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.
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-Kuenneth decomposition is stable under taking the Hilbert cube. This is achieved by considering an explicit resolution of the map $X^3 \dashrightarrow X^{[3]}$. The case of the Hilbert square was taken care of in previous work of ours. The archetypical examples of varieties endowed with a multiplicative Chow-Kuenneth decomposition is given by abelian varieties. Recent work seems to suggest that hyperKa…
Let $X$ be a smooth complex cubic fourfold and let $F$ be the variety of lines of $X$. The variety $F$ is known to be a smooth projective hyperkaehler fourfold, which is moreover endowed with a self rational map $ϕ: F -\rightarrow F$ first constructed by C. Voisin. Here we define a filtration of Bloch--Beilinson type on the Chow group of zero-cycles $CH_0(F)$ which canonically splits under the action of $ϕ$, thereby answering in this case a question of A. Beauville. Moreover, we show that this filtration is of motivic origin, in the sense that it arises from a Chow--Kuenneth decomposition of t…
We answer two questions of Carrell on a singular complex projective variety admitting the multiplicative group action, one positively and the other negatively. The results are applied to Chow varieties and we obtain Chow groups of 0-cycles and Lawson homology groups of 1-cycles for Chow varieties. A short survey on the structure of the Chow varieties is included for comparison and completeness. Moreover, we give counterexamples to Shafarevich's question on the rationality of the irreducible components of Chow varieties.
This 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-Kuenneth 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…
We first present an intersection theory of partial differential varieties with quasi-generic differential hypersurfaces. Then based on the generic intersection theory, we define the partial differential Chow form for an irreducible partial differential variety $V$ of Kolchin polynomial $ω_V(t)=(d+1){t+m\choose m}-{t+m-s\choose m}$. And we establish for the partial differential Chow form most of the basic properties of the ordinary differential Chow form. Furthermore, we prove the existence of a type of partial differential Chow varieties.
We study the Fulton-Macpherson Chow cohomology of affine toric varieties. In particular, we prove that the Chow cohomology vanishes in positive degree. We prove an analogous result for the operational $K$-theory defined by Anderson and Payne.
We show that for a very general principally polarized complex abelian 3-fold, the Chow group of algebraic cycles is infinite modulo every prime number. In particular, this gives the first examples of complex varieties with infinite Chow groups modulo 2.
Conjecturally, Fano varieties of K3 type admit a multiplicative Chow-Künneth decomposition, in the sense of Shen-Vial. We prove this for many of the families of Fano varieties of K3 type constructed by Fatighenti-Mongardi. This has interesting consequences for the Chow ring of these varieties.
The main result of this note is a hard Lefschetz theorem for the Chow groups of generalized Kummer varieties. The same argument also proves hard Lefschetz for Chow groups of Hilbert schemes of abelian surfaces. As a consequence, we obtain new information about certain pieces of the Chow groups of generalized Kummer varieties, and Hilbert schemes of abelian surfaces. The proofs are based on work of Shen-Vial and Fu-Tian-Vial on multiplicative Chow-Künneth decompositions.
Fatighenti and Mongardi have defined Fano varieties of type S6 as zero loci of a certain vector bundle on the Grassmannian $\hbox{Gr}(2,10)$. These varieties have 3 Hodge structures of K3 type in their cohomology. We show that the Chow ring of these varieties also displays "K3 type" behaviour.
We show that the graded Chow rings of two birational irreducible symplectic varieties are isomorphic. This lifts a result known for the cohomology algebras to the level of Chow rings, despite the non-injectivity the cycle class map. In the special case of general Mukai flops, we present an alternative approach based on explicit calculations.
The Chow variety of polynomials that decompose as a product of linear forms has been studied for more than 100 years. Finding equations in the ideal of secant varieties of Chow varieties would enable one to measure the complexity the permanent to prove Valiant's conjecture $\mathbf{VP\neq VNP}$. In this article, I use the method of prolongation to obtain equations for secant varieties of Chow varieties as $GL(V)$-modules.
We show that certain Fano eightfolds (obtained as hyperplane sections of an orthogonal Grassmannian, and studied by Ito-Miura-Okawa-Ueda and by Fatighenti-Mongardi) have a multiplicative Chow-Künneth decomposition. As a corollary, the Chow ring of these eightfolds behaves like that of K3 surfaces.
We give bounds on the order of torsion in the Chow group of zero dimensional cycles for isotropic Grassmannians and Brauer-Severi flag varieties. To do this, we introduce tools to understand the behavior of torsion in Chow groups with coefficients under morphisms of proper varieties
We explain how complexity of rational points on projective varieties can be interpreted via the theories of Chow forms and Okounkov bodies. Precisely, we study discrete measures on filtered linear series and build on work of Boucksom and Chen, Boucksom-et-al and Ferretti. For example, we show how asymptotic volume and Seshadri constants are related to the theories of Chow weights and Okounkov bodies for very ample linear series. We also consider arithmetic applications within the context of Diophantine approximation. In this direction, we establish Roth-type theorems which can be expressed in …
We show that the pseudoeffective cone of $k$-cycles on a complete complexity one $T$-variety is rational polyhedral for any $k$, generated by classes of $T$-invariant subvarieties. When $X$ is also rational, we give a presentation of the Chow groups of $X$ in terms of generators and relations, coming from the combinatorial data defining $X$ as a $T$-variety.
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…
We consider Calabi-Yau $n$-folds $X$ arising from certain hyperplane arrangements. Using Fu-Vial's theory of distinguished cycles for varieties with motive of abelian type, we show that the subring of the Chow ring of $X$ generated by divisors, Chern classes and intersections of subvarieties of positive codimension injects into cohomology. We also prove Voisin's conjecture for $X$, and Voevodsky's smash-nilpotence conjecture for odd-dimensional $X$.
