We discuss the existence of an absolute Chow-Kuenneth decomposition for complete degenerations of families of Abelian threefolds with complex multiplication over a particular Picard Modular Surface studied by Holzapfel. In addition to the work of Gordon, Hanamura and Murre we use Relatively Complete Models in the sense of Mumford-Faltings-Chai of Picard Modular Surfaces in order to describe complete degenerations of families of abelian varieties. We furthermore prove vanishing results for cohomology groups of irreducible representations of certain arithmetic subgroups in SU(2,1) using the non--compact Simpson type correspondence between the $L^2$--Higgs cohomology of the underlying VHS and the $L^2$--de Rham cohomology resp. intersection cohomology of local systems.
We study the notion of a birational Chow-Künneth decomposition, which is essentially a decomposition of the integral birational motive of a variety. The existence of a birational Chow-Künneth decomposition is stably birationally invariant and this notion refines the Chow theoretical decomposition of the diagonal. We show that a birational Chow-Künneth decompostion exists for the following varieties: (a) Jacobian variety; (b) Hilbert scheme of points on a $K3$ surface and (c) The variety of lines on a stably rational cubic threefold or a stably rational cubic fourfold.
We exhibit several families of Fano threefolds with a multiplicative Chow-Künneth decomposition, in the sense of Shen-Vial. As a consequence, a certain tautological subring of the Chow ring of powers of these threefolds injects into cohomology. As a by-product of the argument, we observe that double covers of projective spaces admit a multiplicative Chow-Künneth decomposition.
We show that a smooth projective variety admits a Chow-Kunneth decomposition if the cohomology has level at most one except for the middle degree. This can be extended to the relative case in a weak sense if the morphism has only isolated singularities, the base space is 1-dimensional, and the generic fiber satisfies the above condition.
We construct a relative Chow-Kunneth decomposition for a conic bundle over a surface such that the middle projector gives the Prym variety of the associated double covering of the discriminant of the conic bundle. This gives a refinement (up to an isogeny) of Beauville's theorem on the relation between the intermediate Jacobian of the conic bundle and the Prym variety of the double covering.
Let $Y$ be a smooth dimensionally transverse intersection of the Grassmannian $\hbox{Gr}(2,n)$ with 3 Plücker hyperplanes. We show that $Y$ admits a multiplicative Chow-Künneth decomposition, in the sense of Shen-Vial. As a consequence, a certain tautological subring of the Chow ring of powers of $Y$ injects into cohomology.
Given a smooth projective variety, a Chow-Künneth decomposition is called multiplicative if it is compatible with the intersection product. Following works of Beauville and Voisin, Shen and Vial conjectured that hyper-Kähler varieties admit a multiplicative Chow-Künneth decomposition. In this paper, based on the mysterious link between Fano varieties with cohomology of K3 type and hyper-Kähler varieties, we ask whether Fano varieties with cohomology of K3 type also admit a multiplicative Chow-Künneth decomposition, and provide evidence by establishing their existence for cubic fourfolds and Kü…
In this paper we investigate Murre's conjecture on the Chow--Künneth decomposition for two classes of examples. We look at the universal families of smooth curves over spaces which dominate the moduli space $\cM_g$, in genus at most 8 and show existence of a Chow--Künneth decomposition. The second class of examples include the representation varieties of a finitely generated group with one relation. This is done in the setting of equivariant cohomology and equivariant Chow groups to get equivariant Chow--Künneth decompositions.
Let $Y$ be a smooth complete intersection of three quadrics, and assume the dimension of $Y$ is even. We show that $Y$ has a multiplicative Chow-Künneth decomposition, in the sense of Shen-Vial. As a consequence, the Chow ring of (powers of) $Y$ displays K3-like behaviour. As a by-product of the argument, we also establish a multiplicative Chow-Künneth decomposition for double planes.
Let $Y$ be a smooth complete intersection of a quadric and a cubic in $\mathbb{P}^n$, with $n$ even. We show that $Y$ has a multiplicative Chow-Künneth decomposition, in the sense of Shen-Vial. As a consequence, the Chow ring of (powers of) $Y$ displays K3-like behaviour. As a by-product of the argument, we also establish a multiplicative Chow-Künneth decomposition for the resolution of singularities of a general nodal cubic hypersurface of even dimension.
Let X be the quotient of a smooth projective variety over a field by a finite group action (in which case we say X is pseudo-smooth), such that the singularities of X are isolated k-rational points. Let Y be obtained by blowing up these points on X. Assume further that Y is pseudo-smooth, and that the components of the exceptional divisor are projective spaces. Then, without invoking the theory of finite-dimensional motives or assuming any of the standard conjectures, we show that a Chow-Kunneth decomposition on either X or Y gives rise, by means of an explicit construction, to a Chow-Kunneth …
In this paper, we investigate Murre's conjecture on the existence of a Chow--Kuenneth decomposition for a rational homogeneous bundle $Z\to S$ over a smooth variety, defined over complex numbers. Chow-Künneth decomposition is exhibited for $Z$ whenever $S$ has a Chow--Kuenneth decomposition. The same conclusion holds for a class of log homogeneous varieties, studied by M. Brion.
