) For cyclic coverings, we can realize these embeddings in a rational complex 3-fold. Properties of the Alexander polynomial of are investigated and applied to the calculation of the first Betti number , where is a resolution of singularities of an -sheeted cyclic covering of branched along , and possibly along the line “at infinity”. We prove that is even if is an irreducible Hurwitz curve but, in contrast to the algebraic case, may take any non-negative value in the case when consists of several components.