Hilbert schemes of K3 surfaces are dense in moduli

Markman, Eyal; Mehrotra, Sukhendu

doi:10.48550/arxiv.1201.0031

Cited by 9 publications

(11 citation statements)

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“…Remark 1.6. By [MM12], Hilbert schemes of n points on projective K3 surfaces are dense in the moduli space of hyperkähler varieties of K3 [n] -type.…”

Section: Cones Of Curves and Divisorsmentioning

confidence: 99%

MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations

2014

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We use wall-crossing with respect to Bridgeland stability conditions to systematically study the birational geometry of a moduli space M of stable sheaves on a K3 surface X:(a) We describe the nef cone, the movable cone, and the effective cone of M in terms of the Mukai lattice of X. (b) We establish a long-standing conjecture that predicts the existence of a birational Lagrangian fibration on M whenever M admits an integral divisor class D of square zero (with respect to the Beauville-Bogomolov form). These results are proved using a natural map from the space of Bridgeland stability conditions Stab(X) to the cone Mov(X) of movable divisors on M ; this map relates wall-crossing in Stab(X) to birational transformations of M . In particular, every minimal model of M appears as a moduli space of Bridgeland-stable objects on X.

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“…Remark 1.6. By [MM12], Hilbert schemes of n points on projective K3 surfaces are dense in the moduli space of hyperkähler varieties of K3 [n] -type.…”

Section: Cones Of Curves and Divisorsmentioning

confidence: 99%

MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations

2014

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“…The isometry τ is also constructed in Lemma 3 of [50] and it is shown that it preserves the period. Moreover by proposition 4.6 of [32],…”

Section: Conclusion On the Morphism To The Hilbert Schemementioning

confidence: 90%

“…Λn . Now, as done in Section 4 of [32], we consider A * the dual complex torus of A. Then H 1 (A * , Z) is isomorphic to H 1 (A, Z) * .…”

Section: Conclusion On the Morphism To The Hilbert Schemementioning

confidence: 99%

Integral cohomology of the Generalized Kummer fourfold

Kapfer¹,

Menet²

2016

Preprint

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We describe the integral cohomology of the generalized Kummer fourfold giving an explicit basis, using Hilbert scheme cohomology and tools developed by Hassett and Tschinkel. Then we apply our results to a IHS variety with singularities, obtained by a partial resolution of the generalized Kummer fourfold quotiented by a symplectic involution. We calculate the Beauville-Bogomolov form of this new variety, presenting the first example of such a form that is odd.

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“…In this subsection, we assume that F is a hyperkähler manifold of K3 [2] -type. For any δ ∈ Ω(F ), then δ (or −δ) essentially arises from some S [2] in the deformation equivalent family, see Lemma 3.4 of [16]. Hence for cohomological computations, it is harmless to automatically view an element δ ∈ Ω(F ) as coming from an isomorphism F ∼ = S [2] .…”

Section: The Degree 4 Integral Cohomology Groupmentioning

confidence: 99%

Hyperkähler manifolds of Jacobian type

Shen

2013

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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In this paper we define the notion of a hyperkähler manifold (potentially) of Jacobian type. If we view hyperkähler manifolds as "abelian varieties", then those of Jacobian type should be viewed as "Jacobian varieties". Under a minor assumption on the polarization, we show that a very general polarized hyperkähler fourfold F of K3 [2] -type is not of Jacobian type. As a potential application, we conjecture that if a cubic fourfold is rational then its variety of lines is of Jacobian type. Under some technical assumption, it is proved that the variety of lines on a rational cubic fourfold is potentially of Jacobian type. We also prove the Hodge conjecture in degree 4 for a generic F of K3 [2] -type.

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Hilbert schemes of K3 surfaces are dense in moduli

Cited by 9 publications

References 0 publications

MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations

MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations

Integral cohomology of the Generalized Kummer fourfold

Hyperkähler manifolds of Jacobian type

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