JOURNAL BROWSE
Search
Advanced SearchSearch Tips
SHIODA-TATE FORMULA FOR AN ABELIAN FIBERED VARIETY AND APPLICATIONS
facebook(new window)  Pirnt(new window) E-mail(new window) Excel Download
 Title & Authors
SHIODA-TATE FORMULA FOR AN ABELIAN FIBERED VARIETY AND APPLICATIONS
Oguiso, Keiji;
  PDF(new window)
 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 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;
 Language
English
 Cited by
1.
Fibrations on four-folds with trivial canonical bundles, Geometriae Dedicata, 2014, 171, 1, 93  crossref(new windwow)
2.
Computing Néron–Severi groups and cycle class groups, Compositio Mathematica, 2015, 151, 04, 713  crossref(new windwow)
 References
1.
W. Barth, K. Hulek, C. Peters, and A. Van de Ven, Compact Complex Surfaces, Second edition, Springer-Verlag, Berlin, 2004.

2.
A. Beauville, Variétés Kähleriennes dont la première classe de Chern est nulle, J. Differential Geom. 18 (1983), no. 4, 755–782.

3.
F. Campana, Coréduction algébrique d'un espace analytique faiblement kählérien compact, Invent. Math. 63 (1981), no. 2, 187–223. crossref(new window)

4.
F. Campana, Reduction d'Albanese d'un morphisme propre et faiblement kahlerien. I, Compositio Math. 54 (1985), no. 3, 373–398.

5.
F. Campana, Un critere d'isotrivialite pour les familles de varietes hyperkäleriennes sans facteur algebrique, math.AG/0408148.

6.
G. Cornell and J. H. Silverman, Arithmetic Geometry, Springer-Verlag, New York, 1986.

7.
M. Gross, D. Huybrechts, and D. Joyce, Calabi-Yau Manifolds and Related Geometries, Springer-Verlag, Berlin, 2003.

8.
M. Hindry, A Pacheco, and R. Wazir, Fibrations et conjecture de Tate, J. Number Theory 112 (2005), no. 2, 345–358. crossref(new window)

9.
B. Kahn, Démonstration géométrique du théorème de Lang-Néron, math.AG/0703063.

10.
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. crossref(new window)

11.
Y. Kawamata, On the cone of divisors of Calabi-Yau fiber spaces, Internat. J. Math. 8 (1997), no. 5, 665–687. crossref(new window)

12.
S. Kondō, Automorphisms of algebraic K3 surfaces which act trivially on Picard groups, J. Math. Soc. Japan 44 (1992), no. 1, 75–98. crossref(new window)

13.
D. Matsushita, On fibre space structures of a projective irreducible symplectic manifold, Topology 38 (1999), no. 1, 79–83. crossref(new window)

14.
D. Matsushita, Equidimensionality of Lagrangian fibrations on holomorphic ymplectic manifolds, Math. Res. Lett. 7 (2000), no. 4, 389–391. crossref(new window)

15.
D. Matsushita, Higher direct images of dualizing sheaves of Lagrangian fibrations, Amer. J. Math. 127 (2005), no. 2, 243–259. crossref(new window)

16.
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.

17.
K. Oguiso, Local families of K3 surfaces and applications, J. Algebraic Geom. 12 (2003), no. 3, 405–433. crossref(new window)

18.
J. Sawon, Deformations of holomorphic Lagrangian fibrations, math.AG/0509223. crossref(new window)

19.
C. Schoen, On fiber products of rational elliptic surfaces with section, Math. Z. 197 (1988), no. 2, 177–199. crossref(new window)

20.
T. Shioda, On elliptic modular surfaces, J. Math. Soc. Japan 24 (1972), 20–59. crossref(new window)

21.
T. Shioda, On the Mordell-Weil lattices, Comment. Math. Univ. St. Paul. 39 (1990), no. 2, 211–240.

22.
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. crossref(new window)

23.
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.

24.
F. Campana, Reduction d'Albanese d'un morphisme propre et faiblement kahlerien. II, Compositio Math. 54 (1985), no. 3, 399–416.

25.
Addendum: “On fibre space tructures of a projective irreducible symplectic manifold”, Topology 40 (2001), no. 2, 431–432. crossref(new window)