On the existence of rational curves on projective hyperkähler fourfolds

Abstract

We show that if $X$ is a projective hyperkähler fourfold and there exists a nonzero effective divisor $D$ which is not of bi-elliptic type and contained in the boundary of the nef cone of $X$, then $X$ contains a rational curve. This is a very special case of Oguiso's conjecture for projective hyperkähler fourfolds.

v1, 10 pages, any comments are welcome; v2, the previous version is revised significantly because there is a (bi-elliptic) case missing in proof of the main theorem. This paper will not be published unless the remaining case can be excluded; v3, we revised the proof of Theorem 3.3

References in corpus (1)

• Strictly nef divisors on K-trivial fourfolds