Elliptic curves in Hyper-Kähler varieties

• Verfasst von: Nesterov, Denis [VerfasserIn]
• Verfasst von: Oberdieck, Georg [VerfasserIn]
• Titel: Elliptic curves in Hyper-Kähler varieties
• Verf.angabe: Denis Nesterov, Georg Oberdieck
• E-Jahr: 2021
• Jahr: February 2021
• Umfang: 29 S.
• Fussnoten: Online veröffentlicht: 14. Februar 2020 ; Gesehen am 14.01.2025
• Titel Quelle: Enthalten in: International mathematics research notices
• Ort Quelle: Oxford : Oxford University Press, 1991
• Jahr Quelle: 2021
• Band/Heft Quelle: (2021), 4 vom: Feb., Seite 2962-2990
• ISSN Quelle: 1687-0247
• DOI: doi:10.1093/imrn/rnaa016
• Datenträger: Online-Ressource
• Sprache: eng
• K10plus-PPN: 1914521560

Abstract

We show that the moduli space of elliptic curves of minimal degree in a general Fano variety of lines of a cubic four-fold 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 $3,780$ 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-Kähler 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.