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

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 = of a line in a smooth Lagrangian n-plane ℙn must satisfy (,) = −(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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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