In this text we prove that if an abelian variety $A$ admits of an embedding into the Jacobian of a smooth projective curve $C$, and if we consider $\Th_A$ to be the divisor $\Th_C\cap A$, where $\Th_C$ denotes the theta divisor of $J(C)$, then the embedding of $\Th_A$ into $A$ induces an injective push-forward homomorphism at the level of Chow groups. We show that this is the case for every principally polarized abelian varieties.
Let $X$ be a hyperkähler variety, and let $Z\subset X$ be a Lagrangian subvariety. Conjecturally, $Z$ should have trivial intersection with certain parts of the Chow ring of $X$. We prove this conjecture for certain Hilbert schemes $X$ having a Lagrangian fibration, and $Z\subset X$ a general fibre of the Lagrangian fibration.
Chow varieties are a parameter space for cycles of a given variety of a given codimension and degree. We construct their analog for differential algebraic varieties with differential algebraic subvarieties, answering a question of Gao, Li and Wei. The proof uses the construction of classical algebro-geometric Chow varieties, the theory of characteristic sets of differential varieties, the theory of prolongation spaces, and the theory of differential Chow forms. In the course of the proof several definability results from the theory of algebraically closed fields are required. Elementary proofs…
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.
We study the (covering) gonality of abelian varieties and their orbits of zero-cycles for rational equivalence. We show that any orbit for rational equivalence of zero-cycles of degree $k$ has dimension at most $k-1$. Building on the work of Pirola, we show that very general abelian varieties of dimension $g$ have covering gonality $k\geq f(g)$ where $f(g)$ grows like ${\rm log}\,g$. This answers a question asked by Bastianelli, De Poi, Ein, Lazarsfeld and B. Ullery. We also obtain results on the Chow ring of very general abelian varieties, eg. if $g\geq 2k-1$, for any divisor $D\in {\rm Pic}^…
Let $X$ be a hyperkähler variety, and let $G$ be a group of finite order non-symplectic automorphisms of $X$. Beauville's conjectural splitting property predicts that each Chow group of $X$ should split in a finite number of pieces. The Bloch-Beilinson conjectures predict how $G$ should act on these pieces of the Chow groups: certain pieces should be invariant under $G$, while certain other pieces should not contain any non-trivial $G$-invariant cycle. We can prove this for two pieces of the Chow groups when $X$ is the Hilbert scheme of a $K3$ surface and $G$ consists of natural automorphisms.…
The study of Chow varieties of decomposable forms lies at the confluence of algebraic geometry, commutative algebra, representation theory and combinatorics. There are many open questions about homological properties of Chow varieties and interesting classes of modules supported on them. The goal of this note is to survey some fundamental constructions and properties of these objects, and to propose some new directions of research. Our main focus will be on the study of certain maximal Cohen-Macaulay modules of covariants supported on Chow varieties, and on defining equations and syzygies. We …
The Chow rank of a form is the length of its smallest decomposition into a sum of products of linear forms. For a generic form, this corresponds to finding the smallest secant variety of the Chow variety which fills the ambient space. We determine the Chow rank of generic cubics and quaternary forms by proving nondefectivity of all involved secant varieties. The main new ingredient in our proof is the generalization of a technique by [Brambilla and Ottaviani, On the Alexander--Hirschowitz theorem, J. Pure Appl. Algebra, 2008] that consists of employing Terracini's lemma and Newton's backward d…
Auel-Bigazzi-Böhning-Graf von Bothmer proved that if a proper smooth variety $X$ over a field $k$ of characteristic $p>0$ has universally trivial Chow group of $0$-cycles, the cohomological Brauer group of $X$ is universally trivial as well. In this paper, we generalize their argument to arbitrary unramified mod $p$ étale motivic cohomology groups. We also see that the properness assumption on the variety $X$ can be dropped off by using the Suslin homology together with a certain tame subgroup of the unramified cohomology group.
We propose an explicit conjectural lift of the Neron-Severi Lie algebra of a hyperkähler variety $X$ of $K3^{[2]}$-type to the Chow ring of correspondences ${\rm CH}^\ast(X \times X)$ 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 …
In the present paper we introduce the property AA of a subsemigroup of the endomorphism semigroup of an abelian variety, which holds for semigroup of endomorphisms of an abelian variety defined over a number field, and show that the orbit of any cycle under a semigroup with property AA in the Chow group $\otimes\Q$ has finite dimensional span.
Chow stability is one notion of Mumford's Geometric Invariant Theory for studying the moduli space of polarized varieties. Kapranov, Sturmfels and Zelevinsky detected that Chow stability of polarized toric varieties is determined by its inherent {\it secondary polytope}, which is a polytope whose vertices correspond to regular triangulations of the associated polytope \cite{KSZ}. In this paper, we give a purely convex-geometrical proof that the Chow form of a projective toric variety is $H$-semistable if and only if it is $H$-polystable with respect to the standard complex torus action $H$. Th…
We show that the chow group of $p$-cycles with rational coefficients are isomorphic to the corresponding rational homology groups for smooth complex projective varieties carrying a holomorphic vector field with an isolated zero locus. As applications, we obtain Chow groups and Lawson homology groups with rational coefficients and verify the Friedlander-Mazur conjecture and the Generalized Hodge conjecture for those varieties.
We study $T$-linear schemes, a class of objects that includes spherical and Schubert varieties. We provide a localization theorem for the equivariant Chow cohomology of these schemes that does not depend on resolution of singularities. Furthermore, we give an explicit presentation of the equivariant Chow cohomology of possibly singular complete spherical varieties admitting a smooth equivariant envelope (e.g. group embeddings).
