scholarly journals The Chow ring of hyperkähler varieties of $$K3^{[2]}$$-type via Lefschetz actions

2021 ◽  
Author(s):  
Andreas Kretschmer
Keyword(s):  
Lie Algebra ◽  
Hilbert Scheme ◽  
Fano Variety ◽  
Chow Ring ◽  
Fourier Decomposition ◽  
Hyperkähler Varieties ◽  
Eigenspace Decomposition ◽  
Formula Type ◽  
Cubic Fourfold

AbstractWe propose an explicit conjectural lift of the Neron–Severi Lie algebra of a hyperkähler variety X of $$K3^{[2]}$$ K 3 [ 2 ] -type to the Chow ring of correspondences $$\mathrm{CH}^*(X \times X)$$ CH ∗ ( X × X ) in terms of a canonical lift of the Beauville–Bogomolov class obtained by Markman. We give evidence for this conjecture in the case of the Hilbert scheme of two points of a K3 surface and in the case of the Fano variety of lines of a very general cubic fourfold. Moreover, we show that the Fourier decomposition of the Chow ring of X of Shen and Vial agrees with the eigenspace decomposition of a canonical lift of the cohomological grading operator.

scholarly journals Zero-cycles on double EPW sextics

2020 ◽  
pp. 2050040
Author(s):  
Robert Laterveer ◽  
Charles Vial
Keyword(s):  
Type Theorem ◽  
Chow Group ◽  
K3 Surface ◽  
Chern Classes ◽  
Chow Ring ◽  
Fourier Decomposition ◽  
Complete Family ◽  
Chow Rings ◽  
Rank One ◽  
Hyperkähler Varieties

The Chow rings of hyperKähler varieties are conjectured to have a particularly rich structure. In this paper, we focus on the locally complete family of double EPW sextics and establish some properties of their Chow rings. First, we prove a Beauville–Voisin type theorem for zero-cycles on double EPW sextics; precisely, we show that the codimension-4 part of the subring of the Chow ring of a double EPW sextic generated by divisors, the Chern classes and codimension-2 cycles invariant under the anti-symplectic covering involution has rank one. Second, for double EPW sextics birational to the Hilbert square of a K3 surface, we show that the action of the anti-symplectic involution on the Chow group of zero-cycles commutes with the Fourier decomposition of Shen–Vial.

scholarly journals INTEGRAL CONSTRAINTS ON THE MONODROMY GROUP OF THE HYPERKÄHLER RESOLUTION OF A SYMMETRIC PRODUCT OF A K3 SURFACE

2010 ◽  
Vol 21 (02) ◽  
pp. 169-223 ◽  
Author(s):  
EYAL MARKMAN
Keyword(s):  
Hilbert Scheme ◽  
Prime Power ◽  
Monodromy Group ◽  
Euler Number ◽  
Hodge Structure ◽  
K3 Surface ◽  
Monodromy Operator ◽  
First Case ◽  
Hyperkähler Varieties ◽  
Restriction Homomorphism

Let S[n]be the Hilbert scheme of length n subschemes of a K3 surface S. H2(S[n],ℤ) is endowed with the Beauville–Bogomolov bilinear form. Denote by Mon the subgroup of GL [H*(S[n],ℤ)] generated by monodromy operators, and let Mon2be its image in OH2(S[n],ℤ). We prove that Mon2is the subgroup generated by reflections with respect to +2 and -2 classes (Theorem 1.2). Thus Mon2does not surject onto OH2(S[n],ℤ)/(±1), when n - 1 is not a prime power.As a consequence, we get counterexamples to a version of the weight 2 Torelli question for hyperKähler varieties X deformation equivalent to S[n]. The weight 2 Hodge structure on H2(X,ℤ) does not determine the bimeromorphic class of X, whenever n - 1 is not a prime power (the first case being n = 7). There are at least 2ρ(n - 1) - 1distinct bimeromorphic classes of X with a given generic weight 2 Hodge structure, where ρ(n - 1) is the Euler number of n - 1.The second main result states, that if a monodromy operator acts as the identity on H2(S[n],ℤ), then it acts as the identity on Hk(S[n],ℤ), 0 ≤ k ≤ n + 2 (Theorem 1.5). We conclude the injectivity of the restriction homomorphism Mon → Mon2, if n ≡ 0 or n ≡ 1 modulo 4 (Corollary 1.6).

scholarly journals EPW cubes

2019 ◽  
Vol 2019 (748) ◽  
pp. 241-268 ◽  
Author(s):  
Atanas Iliev ◽  
Grzegorz Kapustka ◽  
Michał Kapustka ◽  
Kristian Ranestad
Keyword(s):  
Moduli Space ◽  
Hilbert Scheme ◽  
Symplectic Manifolds ◽  
K3 Surface ◽  
Degeneracy Loci ◽  
Double Covers ◽  
Formula Type

