Morihiko Saito
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.
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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 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…
Chow Künneth decomposition for étale motives
本文提出 étale 动机的整系数 Chow–Künneth 分解，利用保守函子族处理交换群概形，并构造射影簇的一阶与余一阶整系数子动机。
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.
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-…
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.
Cohomological Chow Groups of codimension one of varieties with isolated singularities
本文研究了具有孤立奇点的代数簇的上同调Chow群，通过不同条件下的计算揭示了其结构特性。
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.
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.
Algebraic cycles and Fano threefolds of genus 7
针对亏格7的素法诺三维簇，构造其乘积上的非挠2-闭链，证明该簇不具备Shen-Vial意义下的乘法周Künneth分解，同时指出所有法诺三维簇在代数等价下均有该分解。
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.
Multiplicative Chow-Künneth decomposition and homology splitting of configuration spaces
本文构造了光滑固有簇点构型空间上同调的分裂，证明了超椭圆Torelli群作用平凡及Leray谱序列立即退化，并据此对$M_{2,n}$和$H_{g,n}$的上同调进行了详细计算。
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 …
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.
Let $X$ be a complex smooth projective variety of dimension $d$. Under some assumption on the cohomology of $X$, we construct mutually orthogonal idempotents in $CH_d(X \times X) \otimes \Q$ whose action on algebraically trivial cycles coincides with the Abel-Jacobi map. Such a construction generalizes Murre's construction of the Albanese and Picard idempotents and makes it possible to give new examples of varieties admitting a self-dual Chow-Künneth decomposition satisfying the motivic Lefschetz conjecture as well as new examples of varieties having a Kimura finite dimensional Chow motive. Fo…
Given any smooth toric surface S, we prove a SYM-HILB correspondence which relates the 3-point, degree zero, extended Gromov-Witten invariants of the n-fold symmetric product stack [Sym^n(S)] of S to the 3-point extremal Gromov-Witten invariants of the Hilbert scheme Hilb^n(S) of n points on S. As we do not specialize the values of the quantum parameters involved, this result proves a strengthening of Ruan's Cohomological Crepant Resolution Conjecture for the Hilbert-Chow morphism from Hilb^n(S) to Sym^n(S) and yields a method of reconstructing the cup product for Hilb^n(S) from the orbifold i…
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.
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.
Chow theory of toric variety bundles
本文研究了托尔奇簇包的Chow理论，通过扇形位移规则和等价乘法性，建立了普通和等价上同调与Minkowski权重及分段多项式函数的对应关系，并应用于对数枚举几何中的平衡条件。
We prove that the group of normalized cohomological invariants of degree 3 modulo the subgroup of semidecomposable invariants of a semisimple split linear algebraic group G is isomorphic to the torsion part of the Chow group of codimension 2 cycles of the respective versal G-flag. In particular, if G is simple, we show that this factor group is isomorphic to the group of indecomposable invariants of G. As an application, we construct nontrivial cohomological classes for indecomposable central simple algebras.
A smooth intersection $Y$ of two quadrics in $\mathbb{P}^{2g+1}$ has Hodge level 1. We show that such varieties $Y$ have 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.
Let $f : X \rightarrow B$ be a proper flat dominant morphism between two smooth quasi-projective complex varieties $X$ and $B$. Assume that there exists an integer $l$ such that all closed fibres $X_b$ of $f$ satisfy $CH_j(X_b) = \Q$ for all $j \leq l$. Then we prove an analogue of the projective bundle formula for $CH_i(X)$ for $i \leq l$. When $B$ is a surface, $X$ is projective and $l = \lfloor \frac{\dim X - 3}{2} \rfloor$, this makes it possible to construct a Chow-Künneth decomposition for $X$ that satisfies Murre's conjectures. For instance we prove Murre's conjectures for complex smoot…
We establish a geometric interpretation of orientifold Donaldson-Thomas invariants of $σ$-symmetric quivers with involution. More precisely, we prove that the cohomological orientifold Donaldson-Thomas invariant is isomorphic to the rational Chow group of the moduli space of $σ$-stable self-dual quiver representations. As an application we prove that the Chow Betti numbers of moduli spaces of stable $m$-tuples in classical Lie algebras can be computed numerically. We also prove a cohomological wall-crossing formula relating semistable Hall modules for different stabilities.
