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Izvestiya: Mathematics, 2005, Volume 69, Issue 4, Pages 667–701
DOI: https://doi.org/10.1070/IM2005v069n04ABEH001651
(Mi im646)
 

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
References:
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
Bibliographic databases:
Document Type: Article
UDC: 514.756.4
MSC: 14F35, 57R17, 14H20
Language: English
Original paper language: Russian
Citation: G.-M. Greuel, Vik. S. Kulikov, “On symplectic coverings of the projective plane”, Izv. Math., 69:4 (2005), 667–701
Citation in format AMSBIB
\Bibitem{GreKul05}
\by G.-M.~Greuel, Vik.~S.~Kulikov
\paper On symplectic coverings of the projective plane
\jour Izv. Math.
\yr 2005
\vol 69
\issue 4
\pages 667--701
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\crossref{https://doi.org/10.1070/IM2005v069n04ABEH001651}
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Linking options:
  • https://www.mathnet.ru/eng/im646
  • https://doi.org/10.1070/IM2005v069n04ABEH001651
  • https://www.mathnet.ru/eng/im/v69/i4/p19
  • This publication is cited in the following 3 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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    Abstract page:609
    Russian version PDF:209
    English version PDF:27
    References:94
    First page:1
     
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