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 Algebraic cycles and Lehn–Lehn–Sorger–van Straten eightfolds

2021 ◽  
pp. 1-24
Author(s):  
Robert Laterveer
Keyword(s):  
Explicit Formula ◽  
Hilbert Scheme ◽  
Algebraic Cycles ◽  
Chow Group ◽  
K3 Surface ◽  
Chow Ring ◽  
Hilbert Scheme Of Points

Abstract This article is about Lehn–Lehn–Sorger–van Straten eightfolds $Z$ and their anti-symplectic involution $\iota$ . When $Z$ is birational to the Hilbert scheme of points on a K3 surface, we give an explicit formula for the action of $\iota$ on the Chow group of $0$ -cycles of $Z$ . The formula is in agreement with the Bloch–Beilinson conjectures and has some non-trivial consequences for the Chow ring of the quotient.

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.

Chow Rings, Decomposition of the Diagonal, and the Topology of Families (AM-187)

2017 ◽  
Author(s):  
Claire Voisin
Keyword(s):  
Recent Work ◽  
Chow Groups ◽  
Ring Structure ◽  
Complete Intersections ◽  
Algebraic Varieties ◽  
Chow Ring ◽  
Hodge Structures ◽  
Chow Rings ◽  
Geometric Methods ◽  
The Right

This book provides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. The book is intended for both students and researchers, and not only presents a survey of the geometric methods developed in the last thirty years to understand the famous Bloch-Beilinson conjectures, but also examines recent work by the author. It focuses on two central objects: the diagonal of a variety—and the partial Bloch-Srinivas type decompositions it may have depending on the size of Chow groups—as well as its small diagonal, which is the right object to consider in order to understand the ring structure on Chow groups and cohomology. An exploration of a sampling of recent works by the author looks at the relation, conjectured in general by Bloch and Beilinson, between the coniveau of general complete intersections and their Chow groups and a very particular property satisfied by the Chow ring of K3 surfaces and conjecturally by hyper-Kähler manifolds. In particular, the book delves into arguments originating in Nori's work that have been further developed by others.

scholarly journals ON THE CHOW RING OF CERTAIN LEHN–LEHN–SORGER–VAN STRATEN EIGHTFOLDS

2021 ◽  
pp. 1-24
Author(s):  
CHIARA CAMERE ◽  
ALBERTO CATTANEO ◽  
ROBERT LATERVEER
Keyword(s):  
Chow Group ◽  
Chow Ring ◽  
Similar Statement ◽  
Intersection Product ◽  
Finite Dimensional

Abstract We consider a 10-dimensional family of Lehn–Lehn–Sorger–van Straten hyperkähler eightfolds, which have a non-symplectic automorphism of order 3. Using the theory of finite-dimensional motives, we show that the action of this automorphism on the Chow group of 0-cycles is as predicted by the Bloch–Beilinson conjectures. We prove a similar statement for the anti-symplectic involution on varieties in this family. This has interesting consequences for the intersection product of the Chow ring of these varieties.

scholarly journals Rank one sheaves over quaternion algebras on Enriques surfaces

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Fabian Reede
Keyword(s):  
Quaternion Algebra ◽  
Double Cover ◽  
Complex Numbers ◽  
K3 Surface ◽  
Enriques Surface ◽  
Quaternion Algebras ◽  
Enriques Surfaces ◽  
Rank One ◽  
Torsion Free

Abstract Let X be an Enriques surface over the field of complex numbers. We prove that there exists a nontrivial quaternion algebra 𝓐 on X. Then we study the moduli scheme of torsion free 𝓐-modules of rank one. Finally we prove that this moduli scheme is an étale double cover of a Lagrangian subscheme in the corresponding moduli scheme on the associated covering K3 surface.

scholarly journals Transfers of Chern classes in BP-cohomology and Chow rings

2000 ◽  
Vol 353 (3) ◽  
pp. 1039-1054 ◽  
Author(s):  
Björn Schuster ◽  
Nobuaki Yagita
Keyword(s):  
Chern Classes ◽  
Chow Rings

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 On the Chow Ring of Cynk–Hulek Calabi–Yau Varieties and Schreieder Varieties

2019 ◽  
Vol 72 (2) ◽  
pp. 505-536 ◽  
Author(s):  
Robert Laterveer ◽  
Charles Vial
Keyword(s):  
Positive Characteristic ◽  
K3 Surfaces ◽  
The Self ◽  
Dimensional Case ◽  
Chern Classes ◽  
Chow Ring ◽  
Cycle Class

AbstractThis note is about certain locally complete families of Calabi–Yau varieties constructed by Cynk and Hulek, and certain varieties constructed by Schreieder. We prove that the cycle class map on the Chow ring of powers of these varieties admits a section, and that these varieties admit a multiplicative self-dual Chow–Künneth decomposition. As a consequence of both results, we prove that the subring of the Chow ring generated by divisors, Chern classes, and intersections of two cycles of positive codimension injects into cohomology via the cycle class map. We also prove that the small diagonal of Schreieder surfaces admits a decomposition similar to that of K3 surfaces. As a by-product of our main result, we verify a conjecture of Voisin concerning zero-cycles on the self-product of Cynk–Hulek Calabi–Yau varieties, and in the odd-dimensional case we verify a conjecture of Voevodsky concerning smash-equivalence. Finally, in positive characteristic, we show that the supersingular Cynk–Hulek Calabi–Yau varieties provide examples of Calabi–Yau varieties with “degenerate” motive.

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.

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