
Mathematics 2004
On Symplectic Coverings of the Projective PlaneDOI: 10.1070/IM2005v069n04ABEH001651 Abstract: We prove that a resolution of singularities of any finite covering of the projective plane branched along a Hurwitz curve $\bar H$ and, maybe, along a line "at infinity" can be embedded as a symplectic submanifold into some projective algebraic manifold equipped with an integer K\"{a}hler symplectic form (assuming that if $\bar H$ has negative nodes, then the covering is nonsingular over them). For cyclic coverings we can realize this embeddings into a rational algebraic 3fold. Properties of the Alexander polynomial of $\bar{H}$ are investigated and applied to the calculation of the first Betti number $b_1(\bar X_n)$ of a resolution $\bar X_n$ of singularities of $n$sheeted cyclic coverings of $\mathbb C\mathbb P^2$ branched along $\bar H$ and, maybe, along a line "at infinity". We prove that $b_1(\bar X_n)$ is even if $\bar H$ is an irreducible Hurwitz curve but, in contrast to the algebraic case, that it can take any nonnegative value in the case when $\bar H$ consists of several irreducible components.
