Luca Barbieri-Viale, Bruno Kahn
Let X be an n-dimensional smooth proper variety over a field admitting resolution of singularities, and Y,Z two disjoint closed subsets of X. We establish an isomorphism M(X-Z,Y) isomorphic to M(X-Y,Z)^*(n)[2n] in Voevodsky's triangulated category of geometric motives. Here, M(X-Z,Y) is the motive of X -Z relative to its closed subset Y.
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We prove formulae for the motives of stacks of coherent sheaves of fixed rank and degree over a smooth projective curve in Voevodsky's triangulated category of mixed motives with rational coefficients.
We prove a formula for the motive of the stack of vector bundles of fixed rank and degree over a smooth projective curve in Voevodsky's triangulated category of mixed motives with rational coefficients.
We study mixed versions of the classical quotient functor from Chow motives to numerical motives. We compare two natural definitions, which turn out to be very different. We investigate fullness, conservativity and exactness of these two functors.
Voevodsky motives and motives with modulus with finite coefficients in characteristic zero
本文证明在特征零域上，以 $\Z/n\Z$ 为系数时，带模的有效 motives 范畴与有效 Voevodsky motives 范畴等价。
Voevodsky motives and motives with modulus in positive characteristic
本文在不假设解析性的情况下，证明了带有模的动机三角范畴与Voevodsky的动机三角范畴在有理系数下等价，展示了正特征下模的乘数在动机范畴中变得不可见。
In this note we describe very explicitly a rich family of mixed motives that generates Voevodsky's $DM^{eff}_{gm}{\mathbb{Q}}$ (as a triangulated category). They "should be" mixed since they have only one non-zero Betti cohomology group. Our method also allows to define a family of direct summands of the numerical motif of any smooth projective variety $P$. Modulo certain standard conjectures, this construction yields the Kunneth decomposition of the diagonal of $P$.
Over a perfect field k, let G be an extension of an abelian variety by the multiplicative group $\G_m$. We compute the motive of G in Voevodsky's category of etale motivic complexes with rational coefficients. The result is a decomposition similar to the classical decomposition of the Chow motive of an abelian variety obtained by Deninger/Murre, Kuennemann, Shermenev and Scholl, and other authors.
We describe Somekawa's K-group associated to a finite collection of semi-abelian varieties (or more general sheaves) in terms of the tensor product in Voevodsky's category of motives. While Somekawa's definition is based on Weil reciprocity, Voevodsky's category is based on homotopy invariance. We apply this to explicit descriptions of certain algebraic cycles.
Artin motives in relative Nori and Voevodsky motives
本文证明特征零域上有限型概形的相对 Artin 动机在 Nori 与 Voevodsky 范畴中等价，并应用于对偶性及动机伽罗瓦群的正合序列类比。
We study the motive of the moduli space of semistable Higgs bundles of coprime rank and degree on a smooth projective curve C over a field k under the assumption that C has a rational point. We show this motive is contained in the thick tensor subcategory of Voevodsky's triangulated category of motives with rational coefficients generated by the motive of C. Moreover, over a field of characteristic zero, we prove a motivic non-abelian Hodge correspondence: the integral motives of the Higgs and de Rham moduli spaces are isomorphic.
We define and study the motive of the moduli stack of vector bundles of fixed rank and degree over a smooth projective curve in Voevodsky's category of motives. We prove that this motive can be written as a homotopy colimit of motives of smooth projective Quot schemes of torsion quotients of sums of line bundles on the curve. When working with rational coefficients, we prove that the motive of the stack of bundles lies in the localising tensor subcategory generated by the motive of the curve, using Bialynicki-Birula decompositions of these Quot schemes. We conjecture a formula for the motive o…
A Voevodsky motive associated to a log scheme
本文为域上 fs 对数概形构造几何 Voevodsky 动机，证明复数域上其 Betti 实现等于对数 Betti 上同调，并应用于管状邻域、极限动机和单值滤子。
Let k be a perfect field. In this paper we prove that biextensions of 1-motives define multilinear morphisms between 1-motives in Voevodsky's triangulated category of effective geometrical motives over k with rational coefficients.
We give a concise exposition of Voevodsky's theory of motives.
After reviewing some basic facts about pure and mixed motives, we explain, following Deligne-Goncharov, how to construct a de Rham realisation functor from the category of geometric mixed motives to the category of bifiltered vector spaces.