Abstract We construct a new 20-dimensional family of projective six-dimensional irreducible holomorphic symplectic manifolds. The elements of this family are deformation equivalent with the Hilbert scheme of three points on a K3 surface and are constructed as natural double covers of special codimension-three subvarieties of the Grassmannian G(3,6) . These codimension-three subvarieties are defined as Lagrangian degeneracy loci and their construction is parallel to that of EPW sextics, we call them the EPW cubes. As a consequence we prove that the moduli space of polarized IHS sixfolds of K3 -type, Beauville–Bogomolov degree 4 and divisibility 2 is unirational.

scholarly journals THE MOTIVE OF THE HILBERT CUBE

2016 ◽  
Vol 4 ◽  
Author(s):  
MINGMIN SHEN ◽  
CHARLES VIAL
Keyword(s):  
Recent Work ◽  
Hilbert Scheme ◽  
Projective Variety ◽  
Abelian Varieties ◽  
Smooth Projective Variety ◽  
Hilbert Cube ◽  
Rational Map ◽  
Chow Rings ◽  
Hyperkähler Varieties

The Hilbert scheme $X^{[3]}$ of length-3 subschemes of a smooth projective variety $X$ is known to be smooth and projective. We investigate whether the property of having a multiplicative Chow–Künneth decomposition is stable under taking the Hilbert cube. This is achieved by considering an explicit resolution of the rational map $X^{3}{\dashrightarrow}X^{[3]}$. The case of the Hilbert square was taken care of in Shen and Vial [Mem. Amer. Math. Soc.240(1139) (2016), vii+163 pp]. The archetypical examples of varieties endowed with a multiplicative Chow–Künneth decomposition is given by abelian varieties. Recent work seems to suggest that hyperKähler varieties share the same property. Roughly, if a smooth projective variety $X$ has a multiplicative Chow–Künneth decomposition, then the Chow rings of its powers $X^{n}$ have a filtration, which is the expected Bloch–Beilinson filtration, that is split.

scholarly journals On the motive of some hyperKähler varieties

2017 ◽  
Vol 2017 (725) ◽  
Author(s):  
Charles Vial
Keyword(s):  
Hilbert Scheme ◽  
Hyperkähler Varieties

AbstractWe show that the motive of the Hilbert scheme of length-

scholarly journals Motivic decompositions for the Hilbert scheme of points of a K3 surface

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Andrei Neguţ ◽  
Georg Oberdieck ◽  
Qizheng Yin
Keyword(s):  
Lie Algebra ◽  
Hilbert Scheme ◽  
K3 Surface ◽  
Hilbert Scheme Of Points

Abstract We construct an explicit, multiplicative Chow–Künneth decomposition for the Hilbert scheme of points of a K3 surface. We further refine this decomposition with respect to the action of the Looijenga–Lunts–Verbitsky Lie algebra.

scholarly journals On the Mumford–Tate conjecture for hyperkähler varieties

2021 ◽  
Author(s):  
Salvatore Floccari
Keyword(s):  
Betti Number ◽  
Hilbert Scheme ◽  
K3 Surface ◽  
Tate Conjecture ◽  
Hilbert Scheme Of Points ◽  
Hyperkähler Varieties

AbstractWe study the Mumford–Tate conjecture for hyperkähler varieties. We show that the full conjecture holds for all varieties deformation equivalent to either an Hilbert scheme of points on a K3 surface or to O’Grady’s ten dimensional example, and all of their self-products. For an arbitrary hyperkähler variety whose second Betti number is not 3, we prove the Mumford–Tate conjecture in every codimension under the assumption that the Künneth components in even degree of its André motive are abelian. Our results extend a theorem of André.

scholarly journals Homological mirror symmetry for generalized Greene–Plesser mirrors

2020 ◽  
Author(s):  
Nick Sheridan ◽  
Ivan Smith
Keyword(s):  
Mirror Symmetry ◽  
Projective Spaces ◽  
Derived Category ◽  
Fano Variety ◽  
Complete Intersections ◽  
Homological Mirror Symmetry ◽  
Higher Dimensional ◽  
Cubic Fourfold

AbstractWe prove Kontsevich’s homological mirror symmetry conjecture for certain mirror pairs arising from Batyrev–Borisov’s ‘dual reflexive Gorenstein cones’ construction. In particular we prove HMS for all Greene–Plesser mirror pairs (i.e., Calabi–Yau hypersurfaces in quotients of weighted projective spaces). We also prove it for certain mirror Calabi–Yau complete intersections arising from Borisov’s construction via dual nef partitions, and also for certain Calabi–Yau complete intersections which do not have a Calabi–Yau mirror, but instead are mirror to a Calabi–Yau subcategory of the derived category of a higher-dimensional Fano variety. The latter case encompasses Kuznetsov’s ‘K3 category of a cubic fourfold’, which is mirror to an honest K3 surface; and also the analogous category for a quotient of a cubic sevenfold by an order-3 symmetry, which is mirror to a rigid Calabi–Yau threefold.

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