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Lagrangian 4-planes in holomorphic symplectic varieties of K3[4]-type

Central European Journal of Mathematics volume 12, pages 952–975 (2014)Cite this article

Abstract

We classify the cohomology classes of Lagrangian 4-planes ℙ4 in a smooth manifold X deformation equivalent to a Hilbert scheme of four points on a K3 surface, up to the monodromy action. Classically, the Mori cone of effective curves on a K3 surface S is generated by nonnegative classes C, for which (C, C) ≥ 0, and nodal classes C, for which (C, C) = −2; Hassett and Tschinkel conjecture that the Mori cone of a holomorphic symplectic variety X is similarly controlled by “nodal” classes C such that (C, C) = −γ, for (·,·) now the Beauville-Bogomolov form, where γ classifies the geometry of the extremal contraction associated to C. In particular, they conjecture that for X deformation equivalent to a Hilbert scheme of n points on a K3 surface, the class C = ℓ of a line in a smooth Lagrangian n-plane ℙn must satisfy (ℓ,ℓ) = −(n + 3)/2. We prove the conjecture for n = 4 by computing the ring of monodromy invariants on X, and showing there is a unique monodromy orbit of Lagrangian 4-planes.

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Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, NY, 10012, USA

    Benjamin Bakker

  2. University of Notre Dame, 275 Hurley, Notre Dame, IN, 46556, USA

    Andrei Jorza

Authors
  1. Benjamin Bakker
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  2. Andrei Jorza
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Correspondence to Benjamin Bakker.

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Bakker, B., Jorza, A. Lagrangian 4-planes in holomorphic symplectic varieties of K3[4]-type. centr.eur.j.math. 12, 952–975 (2014). https://doi.org/10.2478/s11533-013-0389-3

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  • DOI: https://doi.org/10.2478/s11533-013-0389-3

MSC

  • 14C25
  • 14G05
  • 14J282

Keywords

  • Holomorphic symplectic variety
  • Cone of curves