In this thesis we compare V. Voevodsky's geometric motives to the derived category of M. Nori's abelian category of mixed motives by constructing a triangulated tensor functor between them. It will be compatible with the Betti realizations on both sides. We allow an arbitrary noetherian ring of coefficients, but require it to be a field or a Dedekind domain for the tensor structure on derived Nori motives to exist. There are three key ingredients: we present a theory of Nisnevich covers on finite acyclic diagrams of finite correspondences, explain, following D. Rydh, how to interpret finite co…
On the Nori and Hodge realisations of Voevodsky motives
本文通过证明反常Nori动机与混合Hodge模导出范畴的可构造心性质，构造了六运算的∞-范畴提升，建立了Voevodsky étale动机到两者的实现函子，并证明了Artin动机范畴等价及motivic t-结构存在时的等价性。
Following an insight of Kontsevich, we prove that the quotient of Voevodsky's category of geometric mixed motives DM by the endofunctor -Q(1)[2] embeds fully-faithfully into Kontsevich's category of noncommutative mixed motives KMM. We show also that this embedding is compatible with the one between pure motives. As an application, we obtain a precise relation between the Picard groups Pic(-), the Grothendieck groups, the Schur-finitenss, and the Kimura-finitenss of the categories DM and KMM. In particular, the quotient of Pic(DM) by the subgroup of Tate twists Q(i)[2i] injects into Pic(KMM). …
In this self-contained paper we prove that Voevodsky's smooth blowup triangle of motives generalises to a smooth blowup triangle of motives with modulus.
Using the theory of framed correspondences developed by Voevodsky, we introduce and study framed motives of algebraic varieties. They are the major computational tool for constructing an explicit quasi-fibrant motivic replacement of the suspension $\mathbb P^1$-spectrum of any smooth scheme $X\in Sm/k$. Moreover, it is shown that the bispectrum $$(M_{fr}(X),M_{fr}(X)(1),M_{fr}(X)(2),\ldots),$$ each term of which is a twisted framed motive of $X$, has motivic homotopy type of the suspension bispectrum of $X$. Furthermore, an explicit computation of infinite $\mathbb P^1$-loop motivic spaces is …
The Balmer spectrum of Voevodsky motives and pure symbols
本文引入了巴勒默谱中Voevodsky动机范畴的不变量，其值为MilnorK理论中纯符号模块的'轻Rost循环子模块'。证明了各向同性点的闭合性，并引入了边界类型的点概念，表明该类包含各向同性点但不包含微分型点。
Zeta and L functions of Voevodsky motives
本研究为整体域上的几何Voevodsky动机定义了具有欧拉乘积的近L函数，明确了其与Serre zeta函数的差异、乘性及函数域上的有理性与函数方程，核心工具为Ayoub的六/七函子运算。
In this paper, we construct four different theories of integration, two that are for Voevodsky motives, one for mixed $\ell$-adic sheaves, and a fourth theory of integration for rational mixed Hodge structures. We then show that they circumvent some of the complications of classical motivic integration, leading to new arithmetic and geometric results concerning K-equivalent $k$-varieties. For example, in addition to recovering known results regarding K-equivalent smooth projective complex varieties, we show that K-equivalent smooth projective $\mathbb{F}_q$-varieties have isomorphic rational $…
On the Motivic Homotopy Type of Algebraic Stacks
本文构造了在Morel-Voevodsky A1-同伦范畴中为局部满射的代数栈的光滑展示，并证明光滑栈的动机具有与光滑概形动机相似的许多性质。
This small note contains some easy examples of quartic hypersurfaces that have finite-dimensional motive. As an illustration, we verify a conjecture of Voevodsky (concerning smash-equivalence) for some of these special quartics.
