This article is cited in
3 papers
On symplectic coverings of the projective plane
G.-M. Greuela,
Vik. S. Kulikovb a Technical University of Kaiserslautern
b Steklov Mathematical Institute, Russian Academy of Sciences
Abstract:
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.
UDC:
514.756.4
MSC: 14F35,
57R17,
14H20 Received: 23.11.2004
DOI:
10.4213/im646
Fulltext:
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