Families of rational curves on holomorphic symplectic varieties and applications to zero-cycles

Abstract

We study families of rational curves on irreducible holomorphic symplectic varieties. We give a necessary and sufficient condition for a sufficiently ample linear system on a holomorphic symplectic variety of $K{3}^{[n]}$-type to contain a uniruled divisor covered by rational curves of primitive class. In particular, for any fixed $n$, we show that there are only finitely many polarization types of holomorphic symplectic variety of $K{3}^{[n]}$-type that do not contain such a uniruled divisor. As an application we provide a generalization of a result due to Beauville-Voisin on the Chow group of 0-cycles on such varieties.

This paper replaces, corrects and improves arxiv:1401.4071 by the first and the third authors, which will be subsequently withdraw. Nevertheless we kept the same title. Final version, to appear in Compositio Mathematica