Noemie C. Combe, Yuri I. Manin, Matilde Marcolli
In this note, we sketch an approach to the problems of equivariant birational geometry developed by M. Kontsevich and Yu. Tschinkel, where Burnside invariants were introduced. We are making explicit the role of Nori constructions in this environment.
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We introduce the notion of algebraic cogroup over a subfield $k$ of the complex numbers, and use it to prove that every Nori motive over $k$ is isomorphic to a quotient of a motive of the form $H^n(X, Y)(i)$.
Nori motives (and mixed Hodge modules) with integral coefficients
本文构建了具有整数系数的Nori动机sheaves的abelian范畴，研究了Nori动机的派生范畴，并构造了étale motives的代数，提供了混合Hodge模在实数方案上的类似构造。
Motivic nearby functors on perverse Nori motives
本文证明若干候选动机邻近函子在怪态诺里动机上一致，借助六函子体系构造典范函子并推导Kontsevich-Soibelman动机积分恒等式，还证明几何来源怪态层的Beilinson等价版本。
The motivic Satake equivalence using perverse Nori motives
本文提出了一种基于 perverse Nori 动机的 Satake 等价性理论，扩展了 Mirković-Vilonen 的几何 Satake 等价性。
Let EHM be Nori's category of effective homological mixed motives. In this paper, we consider the thick abelian subcategory EHM_1 generated by the i-th relative homology of pairs of varieties for i = 0,1. We show that EHM_1 is naturally equivalent to the abelian category M_1 of Deligne 1-motives with torsion; this is our main theorem. Along the way, we obtain several interesting results. Firstly, we realize M_1 as the universal abelian category obtained, using Nori's formalism, from the Betti representation of an explicit diagram of curves. Secondly, we obtain a conceptual proof of a theorem o…
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。
In this paper, we prove a motivic enhancement of the theorem of the fixed part in Hodge theory due to Deligne. In the pure motivic case, this was done for the first time by André in [And96]. Our main result is an extension to the mixed case, which strengthens a result by Arapura [Ara13] and also provides an alternative and simpler proof, in the framework of Nori motives, of a result by Ayoub [Ayo14].
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…
We prove that for $1$-motives defined over an algebraically closed subfield of $\C$, viewed as Nori motives, the motivic Galois group is the Mumford-Tate group. In particular, the Hodge realization of the tannakian category of (Nori) motives generated by $1$-motives is fully faithful.
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。
Supercycles, stable supermaps and SUSY Nori motives
本文定义了稳定的超曲线和稳定的超映射，并基于这些概念发展了SUSY曲线带孔的诺里动机理论，需先构建超循环的基础理论。
We construct a tensor product on Freyd's universal abelian category attached to an additive tensor category or a tensor quiver and establish a universal property. This is used to give an alternative construction for the tensor product on Nori motives.
In the late 90s M. V. Nori constructed a category of motives in charakteristic 0. Using a directed graph with a representation into noetherian R-Modules, he defined a universal R-linear abelian category, called diagram category.The following paper describes this construction and gives proof to the necessary universal property. In addition we establish a criteria for the diagram category to be a category of finitely generated modules over a certain coalgebra. Finally we sketch the connection between the diagram category and the representations of affine group schemes (Tannaka-Dualism) and defin…
We show that the spectrum of Kontsevich's algebra of formal periods is a torsor under the motivic Galois group for mixed motives over the rational numbers. This assertion is stated without proof by Kontsevich and originally due to Nori. In a series of appendices, we also provide the necessary details on Nori's category of motives.
This is a considerably expanded version of the "pure" part of our 2002 preprint. We define a category of pure birational motives over a field, depending on the choice of an adequate equivalence relation on algebraic cycles. It is obtained by "killing" the Lefschetz motive in the corresponding category of effective motives. For rational equivalence, it encompasses Bloch's decomposition of the diagonal. We study the induced Chow-Künneth decompositions in this category, and establish relationships with Rost's cycle modules and the Albanese functor for smooth projective varieties.
Let $k$ be a number field. We describe the category of Laumon 1-isomotives over $k$ as the universal category in the sense of Nori associated with a quiver representation built out of smooth proper $k$-curves with two disjoint effective divisors and a notion of $H^1_\dR$ for such "curves with modulus". This result extends and relies on the theorem of J. Ayoub and L. Barbieri-Viale that describes Deligne's category of 1-isomotives in terms of Nori's Abelian category of motives.
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".
Birational motives and the norm residue isomorphism theorem
该研究揭示了Suslin-Voevodsky-Rost范数剩余同构定理与双有理动机理论的关系，并给出该定理向高阶jets的推广。
We construct geometric lifts of the Bost-Connes algebra to Grothendieck rings and to the associated assembler categories and spectra, as well as to certain categories of Nori motives. These categorifications are related to the integral Bost-Connes algebra via suitable Euler characteristic type maps and zeta functions, and in the motivic case via fiber functors. We also discuss aspects of F1-geometry, in the framework of torifications, that fit into this general setting.
