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arXiv · 1805.07001

Rational curves in holomorphic symplectic varieties and Gromov-Witten invariants

Georg Oberdieck·Junliang Shen·Qizheng Yin
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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 $K3^{[n]}$ 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.

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Georg Oberdieck, Junliang Shen, Qizheng Yin. 2018-05-18. Rational curves in holomorphic symplectic varieties and Gromov-Witten invariants. https://arxiv.org/abs/1805.07001

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