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On symplectic coverings of a projective plane. (English. Russian original) Zbl 1093.14023

Izv. Math. 69, No. 4, 667-701 (2005); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 69, No. 4, 19-58 (2005).
The authors 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. For cyclic coverings they 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”. They 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.

MSC:

14E20 Coverings in algebraic geometry
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
57R17 Symplectic and contact topology in high or arbitrary dimension
14H20 Singularities of curves, local rings
14J80 Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants)
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