Tensor structure on perverse Nori motives
本文研究了 perverse Nori 动机上的张量结构，构建了闭合 monoidal 结构，并定义了 Chern 类，推导出 motivic 版的相对 Hard Lefschetz 定理。
The first purpose of this note is to give a partial exposition of the $l$-adic realization of Nori's category of motives. The second one is to give a simple description of the second $l$-adic Abel-Jacobi map.
We give a new construction, based on categorical logic, of Nori's $\mathbb Q$-linear abelian category of mixed motives associated to a cohomology or homology functor with values in finite-dimensional vector spaces over $\mathbb Q$. This new construction makes sense for infinite-dimensional vector spaces as well, so that it associates a $\mathbb Q$-linear abelian category of mixed motives to any (co)homology functor, not only Betti homology (as Nori had done) but also, for instance, $\ell$-adic, $p$-adic or motivic cohomology. We prove that the $\mathbb Q$-linear abelian categories of mixed mot…
We introduce a new ascending filtration, that we call the co-radical filtration in analogy with the basic theory of co-algebras, on the Chow groups of pointed smooth projective varieties. In the case of zero-cycles on projective hyper-Kähler manifolds, we conjecture it agrees with a filtration introduced by Voisin. This is established for moduli spaces of stable objects on K3 surfaces, for generalized Kummer varieties and for the Fano variety of lines on a smooth cubic fourfold. Our overall strategy is to view the birational motive of a smooth projective variety as a co-algebra object with res…
Uniform Rost nilpotence and birational motives
本文为基变换理想建立显式幂零性界，证明有效泛下降，改进均匀界，并推广检测子至理想层面，应用于三维簇。
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.
In this note, we prove that the F-fundamental group scheme is birational invariant for smooth projective varieties. We prove that the F-fundamental group scheme is naturally a quotient of the Nori fundamental group scheme. For elliptic curves, it turns out that the F-fundamental group scheme and the Nori fundamental group scheme coincides. We also consider an extension of the Nori fundamental group scheme in positive characteristic using semi-essentially finite vector bundles and prove that in this way, we do not get a non-trivial extension of the Nori fundamental group scheme for elliptic cur…
In this mostly expository note we explain how Nori's theory of motives achieves the aim of establishing a Galois theory of periods, at least under the period conjecture. We explain and compare different notions periods, different versions of the period conjecture and view the evidence by explaining the examples of Artin motive, mixed Tate motives and 1-motives.
Torsion birational motives of surfaces and unramified cohomology
该研究探讨了曲面的torsion birational motives与unramified cohomology的关系，并扩展了Kahn的结果，同时给出了违反积分Hodge猜想的例子。
We will consider an explicit birational map between a quadric and the projective variety X(J) of traceless rank one elements in a simple reduced Jordan algebra J. X(J) is a homogeneous G-variety for the automorphism group G=Aut(J). We will show that the birational map is a blow up followed by a blow down. This will allow us to use the blow up formula for motives together with Vishik's work on the motives of quadrics to give a motivic decomposition of X(J).
In this paper we demonstrate that 'non-commutative localizations' of arbitrary additive categories (generalizing those defined by Cohn for rings) are closely (and naturally) related with weight structures. Localizing an arbitrary triangulated $C$ by a set $S$ of morphisms in the heart of a weight structure $w$ for it one obtains a triangulated category endowed with a weight structure $w'$. The heart of $w'$ is a certain idempotent completion of the non-commutative localization of the heart of $w$ by $S$. The latter is the natural categorical version of Cohn's localizations of rings i.e. the fu…
We explain how some results of M. Nori (on motives) and F. Ivorra (on perverse motives) can be used to define "motivic" versions of Lyubeznik numbers, a set of numerical invariants for local rings. We also discuss on additional structures that might be put on some local cohomology modules (besides those of D- or F-module). This note is mainly expository, it is an expanded version of my talk at the FACARD workshop in Barcelona, on January 2019.
For ordinary flops, the correspondence defined by the graph closure is shown to give equivalence of Chow motives and to preserve the Poincaré pairing. In the case of simple ordinary flops, this correspondence preserves the big quantum cohomology ring after an analytic continuation over the extended Kähler moduli space. For Mukai flops, it is shown that the birational map for the local models is deformation equivalent to isomorphisms. This implies that the birational map induces isomorphisms on the full quantum rings and all the quantum corrections attached to the extremal ray vanish.
We give a survey of the background and recent development on the $K$-equivalence relation among birational manifolds. After a brief historical sketch of birational geometry, we define the $K$-partial ordering and $K$-equivalence in a birational class and discuss geometric situations that lead to these notions. One application to the filling-in problem for threefolds is given. We discuss the motivic aspect of $K$-equivalence relation. We believe that $K$-equivalent manifolds have the same Chow motive though we are unable to prove it at this moment. Instead we discuss various approaches toward t…
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