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 Izv. RAN. Ser. Mat., 1995, Volume 59, Issue 1, Pages 157–184 (Mi izv7)

Completely integrable projective symplectic 4-dimensional varieties

D. G. Markushevich

Université Claude Bernard Lyon 1

Abstract: Families of Liouville tori on a completely integrable compact complex symplectic manifold are considered as a tool for constructing such manifolds: given a family of $n$-dimensional tori with degenerations over an $n$-dimensional base, find conditions which guarantee the existence of a symplectic structure on this family such that the generic fiber is maximal isotropic. This question is studied for families of Jacobians of genus 2 curves in terms of the relative compactified Jacobian and point Hilbert scheme. The question on possible bases of families of Liouville tori is investigated in using Fujita–Kawamata–Viehweg–Kollár results on positivity properties of direct images of relative dualizing sheaves. In the case when the base surface is the projective plane, it is proved that the family of Jacobians is Liouville iff it is the Mukai transform of the Fujiki–Beauville 4-fold built from a hyperelliptic K3 surface.
Bibliography: 44 titles.

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English version:
Izvestiya: Mathematics, 1995, 59:1, 159–187

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MSC: 14M99
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Citation: D. G. Markushevich, “Completely integrable projective symplectic 4-dimensional varieties”, Izv. RAN. Ser. Mat., 59:1 (1995), 157–184; Izv. Math., 59:1 (1995), 159–187

Citation in format AMSBIB
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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. Matsushita D., “On Fibre Space Structures of a Projective Irreducible Symplectic Manifold”, Topology, 38:1 (1999), 79–83
2. Sawon J., “Derived Equivalence of Holomorphic Symplectic Manifolds”, Algebraic Structures and Moduli Spaces, CRM Proceedings & Lecture Notes, 38, eds. Johnson W., Markman E., Amer Mathematical Soc, 2004, 193–211
3. Sawon J., “Lagrangian Fibrations on Hilbert Schemes of Points on K3 Surfaces”, J. Algebr. Geom., 16:3 (2007), 477–497
4. Sawon J., “Twisted Fourier-Mukai transforms for holomorphic symplectic four-folds”, Advances in Mathematics, 218:3 (2008), 828–864
5. J. Sawon, “Foliations on Hypersurfaces in Holomorphic Symplectic Manifolds”, Internat Math Res Notices, 2009
6. Justin Sawon, “Fibrations on four-folds with trivial canonical bundles”, Geom Dedicata, 2013
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