On the Chow Ring of Fano Fourfolds of K3 type
本文证明了一类新构造的 K3 型 Fano 簇具有乘性 Chow-Künneth 分解，使其 Chow 环表现如 K3 曲面，并给出了爆破射影簇 Franchetta 性质的判据。
In this paper, we prove that Ruan's Cohomological Crepant Resolution Conjecture holds for the Hilbert-Chow morphisms. There are two main ideas in the proof. The first one is to use the representation theoretic approach proposed in [QW] which involves vertex operator techniques. The second is to prove certain universality structures about the 3-pointed genus-0 extremal Gromov-Witten invariants of the Hilbert schemes by using the indexing techniques from [LiJ], the product formula from [Beh2] and the co-section localization from [KL1, KL2, LL]. We then reduce Ruan's Conjecture from the case of a…
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.
For every $n$-dimensional smooth projective variety $X$ over ${\bf C}$, the motive $M(X)$ is expected to admit a Chow-Künneth decomposition $M_0(X)\oplus\cdots \oplus M_{2n}(X)$. Inspired by the slice filtration of $M(X)$ we propose the definitions of $M_2(X)$ and $M_{2n-2}(X)$.
We develop a theory of abstract arithmetic Chow rings where the role of the fibers at infinity is played by a complex of abelian groups that computes a suitable cohomology theory. This theory allows the construction of many variants of the arithmetic Chow groups with different properties. As particular cases of this formalism we recover the original arithmetic intersection theory of Gillet and Soulé for projective varieties, we introduce a theory of arithmetic Chow groups which are covariant with respect to arbitrary proper morphisms, and we develop a theory of arithmetic Chow rings using a co…
A theorem of Esnault, Srinivas and Viehweg asserts that if the Chow group of 0-cycles of a smooth complete complex variety decomposes, then the top-degree coherent cohomology group decomposes similarly. In this note, we prove that (a weak version of) the converse holds for varieties of dimension at most 5 that have finite-dimensional motive and satisfy the Lefschetz standard conjecture. The proof is based on Vial's construction of a refined Chow-Kunneth decomposition for these varieties.
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…
We study a generalization of a conjecture made by Beauville on the Chow ring of hyper-Kähler algebraic varieties. Namely we prove in a number of cases that polynomial cohomological relations involving only CH^1(X) and the Chern classes of X are satisfied in CH(X). These cases are : punctual Hilbert schemes of a K3 surface S parameterizing subschemes of length n, for n<2b_2(S)_tr+5; Fano varieties of lines in a cubic fourfold.
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.
Real Regulators for Products of Elliptic Curves
本文在Kunneth分解假设下，证明了对足够多的椭圆曲线乘积，实调节子的Hodge-${\mathcal D}$-猜想不成立，给出了反例。
Divided powers on abelian varieties
在可分闭域上的阿贝尔簇的平展Chow群中证明分次幂的存在性，从而在特征之外且忽略2-挠时得到Fourier变换的整提升。方法是将Deninger-Murre Chow-Künneth投射子提升为整投射子并推导结论。
We explain how to exploit Rost's theory of Chow groups with coefficients to carry some computations of cohomological invariants (i.e. characteristic classes for G-torsors in Galois cohomology, for an algebraic group G). In particular, we use the idea of the ``stratification method'' introduced by Vezzosi. We recover a number of known results, with very different proofs. We obtain some new information on spin groups..
Given a smooth projective variety $M$ endowed with a faithful action of a finite group $G$, following Jarvis-Kaufmann-Kimura and Fantechi-Göttsche, we define the orbifold motive (or Chen-Ruan motive) of the quotient stack $[M/G]$ as an algebra object in the category of Chow motives. Inspired by Ruan, one can formulate a motivic version of his Cohomological HyperKähler Resolution Conjecture. We prove this motivic version, as well as its K-theoretic analogue conjectured by Jarvis-Kaufmann-Kimura, in two situations related to an abelian surface $A$ and a positive integer $n$. Case (A) concerns Hi…
We prove a decomposition theorem for the cohomological Chow group of 0-cycles on the double of a quasi-projective $R_1$-scheme over a field along a closed subscheme, in terms of the Chow groups, with and without modulus, of the scheme. This yields a significant generalization of the decomposition theorem of Binda-Krishna. As applications, we prove a moving lemma for Chow groups with modulus and an analogue of Bloch's formula for 0-cycles with modulus on singular surfaces. The latter extends a previous result of Binda-Krishna-Saito.
