Picard number of the generic fiber of an abelian fibered hyperkähler manifold

摘要

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. Mathematics Subject Classification (2000) 14D06 - 14J40 - 14J28