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

TL;DR

This paper proves that the Picard number of the generic fiber of an abelian fibered hyperkähler manifold over projective space is always one, and constructs examples with maximal Mordell-Weil rank.

Contribution

It establishes a universal Picard number result for generic fibers and constructs a hyperkähler manifold with maximal Mordell-Weil rank among known examples.

Findings

Picard number of generic fiber is always one.

Constructed hyperkähler manifold with Mordell-Weil rank 20.

Applications to the Mordell-Weil group.

Abstract

We shall show that the Picard number of the generic fiber, in the sense of scheme, of an abelian fibered hyperk\"ahler 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 hyperk\"ahler manifold of Mordell-Weil rank 20, which is the maximum possible among all known ones.