We first recall the construction of the Chow motive modelling intersection cohomology of a proper surface and study its fundamental properties. Using Voevodsky's category of effective geometrical motives, we then study the motive of the exceptional divisor in a non-singular blow-up. If all geometric irreducible components of the divisor are of genus zero, then Voevodsky's formalism allows us to construct certain one-extensions of Chow motives, as canonical sub-quotients of the motive with compact support of the smooth part of the surface. Specializing to Hilbert--Blumenthal surfaces, we recove…
We describe the Voevodsky's category $DM^{eff}_{gm}$ of motives in terms of Suslin complexes of smooth projective varieties. This shows that Voeovodsky's $DM_{gm}$ is anti-equivalent to Hanamura's one. We give a description of any triangulated subcategory of $DM^{eff}_{gm}$ (including the category of effective mixed Tate motives). We descibe 'truncation' functors $t_N$ for $N>0$. $t=t_0$ generalizes the weight complex of Soule and Gillet; its target is $K^b(Chow_{eff})$; it calculates $K_0(DM^{eff}_{gm})$, and checks whether a motive is a mixed Tate one. $t_N$ give a weight filtration and a 'm…
This small note presents in any dimension a family of cubics that have finite-dimensional motive (in the sense of Kimura). As an illustration, we verify a conjecture of Voevodsky for these cubics, and a conjecture of Murre for the Fano variety of lines of these cubics.
A natural place to study the Chow ring of the classifying space $BG$, for $G$ a linear algebraic group, is Voevodsky's triangulated category of motives, inside which Morel and Voevodsky, and Totaro have defined motives $M(BG)$ and $M^c(BG)$, respectively. We show that, for any group $G$ of order $p^3$ over a field of characteristic not $p$ which contains a primitive $p^2$-th root of unity, the motive $M(BG)$ is a mixed Tate motive. We also show that, for a finite group $G$ over a field of characteristic zero, $M(BG)$ is a mixed Tate motive if and only $M^c(BG)$ is a mixed Tate motive.
Comparison of motives with rational coefficients
本文证明在准优秀基方案上，Voevodsky的有理动机DM(S, Q)与Morel/Beilinson/Ayoub的有理动机等价，并部分验证了Voevodsky的猜想。
We prove that the embedding of the derived category of 1-motives into the triangulated category of effective Voevodsky motives, as well as its left adjoint functor $LAlb$, commute with the Hodge realization.
This survey covers some of the recent developments on noncommutative motives and their applications. Among other topics, we compute the additive invariants of relative cellular spaces and orbifolds; prove Kontsevich's semi-simplicity conjecture; prove a far-reaching noncommutative generalization of the Weil conjectures; prove Grothendieck's standard conjectures of type C+ and D, Voevodsky's nilpotence conjecture, and Tate's conjecture, in several new cases; embed the (cohomological) Brauer group into secondary K-theory; construct a noncommutative motivic Gysin triangle; compute the localizing …
We develop birational versions of Voevodsky's triangulated categories of motives over a field, and relate them with the pure birational motives studied in arXiv:0902.4902 [math.AG]. We also get an interpretation of unramified cohomology in this framework, leading to "higher derived functors of unramified cohomology".
Using Dold--Puppe category approach to the duality in topology, we prove general duality theorem for the category of motives. As one of the applications of this general result we obtain, in particular, a generalization of Friedlander--Voevodsky's duality to the case of arbitrary base field characteristic.
Triangulated Categories of Big Motives via Enriched Functors
本文利用丰富函子的Grothendieck范畴同调代数，为Voevodsky的合理对应大动机范畴构造了两个模型。
We construct a functor from the triangulated category of Voevodsky motives to a certain derived category of mixed Hodge structures enriched with integral weight filtration. We use this construction to prove a strong integral version of the Deligne conjecture on 1-motives which was known previously only up to isogeny.
In this paper we describe the category of motives for an elliptic curve in the sense of Voevodsky as a derived category of dg modules over a commutative differential graded algebra in the category of representations over some reductive group.
We show that the pairing on de Rham realizations of 1-motives in "Theorie di Hodge III", IHES 44, can be defined over any base scheme and we prove that it gives rise to a perfect duality if one is working with a 1-motive and its Cartier dual. Furthermore we study universal extensions of 1-motives and their relation with $\natural$-extensions.
For every smooth and separated Deligne-Mumford stack $F$, we associate a motive $M(F)$ in Voevodsky's category of mixed motives with rational coefficients $\mathbf{DM}^{\eff}(k,\mathbb{Q})$. When $F$ is proper over a field of characteristic 0, we compare $M(F)$ with the Chow motive associated to $F$ by Toen (\cite{t}). Without the properness condition we show that $M(F)$ is a direct summand of the motive of a smooth quasi-projective variety.
