Abstract
We shall show that the Picard number of the generic fiber of an abelian fibered hyperkähler manifold over the projective space is always one. We then give a few applications for the Mordell-Weil group. In particular, by deforming O’Grady’s 10-dimensional manifold, we construct an abelian fibered hyperkähler manifold of Mordell-Weil rank 20, which is the maximum possible among all known ones.
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References
Barth W., Hulek K., Peters C., Van de Ven A.: Compact Complex Surfaces. Springer, Berlin (2004)
Beauville A.: Variétés Kähleriennes dont la première classe de Chern est nulle. J. Differ. Geom. 18, 755–782 (1983)
Fantechi B., Göttsche L., Illusie L., Kleiman S.L., Nitsure N., Vistoli A.: Fundamental algebraic geometry. Grothendieck, FGA explained. Mathematical Surveys and Monographs, vol. 123. American Mathematical Society, Providence (2005)
Gross, M., Huybrechts, D., Joyce, D.: Calabi-Yau manifolds and related geometries. Lectures from the Summer School held in Nordfjordeid, June 2001. Universitext. Springer, Berlin (2003)
Grothendieck A.: Fondements de la Géométrie Algébrique. Secretariat mathématique, Paris (1962)
Hindry M., Pacheco A., Wazir R.: Fibrations et conjecture de Tate. J. Number Theory 112, 345–368 (2005)
Huybrechts, D.: Compact hyperkähler manifolds: basic results. Invent. Math. 135, 63–113 (1999) (Erratum: Compact hyperkähler manifolds: basic results. Invent. Math. 152, 209–212 (2003))
Huybrechts, D., Lehn, M.: The geometry of moduli spaces of sheaves. Aspects of Mathematics, vol. E31. Friedr. Vieweg and Sohn, Braunschweig (1997)
Hwang J.M.: Base manifolds for fibrations of projective irreducible symplectic manifolds. Invent. math. 174, 625–644 (2008) arXiv:0711.3224
Hwang, J.M., Oguiso, K.: Characteristic foliation on the discriminantal hypersurface of a holomorphic Lagrangian fibration. arXiv:0710.2376
Kahn, B.: Démonstration géométrique du Théorème de Lang-Néron. math.AG/0703063
Lehn M., Sorger C.: La singularité de O’Grady. J. Algebra Geom. 15, 753–770 (2006)
Matsushita, D.: On fibre space structures of a projective irreducible symplectic manifold. Topology 38, 79–83 (1999) (Addendum: On fibre space structures of a projective irreducible symplectic manifold. Topology 40, 431–432 (2001))
Matsushita D.: Equidimensionality of Lagrangian fibrations on holomorphic symplectic manifolds. Math. Res. Lett. 7, 389–391 (2000)
Matsushita D.: Higher direct images of dualizing sheaves of Lagrangian fibrations. Am. J. Math. 127, 243–259 (2005)
O’Grady K.G.: Desingularized moduli spaces of sheaves on a K3. J. Reine Angew. Math. 512, 49–117 (1999)
Oguiso K.: Local families of K3 surfaces and applications. J. Algebra Geom. 12, 405–433 (2003)
Oguiso, K.: Shioda-Tate formula for an abelian fibered variety and applications, mathAG/0703245 (to appear in Korean J. Math.)
Rapagnetta A.: Topological invariants of O’Grady’s six dimensional irreducible symplectic variety. Math. Z. 256, 1–34 (2007)
Rapagnetta A.: On the Beauville form of the known irreducible symplectic varieties. Math. Ann. 340, 77–95 (2008)
Sawon J.: Abelian fibred holomorphic symplectic manifolds. Turkish J. Math. 27, 197–230 (2003)
Sawon, J.: Deformations of holomorphic Lagrangian fibrations. math.AG/0509223
Voisin, C.: Sur la stabilité des sous-variétés lagrangiennes des variétés symplectiques holomorphes, Complex projective geometry. London Math. Soc. Lecture Note Ser., vol. 179, pp. 294–303. Cambridge University Press, Cambridge (1992)
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Oguiso, K. Picard number of the generic fiber of an abelian fibered hyperkähler manifold. Math. Ann. 344, 929–937 (2009). https://doi.org/10.1007/s00208-009-0335-7
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DOI: https://doi.org/10.1007/s00208-009-0335-7