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http://dx.doi.org/10.4134/JKMS.2009.46.2.237

SHIODA-TATE FORMULA FOR AN ABELIAN FIBERED VARIETY AND APPLICATIONS  

Oguiso, Keiji (DEPARTMENT OF ECONOMICS KEIO UNIVERSITY, SCHOOL OF MATHEMATICS KOREA INSTITUTE FOR ADVANCED STUDY)
Publication Information
Journal of the Korean Mathematical Society / v.46, no.2, 2009 , pp. 237-248 More about this Journal
Abstract
We give an explicit formula for the Mordell-Weil rank of an abelian fibered variety and some of its applications for an abelian fibered $hyperk{\ddot{a}}hler$ manifold. As a byproduct, we also give an explicit example of an abelian fibered variety in which the Picard number of the generic fiber in the sense of scheme is different from the Picard number of generic closed fibers.
Keywords
Mordell-Weil group;
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1 F. Campana, Reduction d'Albanese d'un morphisme propre et faiblement kahlerien. II, Compositio Math. 54 (1985), no. 3, 399–416.
2 Addendum: “On fibre space tructures of a projective irreducible symplectic manifold”, Topology 40 (2001), no. 2, 431–432.   DOI   ScienceOn
3 C. Voisin, Sur la stabilité des sous-variétés lagrangiennes des variétés symplectiques holomorphes, Complex projective geometry (Trieste, 989/Bergen, 1989), 294–303, London Math. Soc. Lecture Note Ser., 179, Cambridge Univ. Press, Cambridge, 1992.
4 M. Gross, D. Huybrechts, and D. Joyce, Calabi-Yau Manifolds and Related Geometries, Springer-Verlag, Berlin, 2003.
5 W. Barth, K. Hulek, C. Peters, and A. Van de Ven, Compact Complex Surfaces, Second edition, Springer-Verlag, Berlin, 2004.
6 A. Beauville, Variétés Kähleriennes dont la première classe de Chern est nulle, J. Differential Geom. 18 (1983), no. 4, 755–782.   DOI
7 F. Campana, Un critere d'isotrivialite pour les familles de varietes hyperkäleriennes sans facteur algebrique, math.AG/0408148.
8 C. Schoen, On fiber products of rational elliptic surfaces with section, Math. Z. 197 (1988), no. 2, 177–199.   DOI
9 G. Cornell and J. H. Silverman, Arithmetic Geometry, Springer-Verlag, New York, 1986.
10 M. Hindry, A Pacheco, and R. Wazir, Fibrations et conjecture de Tate, J. Number Theory 112 (2005), no. 2, 345–358.   DOI   ScienceOn
11 B. Kahn, Démonstration géométrique du théorème de Lang-Néron, math.AG/0703063.
12 Y. Kawamata, Crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces, Ann. of Math. (2) 127 (1988), no. 1, 93–163.   DOI   ScienceOn
13 Y. Kawamata, On the cone of divisors of Calabi-Yau fiber spaces, Internat. J. Math. 8 (1997), no. 5, 665–687.   DOI   ScienceOn
14 B. Moishezon, On n-dimensional compact complex manifolds having n algebraically independent meromorphic functions. I, Izv. Akad. Nauk SSSR Ser. Mat. 30 (1966), 133–174.
15 F. Campana, Coréduction algébrique d'un espace analytique faiblement kählérien compact, Invent. Math. 63 (1981), no. 2, 187–223.   DOI
16 S. Kondō, Automorphisms of algebraic K3 surfaces which act trivially on Picard groups, J. Math. Soc. Japan 44 (1992), no. 1, 75–98.   DOI
17 D. Matsushita, On fibre space structures of a projective irreducible symplectic manifold, Topology 38 (1999), no. 1, 79–83.   DOI   ScienceOn
18 D. Matsushita, Equidimensionality of Lagrangian fibrations on holomorphic ymplectic manifolds, Math. Res. Lett. 7 (2000), no. 4, 389–391.   DOI
19 D. Matsushita, Higher direct images of dualizing sheaves of Lagrangian fibrations, Amer. J. Math. 127 (2005), no. 2, 243–259.   DOI
20 F. Campana, Reduction d'Albanese d'un morphisme propre et faiblement kahlerien. I, Compositio Math. 54 (1985), no. 3, 373–398.
21 T. Shioda, On the Mordell-Weil lattices, Comment. Math. Univ. St. Paul. 39 (1990), no. 2, 211–240.
22 K. Oguiso, Local families of K3 surfaces and applications, J. Algebraic Geom. 12 (2003), no. 3, 405–433.   DOI   ScienceOn
23 J. Sawon, Deformations of holomorphic Lagrangian fibrations, math.AG/0509223.   DOI   ScienceOn
24 T. Shioda, On elliptic modular surfaces, J. Math. Soc. Japan 24 (1972), 20–59.   DOI
25 T. Shioda, Mordell-Weil lattices for higher genus fibration over a curve, New trends in algebraic geometry (Warwick, 1996), 359–373, London Math. Soc. Lecture Note Ser., 264, Cambridge Univ. Press, Cambridge, 1999.   DOI