On symplectic coverings of the projective plane

G.-M. Greuel ; Vik. S. Kulikov

Izvestiya. Mathematics , Tome 69 (2005) no. 4, pp. 667-701

Résumé

We prove that a resolution of singularities of any finite covering of the projective complex plane branched along a Hurwitz curve $\overline H$, 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 $\overline H$ 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 $\overline H$ are investigated and applied to the calculation of the first Betti number $b_1(\overline X_n)$, where $\overline X_n$ is a resolution of singularities of an $n$-sheeted cyclic covering of $\mathbb C\mathbb P^2$ branched along $\overline H$, and possibly along the line “at infinity”. We prove that $b_1(\overline X_n)$ is even if $\overline H$ is an irreducible Hurwitz curve but, in contrast to the algebraic case, $b_1(\overline X_n)$ may take any non-negative value in the case when $\overline H$ consists of several components.