Rational curves in holomorphic symplectic varieties and Gromov-Witten invariants

Abstract

We use Gromov-Witten theory to study rational curves in holomorphic symplectic varieties. We present a numerical criterion for the existence of uniruled divisors swept out by rational curves in the primitive curve class of a very general holomorphic symplectic variety of $K{3}^{[n]}$ type. We also classify all rational curves in the primitive curve class of the Fano variety of lines in a very general cubic $4$-fold, and prove the irreducibility of the corresponding moduli space. Our proofs rely on Gromov-Witten calculations by the first author, and in the Fano case on a geometric construction of Voisin. In the Fano case a second proof via classical geometry is sketched.

Minor changes. Replace Proposition 3.2 in v3, whose proof contains a gap, by Lemma 3.2. Main theorems 0.1 and 0.2 remain unchanged. Modifications comparing to v3 are summarized in Footnote 3 (Page 4)