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

Benjamin Bakker; Andrei Jorza

Open Mathematics (2014)

  • Volume: 12, Issue: 7, page 952-975
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Benjamin Bakker, and Andrei Jorza. "Lagrangian 4-planes in holomorphic symplectic varieties of K3[4]-type." Open Mathematics 12.7 (2014): 952-975. <http://eudml.org/doc/269615>.

@article{BenjaminBakker2014,
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.},
author = {Benjamin Bakker, Andrei Jorza},
journal = {Open Mathematics},
keywords = {Holomorphic symplectic variety; Cone of curves; holomorphic symplectic variety; cone of curves; Lagrangian planes},
language = {eng},
number = {7},
pages = {952-975},
title = {Lagrangian 4-planes in holomorphic symplectic varieties of K3[4]-type},
url = {http://eudml.org/doc/269615},
volume = {12},
year = {2014},
}

TY - JOUR
AU - Benjamin Bakker
AU - Andrei Jorza
TI - Lagrangian 4-planes in holomorphic symplectic varieties of K3[4]-type
JO - Open Mathematics
PY - 2014
VL - 12
IS - 7
SP - 952
EP - 975
AB - 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.
LA - eng
KW - Holomorphic symplectic variety; Cone of curves; holomorphic symplectic variety; cone of curves; Lagrangian planes
UR - http://eudml.org/doc/269615
ER -

References

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