Central European Journal of Mathematics

, Volume 12, Issue 7, pp 952–975 | Cite as

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

  • Benjamin BakkerEmail author
  • Andrei Jorza
Research Article


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 ℙ n 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.


Holomorphic symplectic variety Cone of curves 


14C25 14G05 14J282 


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Copyright information

© Versita Warsaw and Springer-Verlag Wien 2013

Authors and Affiliations

  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  2. 2.University of Notre DameNotre DameUSA

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