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 Izv. RAN. Ser. Mat., 2005, Volume 69, Issue 4, Pages 19–58 (Mi izv646)

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.

DOI: https://doi.org/10.4213/im646

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English version:
Izvestiya: Mathematics, 2005, 69:4, 667–701

Bibliographic databases:

UDC: 514.756.4
MSC: 14F35, 57R17, 14H20

Citation: G.-M. Greuel, Vik. S. Kulikov, “On symplectic coverings of the projective plane”, Izv. RAN. Ser. Mat., 69:4 (2005), 19–58; Izv. Math., 69:4 (2005), 667–701

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/izv646
• https://doi.org/10.4213/im646
• http://mi.mathnet.ru/eng/izv/v69/i4/p19

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Vik. S. Kulikov, “Alexander polynomials of Hurwitz curves”, Izv. Math., 70:1 (2006), 69–86
2. Vik. S. Kulikov, “Hurwitz curves”, Russian Math. Surveys, 62:6 (2007), 1043–1119
3. Vik. S. Kulikov, “Alexander modules of irreducible $C$-groups”, Izv. Math., 72:2 (2008), 305–344
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