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 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 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 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 LEHN’S FORMULA IN CHOW AND CONJECTURES OF BEAUVILLE AND VOISIN

2020 ◽  
pp. 1-39
Author(s):  
Davesh Maulik ◽  
Andrei Neguţ
Keyword(s):  
Hilbert Scheme ◽  
Cohomology Ring ◽  
K3 Surface ◽  
Projective Surface ◽  
Weak Version ◽  
Chow Ring ◽  
Hilbert Scheme Of Points ◽  
Chern Characters ◽  
Hyperkähler Manifold ◽  
Voisin Conjecture

The Beauville–Voisin conjecture for a hyperkähler manifold $X$ states that the subring of the Chow ring $A^{\ast }(X)$ generated by divisor classes and Chern characters of the tangent bundle injects into the cohomology ring of  $X$ . We prove a weak version of this conjecture when $X$ is the Hilbert scheme of points on a K3 surface for the subring generated by divisor classes and tautological classes. This in particular implies the weak splitting conjecture of Beauville for these geometries. In the process, we extend Lehn’s formula and the Li–Qin–Wang $W_{1+\infty }$ algebra action from cohomology to Chow groups for the Hilbert scheme of an arbitrary smooth projective surface  $S$ .

scholarly journals MATRIX THEORY, HILBERT SCHEME AND INTEGRABLE SYSTEM

1998 ◽  
Vol 13 (34) ◽  
pp. 2731-2742 ◽  
Author(s):  
YUTAKA MATSUO
Keyword(s):  
Hilbert Scheme ◽  
Matrix Theory ◽  
Fock Space ◽  
Homology Class ◽  
Orthogonal Basis ◽  
Inner Product ◽  
Virasoro Symmetry ◽  
Hilbert Scheme Of Points ◽  
The Matrix ◽  
Boson Fock Space

We give a reinterpretation of the matrix theory discussed by Moore, Nekrasov and Shatashivili (MNS) in terms of the second quantized operators which describes the homology class of the Hilbert scheme of points on surfaces. It naturally relates the contribution from each pole to the inner product of orthogonal basis of free boson Fock space. These bases can be related to the eigenfunctions of Calogero–Sutherland (CS) equation and the deformation parameter of MNS is identified with coupling of CS system. We discuss the structure of Virasoro symmetry in this model.

scholarly journals The integral cohomology of the Hilbert scheme of points on a surface

2020 ◽  
Vol 8 ◽  
Author(s):  
Burt Totaro
Keyword(s):  
Natural Number ◽  
Hilbert Scheme ◽  
Obstruction Theory ◽  
Projective Surface ◽  
Hilbert Schemes ◽  
Complex Projective Surface ◽  
Smooth Complex ◽  
Hilbert Scheme Of Points ◽  
Complex Projective ◽  
Torsion Free

Abstract We show that if X is a smooth complex projective surface with torsion-free cohomology, then the Hilbert scheme $X^{[n]}$ has torsion-free cohomology for every natural number n. This extends earlier work by Markman on the case of Poisson surfaces. The proof uses Gholampour-Thomas’s reduced obstruction theory for nested Hilbert schemes of surfaces.

On a cell decomposition of the Hilbert scheme of points in the plane

1988 ◽  
Vol 91 (2) ◽  
pp. 365-370 ◽  
Author(s):  
Geir Ellingsrud ◽  
Stein Arild Str�mme
Keyword(s):  
Hilbert Scheme ◽  
Cell Decomposition ◽  
Hilbert Scheme Of Points ◽  
A Cell
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