In Voevodsky's theory of motives, the Nisnevich topology on smooth schemes is used as an important building block. In this paper, we introduce a Grothendieck topology on proper modulus pairs, which will be used to construct a non-homotopy invariant generalization of motives. We also prove that the topology satisfies similar properties to the Nisnevich topology.
We introduce a tower of localizing subcategories in Voevodsky's big (closed under infinite coproducts) triangulated category of motives. We show that the tower induces an interesting finite filtration on the motivic cohomology groups of smooth schemes over a perfect field. With rational coefficients, this finite filtration satisfies several of the properties of the still conjectural Bloch-Beilinson-Murre filtration.
K3 surfaces and cubic fourfolds with Abelian motive
本文证明高代数2-圈秩的三次四维流形及其相关超凯勒簇具有阿贝尔且有限维的Chow动机，并从四维K3曲面族构造出两族动机有限维的Fano簇。
Motives with modulus, I: Modulus sheaves with transfers for non-proper modulus pairs
本研究构建了推广Voevodsky带转移层理论的带转移模层理论，为[KMSY20]中的带模动机理论奠定基础。
Grothendieck first defined the notion of a "motif" as a way of finding a universal cohomology theory for algebraic varieties. Although this program has not been realized, Voevodsky has constructed a triangulated category of geometric motives over a perfect field, which has many of the properties expected of the derived category of the conjectural abelian category of motives. The construction of the triangulated category of motives has been extended by Cisinski-Déglise to a triangulated category of motives over a base-scheme $S$. Recently, Bondarko constructed a DG category of motives, whose ho…
The category of effective $Witt$-motives $DWM^-(k)$ with functor $WM\colon Sm_k\to DWM^-(k)$ defining motives of smooth affine varieties for perfect field $k$, $char k\neq 2$ is constructed. In the construction Voevodsky-Suslin method is applyed to a category of $Witt$-correspondence between affine smooth varieties $WCor_k$ that morphisms are defined by class in $Witt$-group of quadratic space $(P,q_P)$ with $P$ being $k[X\times Y]$-module finitely generated projective over $k[X]$ and $q_P\colon P\to Hom_{k[X]}(P,k[X])$ being $k[X\times Y]$-liner isomorphism. And the natural isomorphism $$Hom_…
Big pictures of motivic and classical homotopy theories
本文通过与经典对应物比较，探讨motivic同伦理论的全景及其与Voevodsky motives的联系。
In the article [GS96], Gillet and Soulé define a weight complex on the category of Voevodsky motives over a field of characteristic 0. In [Bon07], Bondarko generalizes this construction for any f-category with a bounded weight structure, as is the case for Beilinson motives (following Cisinski-Déglise ; [CD09]). The first purpose of this note is to generalize [GS96, thm. 2] in the world of Beilinson motives. This done, we will naturally be led to define the motivic Euler characteristic dual to that considered by Bondarko in [Bon10]. This fact will motivate the second line of this note : provin…
Strict homotopy invariance via compactified homotopies and correspondences, and fibres of essentially smooth schemes over one-dimensional base schemes
本研究发展一维基概形上的紧化对应与同伦技术，阐明Voevodsky严格同伦不变性定理等的特征假设性质，还得到广义动机上同伦Cousin复形相关猜想的类似结论及一类无环性结果。
Some remarks on Chow correspondences
本文在Voevodsky的三角动机范畴中研究了Chow环中几种适当的等价关系及其性质。
Voevodsky's derived category of motives is the main arena today for the study of algebraic cycles and motivic cohomology. In this paper we study whether the inclusions of three important subcategories of motives have a left or right adjoint. These adjoint functors are useful constructions when they exist, describing the best approximation to an arbitrary motive by a motive in a given subcategory. We find a fairly complete picture: some adjoint functors exist, including a few which were previously unexplored, while others do not exist because of the failure of finite generation for Chow groups …
In this short note, making use of the recent theory of noncommutative mixed motives, we prove that the Voevodsky's mixed motive of a quadric fibration over a smooth curve is Kimura-finite.
