An inequality on the Hodge number $h^{1,1}$ of a fibration and the Mordell-Weil rankthe Mordell-Weil rank

An inequality on the Hodge number $h^{1,1}$ of a fibration and the Mordell-Weil rankthe Mordell-Weil rank

An this paper, we establish some formulas on the Mordell–Weil rank and the Hodge number $h^{1,1}$ for a fibration.

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  • [1] Gong C, Lu X, Tan SL. Families of curves over $P^1$ with 3 singular fibers. C R Acad Sci Paris Ser I 2013; 351: 375-380.
  • [2] Gong C, Lu J, Tan SL. On families of curves over $P^1$ with two singular fibers. Osaka J Math 2016; 53: 83-99.
  • [3] Gong C, Lu J, Tan SL. On the classification and Mordell-Weil groups of families of curves with two singular fibers. Available at http://math.ecnu.edu.cn/ jlu/7 classification.pdf.
  • [4] Griffiths P, Harris J. Principles of Algebraic Geometry. New York, NY, USA: John Wiley & Sons, Inc., 1978.
  • [5] Jiang Z, Sun H. Cohomological support loci of varieties of Albanese fiber dimension one. T Am Math Soc 2015; 367: 103-119.
  • [6] Kleiman S. The Picard Scheme. Providence, RI, USA: American Mathematical Society, 2005.
  • [7] Lu J, Tan SL, Yu F, Zuo K. A new inequality on the Hodge number h1,1 of algebraic surfaces. Math Z 2014; 276: 543-555.
  • [8] Mok N. Aspects of K ̈ahler geometry on arithmetic varieties. Several complex variables and complex geometry, part 2. In: Proceedings of Symposia in Pure Mathematics (Santa Cruz, CA, USA, 1989). Providence, RI, USA: American Mathematical Society, 1991, pp. 335-396.
  • [9] Mok N, To W. Eigensections on Kuga families of abelian varieties and finiteness of their Mordell-Weil groups. J Reine Angew Math 1993; 444: 29-78.
  • [10] Oguiso K. Shioda-Tate formula for an Abelian fibered variety and applications. J Korean Math Soc 2009; 46: 237-248.
  • [11] Oguiso K. Picard number of the generic fiber of an abelian fibered hyperk ̈ahler manifold. Math Ann 2009; 344: 929-937.
  • [12] Shioda T. On elliptic modular surfaces. J Math Soc Japan 1972; 24: 20-59.
  • [13] Shioda T. Mordell-Weil lattices for higher genus fibration over a curve. In: New Trends in Algebraic Geometry. Cambridge, UK: Cambridge University Press, pp. 359-373.
  • [14] Sun H. Pluricanonical maps of varieties of Albanese fiber dimension two. Math Z 2014; 277: 739-747.