We show that Gushel-Mukai fivefolds admit a multiplicative Chow-Künneth decomposition, in the sense of Shen-Vial. As a consequence, a certain tautological subring of the Chow ring of powers of these varieties injects into cohomology.
We give the first examples of finite groups G such that the Chow ring of the classifying space BG depends on the base field, even for fields containing the algebraic closure of Q. As a tool, we give several characterizations of the varieties which satisfy Kunneth properties for Chow groups or motivic homology. We define the (compactly supported) motive of a quotient stack in Voevodsky's derived category of motives. This makes it possible to ask when the motive of BG is mixed Tate, which is equivalent to the motivic Kunneth property. We prove that BG is mixed Tate for various "well-behaved" fin…
This note is about a $16$-dimensional family of surfaces of general type with $p_g=2$ and $q=0$ and $K^2=1$, called "special Horikawa surfaces". These surfaces, studied by Pearlstein-Zhang and by Garbagnati, are related to K3 surfaces. We show that special Horikawa surfaces have a multiplicative Chow-Künneth decomposition, in the sense of Shen-Vial. As a consequence, the Chow ring of special Horikawa surfaces displays K3-like behaviour.
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.
The purpose of this article is to construct a Hecke-equivariant Chow motive whose realizations equal interior (or intersection) cohomology of Picard surfaces with regular algebraic coefficients. As a consequence, we are able to define Grothendieck motives for Picard modular forms.
We prove that the Chow motive with integral coefficient of a geometrically rational surfaces~$S$ over a perfect field~$k$ is zero dimensional if and only if the Picard group of~$\bar{k}\times_{k}S$, where~$\bar{k}$ is an algebraic closure of~$k$, is a direct summand of a $\Gal (\bar{k}/k)$-permutation module, and~$S$ possesses a zero cycle of degree one. As shown by Colliot-Thélène in a letter to the author (which we have reproduced in the appendix) this is in turn equivalent to~$S$ having a zero cycle of degree~$1$ and $\CH_{0}(k(S)\times_{k}S)$ being torsion free.
We study the Fulton-Macpherson operational Chow rings of good moduli spaces of properly stable, smooth, Artin stacks. Such spaces are étale locally isomorphic to geometric invariant theory quotients of affine schemes, and are therefore natural extensions of GIT quotients. Our main result is that, with rational coefficients, every operational class can be represented by a so-called topologically strong cycle on the corresponding stack. Moreover, this cycle is unique modulo rational equivalence on the stack. Using out methods, we prove that if $X$ is the good moduli space of a properly stable, s…
Given a connected algebraic group G over an algebraically closed field and a G-homogeneous space X, we describe the Chow ring of G and the rational Chow ring of X, with special attention to the Picard group. Also, we investigate the Albanese and the "anti-affine" fibrations of G and X.
Let $X$ be a smooth projective variety over a field of characteristic $0$ and $L$ an ample divisor. In this paper we study the Weak Lefschetz conjecture for Chow groups using the technique employed by Grothendieck in his study of the problem for Picard groups, and using Bloch's formula to interpret Chow groups in terms of Milnor $K$- theory we prove the infinitesimal part of the conjecture.
We provide a general method for computing rational Chow rings of moduli of smooth complete intersections. We specialize this result in different ways: to compute the integral Picard group of the associated stack ; to obtain an explicit presentation of rational Chow rings of moduli of smooth complete intersections of codimension two; to prove old and new results on moduli of smooth curves of genus $\leq 5$ and polarized K3 surfaces of degree $\leq 8$.
A remarkable result of Peter O'Sullivan asserts that the algebra epimorphism from the rational Chow ring of an abelian variety to its rational Chow ring modulo numerical equivalence admits a (canonical) section. Motivated by Beauville's splitting principle, we formulate a conjectural Section Property which predicts that for smooth projective holomorphic symplectic varieties there exists such a section of algebra whose image contains all the Chern classes of the variety. In this paper, we investigate this property for (not necessarily symplectic) varieties with Chow motive of abelian type. We i…
We prove that all points of a toroidal compactification lying over 0-dimensional cusps are rationally equivalent in the integral Chow group for most classical modular varieties (Siegel, Hilbert, orthogonal, Hermitian, quaternionic). This gives a generalization, and even strengthening, of the Manin-Drinfeld theorem in higher dimension from the viewpoint of algebraic cycles. The same result no longer holds for Picard modular varieties, but for them we prove that the difference of any two "special" boundary points, which are dense in the boundary, is torsion.
