November 2023 Density of Noether–Lefschetz loci of polarized irreducible holomorphic symplectic varieties and applications
Giovanni Mongardi, Gianluca Pacienza
Author Affiliations +
Kyoto J. Math. 63(4): 749-781 (November 2023). DOI: 10.1215/21562261-2023-0002


In this paper, we derive from deep results due to Clozel and Ullmo a sharp density result of Noether–Lefschetz loci inside the moduli space of marked (polarized) irreducible holomorphic symplectic (IHS) varieties. In particular, we obtain the density of Hilbert schemes of points on projective K3 surfaces and of projective generalized Kummer varieties in their moduli spaces. We present applications to the existence of rational curves on projective deformations of such varieties, to the study of the Mori cone of curves and of the associated extremal birational contractions, and a refinement of Hassett’s result on cubic fourfolds whose Fano variety of lines is isomorphic to a Hilbert scheme of two points on a K3 surface. We also discuss Voisin’s conjecture on the existence of coisotropic subvarieties on IHS varieties and relate it to a stronger statement on Noether–Lefschetz loci in their moduli spaces.


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Giovanni Mongardi. Gianluca Pacienza. "Density of Noether–Lefschetz loci of polarized irreducible holomorphic symplectic varieties and applications." Kyoto J. Math. 63 (4) 749 - 781, November 2023.


Received: 20 April 2020; Revised: 22 June 2021; Accepted: 25 October 2021; Published: November 2023
First available in Project Euclid: 18 September 2023

MathSciNet: MR4643003
Digital Object Identifier: 10.1215/21562261-2023-0002

Primary: 14C99
Secondary: 14J28 , 14J35 , 14J40

Keywords: coisotropic subvarieties , cubic fourfolds , generalized Kummer varieties , Hilbert schemes of points on K3 surfaces , holomorphic symplectic varieties , Rational curves

Rights: Copyright © 2023 by Kyoto University


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Vol.63 • No. 4 • November 2023
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