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arXiv · 1906.05817

Elliptic curves in hyper-K\"ahler varieties

Denis Nesterov·Georg Oberdieck
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Abstract

We show that the moduli space of elliptic curves of minimal degree in a general Fano variety of lines of a cubic fourfold is a non-singular curve of genus $631$. The curve admits a natural involution with connected quotient. We find that the general Fano contains precisely $3780$ elliptic curves of minimal degree with fixed (general) $j$-invariant. More generally, we express (modulo a transversality result) the enumerative count of elliptic curves of minimal degree in hyper-K\"ahler varieties with fixed $j$-invariant in terms of Gromov--Witten invariants. In $K3[2]$-type this leads to explicit formulas of these counts in terms of modular forms.

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Denis Nesterov, Georg Oberdieck. 2019-06-13. Elliptic curves in hyper-K\"ahler varieties. https://arxiv.org/abs/1906.05817

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