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This article is cited in 3 scientific papers (total 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.
Received: 23.11.2004
Citation:
G.-M. Greuel, Vik. S. Kulikov, “On symplectic coverings of the projective plane”, Izv. Math., 69:4 (2005), 667–701
Linking options:
https://www.mathnet.ru/eng/im646https://doi.org/10.1070/IM2005v069n04ABEH001651 https://www.mathnet.ru/eng/im/v69/i4/p19
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Abstract page: | 609 | Russian version PDF: | 209 | English version PDF: | 27 | References: | 94 | First page: | 1 |
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