We adapt for algebraically closed fields $k$ of characteristic greater than $2$ two results of Voisin, on the decomposition of the diagonal of a smooth cubic hypersurface $X$ of dimension $3$ over $\mathbb C$, namely: the equivalence between Chow-theoretic and cohomological decompositions of the diagonal of those hypersurfaces and the fact that the algebraicity (with $\mathbb Z_2$-coefficients) of the minimal class $θ^4/4!$ of the intermediate jacobian $J(X)$ of $X$ implies the Chow-theoretic decomposition of the diagonal of $X$. Using the second result, the Tate conjecture for divisors on sur…
We study a variant of the semi-stable Cohomological Hall algebra which we construct using equivariant Chow groups. This algebra, we call it the semi-stable ChowHa, arises as a quotient of the CoHa. Smooth models of quiver moduli give rise to modules over the semi-stable ChowHa. We prove that these modules are cyclic and we compute a presentation using Harder-Narasimhan methods.
We describe algebraically defined cohomological and homological Albanese and Picard 1-motives (or mixed motives) of any algebraic variety in characteristic zero, generalizing the classical Albanese and Picard varieties. We compute Hodge, l-adic and De Rham realizations proving Deligne's conjecture for the concerned mixed Hodge structures. We investigate functoriality, universality, homotopical invariance and invariance under formation of projective bundles. We compare our cohomological and homological 1-motives for normal schemes. For proper schemes, we obtain an Abel-Jacobi map from the (Levi…
Given a smooth variety $X$ and an effective Cartier divisor $D \subset X$, we show that the cohomological Chow group of 0-cycles on the double of $X$ along $D$ has a canonical decomposition in terms of the Chow group of 0-cycles ${\rm CH}_0(X)$ and the Chow group of 0-cycles with modulus ${\rm CH}_0(X|D)$ on $X$. When $X$ is projective, we construct an Albanese variety with modulus and show that this is the universal regular quotient of ${\rm CH}_0(X|D)$. As a consequence of the above decomposition, we prove the Roitman torsion theorem for the 0-cycles with modulus. We show that ${\rm CH}_0(X|…
Let $X$ be a smooth projective surface over an algebraically closed field $k$ such that $char(k) \neq 2$. Let $X^{[d]}$ denote the punctual Hilbert scheme of zero dimensional quotients of degree $d$ and $X^{(d)}$ denote the symmetric product of $X$. For $\ell \neq 2$, we give a formula for the $\ell$-primary part of the Brauer group of $X^{[2]}$. We show that the Hilbert to Chow morphism induces an isomorphism of cohomological Brauer groups for $d=2$ and a similar result for $d \geq 3$. Let $Q(r,d)$ denote the punctual Quot-scheme parametrising zero dimensional quotients of $\mathcal{O}_X^{ \o…
Divisibility and torsion in higher Chow groups over arithmetic fields
本文研究算术域上光滑概形的高阶Chow群的可除性与无挠性，并应用于有限域、局部域和整体域。
From $3$-spin to Negative $r$-spin: Limits of Cohomological Field Theories and Tautological Relations
本文证明负 r-自旋同调关系可由 Pixton 的 3-自旋关系推出，并由此推导其在同调 Chow 环中成立，特别地 r=2 时得到 Kazarian--Norbury K-关系在 Chow 环中成立。
Motivic multiplicativity of complete intersections
本文针对带CK分解的光滑射影簇引入动机多重扭曲可乘性等概念，确定了光滑加权射影空间中Fano或Calabi-Yau完全交的动机2重可乘性缺陷，证实了Voisin的猜想并得到Franchetta性质等推论。
Let p be a prime number. We give a conjecture of a sheaf-theoretic nature which is equivalent to the strong form of the Tate conjecture for smooth, projective varieties X over F_p: for all n>0, the order of pole of the Hasse-Weil zeta function of X at s=n equals the rank of the group of algebraic cycles of codimension n modulo numerical equivalence. Our main result is that this conjecture implies other well-known conjectures in characteristic p, among which: - The (weak) Tate conjecture for smooth, projective varieties X over any finitely generated field of characteristic p: given a prime l di…
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