scholarly journals A Lie algebra action on the Chow ring of the Hilbert scheme of points of a K3 surface

2021 ◽  
Vol 96 (1) ◽  
pp. 65-77
Author(s):  
Georg Oberdieck
Keyword(s):  
Lie Algebra ◽  
Hilbert Scheme ◽  
K3 Surface ◽  
Chow Ring ◽  
Hilbert Scheme Of Points

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 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 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 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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