On the existence of rational curves on projective hyperk\"ahler fourfolds

Haidong Liu

On the existence of rational curves on projective hyperk\"ahler fourfolds

Algebraic Geometry 2021-12-24 v3

Authors: Haidong Liu

Abstract

We show that if $X$ is a projective hyperk\"ahler 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 hyperk\"ahler fourfolds.

Keywords

projective hyperkahler fourfolds rational curves effective divisor nef cone oguiso conjecture

Cite

@article{arxiv.2111.04270,
  title  = {On the existence of rational curves on projective hyperk\"ahler fourfolds},
  author = {Haidong Liu},
  journal= {arXiv preprint arXiv:2111.04270},
  year   = {2021}
}

Comments

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 This paper will not be published unless the remaining case can be excluded; v3, we revised the proof of Theorem 3.3

On the existence of rational curves on projective hyperk\"ahler fourfolds

Abstract

Keywords

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