This thesis presents a way to apply this theorem of Gabber to a large portion of Voevodsky's work in order to lift the assumption that resolution of singularities holds. This gives unconditional versions of many of his and others' theorems provided we work Z[1/p] linearly, where p is the exponential characteristic of the base field. One example of the many applications we give is a partial answer to a 1980 conjecture of Weibel. Another is the removal of the hypothesis of resolution of singularities from a result of Suslin that compares Bloch's higher Chow groups and etale cohomology. Voevodsky…
The machinery of framed (pre)sheaves was developed by Voevodsky [V1]. Based on the theory, framed motives of algebraic varieties are introduced and studied in [GP1]. An analog of Voevodsky's Cancellation Theorem [V1] is proved in this paper for framed motives stating that a natural map of framed $S^1$-spectra $$M_{fr}(X)(n)\to\underline{\textrm{Hom}}(\mathbb G,M_{fr}(X)(n+1)),\quad n\geq 0,$$ is a schemewise stable equivalence, where $M_{fr}(X)(n)$ is the $n$th twisted framed motive of $X$. This result is also necessary for the proof of the main theorem of [GP1] computing fibrant resolutions o…
The theory of framed motives by Garkusha and Panin gives computations in the stable motivic homotopy category $\mathbf{SH}(k)$ in terms of Voevodsky's framed correspondences. In particular the motivically fibrant $Ω$-resolution in positive degrees of the motivic suspension spectrum $Σ_{\mathbb P^1}^\infty X_+$, where $X_+=X\amalg *$, for a smooth scheme $X\in \mathrm{Sm}_k$ over an infinite perfect field $k$, is computed. The computation by Garkusha, Neshitov and Panin of the framed motives of relative motivic spheres $(\mathbb A^l\times X,(\mathbb A^l-0)\times X)$, $X\in \mathrm{Sm}_k$, is on…
Profinite Borel completeness and smooth Artin motives
本文研究有限群的Borel完备性与光滑Artin动机，探讨了有限群和 profinite 群的Borel完备性概念，并通过Voevodsky定理扩展了Nisnevich情况下的光滑Artin动机与Bredon上同调谱的关系。
The goal of this paper is to define a certain Chow weight structure for the category of Voevodsky's motivic complexes with integral coefficients (as described by Cisinski and Deglise) over any excellent finite-dimensional separated scheme $S$. Our results are parallel to (though substantially weaker than) the corresponding 'rational coefficient' statements proved by D. Hebert and the author. As an immediate consequence of the existence of 'weights', we obtain certain (Chow)-weight spectral sequences and filtrations for any (co)homology of $S$-motives.
We study Grothedieck groups of triangulated categories using weight structures, weight complexes, and the corresponding pure (co)homological functors. We prove some general statements on $K_0$ of weighted categories and apply it to Voevodsky motives endowed with so-called Chow weight structures. We obtain certain "motivic substitutes" for smooth compactifications of smooth varieties over arbitrary perfect fields; this enables us to make certain unramified cohomology and Euler characteristic calculations that are closely related to results of T. Ekedahl and B. Kahn.
Let X and Y be smooth complex projective varieties. Orlov conjectured that if X and Y are derived equivalent then their motives M(X) and M(Y) are isomorphic in Voevodsky's triangulated category of geometrical motives with rational coefficients. In this paper we prove the conjecture in the case X is a K3 surface admitting an elliptic fibration (a case that always occurs if the Picard rank of X is at least 5) with finite-dimensional Chow motive. We also relate this result with a conjecture by Huybrechts showing that, for a K3 surface with a symplectic involution, the finite-dimensionality of its…
The category of effective Grothendieck-Witt-motives $\mathbf{DM}^{GW}_{\mathrm{eff},-}(k)$ (and Witt-motives $\mathbf{DM}^W_{\mathrm{eff},-}(k)$) by Voevodsky-Suslin method starting with some category of GW-correspondences (and Witt-correspondences) over a perfect field $k$, $char\,k\neq 2$, is defined. The functor $M^{GW}_{eff}\colon Sm_k\to \mathbf{DM}^{GW}_{\mathrm{eff},-}(k)$ of Grothendieck-Witt-motives of smooth varieties is computed and it is proved that for any smooth scheme $X$ and homotopy invariant sheave with GW-transfers $F$ $$ Hom_{\mathbf{DM}^{GW}_{\mathrm{eff},-}(k)}(M^{GW}_{ef…
On motives of parabolic Higgs bundles and parabolic connections
本文研究紧 Riemann 曲面上抛物型 Higgs 丛与抛物型联络的模空间，证明一般权重下二者在 Grothendieck 动机类、$E$-多项式及 Voevodsky、Chow 动机上等价，并推广至固定行列式情形。
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