54 pages, fully revised exposition with a new "structure theorem" for 1-motivic sheaves, see Theorem 1.3.10, including finitely presented (or constructible) 1-motivic sheaves
发表信息：
Journal of Pure and Applied Algebra Vol. 213, No. 5 (2009) 809-839
We introduce n-generated sheaves and n-motivic sheaves, describing completely for n = 0, 1 and proposing a conjecture for n > 1. We then obtain functors Lπ_0 and LAlb on DM_{eff}(k) deriving π_0 and Alb. The functor LAlb extends the one constructed (by the second author jointly with B.Kahn) to non-necessarily geometric motives. These functors are then used to define higher Néron-Severi groups and higher Albanese sheaves. The latter may be considered as an algebraic avatar of Deligne (co)homology.
For every noetherian smooth and separated scheme over an algebraically closed field, we define its derived Albanese in Voevodsky's triangulated category of effective Nisnevich motives. To justify our definition, we relate the derived Albanese with the Albanese scheme. We also prove that the derived Albanese satisfies the Nisnevich descent property.
Periods of moduli spaces of stable sheaves on K3 surfaces were computed by Mukai, O'Grady and the author. In this paper, we shall treat moduli spaces of stable sheaves on abelian surfaces.
We define higher pro-Albanese functors for every effective log motive over a field $k$ of characteristic zero, and we compute them for every smooth log smooth scheme $X=(\underline{X}, \partial X)$. The result involves an inverse system of the coherent cohomology of the underlying scheme as well as a pro-group scheme $\mathrm{Alb}^{\log}(X)$ that extends Serre's semi-abelian Albanese variety of $\underline{X}-|\partial X|$. This generalizes the higher Albanese sheaves of Ayoub, Barbieri-Viale and Kahn.
For a smooth projective complex variety whose Albanese morphism is finite, we show that every Bridgeland stability condition on its bounded derived category of coherent sheaves is geometric, in the sense that all skyscraper sheaves are stable with the same phase. Furthermore, we describe the stability manifolds of irregular surfaces and abelian threefolds with Picard rank one, and show that they are connected and contractible.
We study the moduli space of rank 0 semistable sheaves on some rational surfaces. We show the irreducibility and stable rationality of them under some conditions. We also compute several (virtual) Betti numbers of those moduli spaces by computing their motivic measures.
We extend the construction of the category of 1-motivic sheaves (introduced by Barbieri-Viale and Kahn) allowing quotients of connected algebraic k-groups by formal k-groups. We show that its bounded derived category is equivalent to the bounded derived category of the category of generalized 1-motives with torsion introduced in a previous paper by Barbieri-Viale and the author.
We generalize the construction of the category of 1-motives with torsion ${}^tM_1$ (introduced by Barbieri-Viale, Rosenschon and Saito) as well as the construction of the category of 1-motivic sheaves ${\rm Shv}_1$ (defined by Barbieri-Viale and Kahn) to perfect fields $k$ (without inverting the exponential characteristic). For $k$ transcendental over the prime field we extend a result of Barbieri-Viale and Kahn, showing that ${}^tM$ and ${\rm Shv}_1$ have equivalent bounded derived categories.
Let $\mathcal{M}(d,χ)$ be the moduli stack of stable sheaves of rank 0, Euler characteristic $χ$ and first Chern class $dH~(d>0)$, with $H$ the hyperplane class in $\mathbb{P}^2$. We compute the $A$-valued motivic measure $μ_A(\mathcal{M}(d,χ))$ of $\mathcal{M}(d,χ)$ and get explicit formula in codimension $D:=ρ_d-1$, where $ρ_d$ is $d-1$ for $d=p$ or $2p$ with $p$ prime, and $7$ otherwise. As a corollary, we get the last $2(D+1)$ Betti numbers of the moduli scheme $M(d,χ)$ when $d$ is coprime to $χ$.
These lecture notes cover four topics. There is a proof of the fact that the functors represented by the motivic Eilenberg-Maclane spaces on the motivic homotopy category coincide with the motivic cohomology defined in terms of the motivic complexes. There is a description of the equivariant motivic homotopy category for a finite flat group scheme (over a noetherian base) together with a new characterization of A^1-equivalences. There is a part where we introduce a class of sheaves called solid sheaves. Finally there is a part where we study functors of the form X -> X/G and X -> X^W and show …
The goal of this paper is to construct a category of motivic "sheaves" on an algebraic variety defined over a subfield of C, using Nori's method. This categoryis abelian and it possesses faithful exact realization functors to the categoriesof constructible sheaves for the classical and etale topologies. Moreover, there is a tannakian subcategory of motivic local systems with a realization functor into the category of variations of mixed Hodge structures. Conversely, all basic geometric examples of the latter come from this motivic category.
The goal of this paper is to prove: if certain 'standard' conjectures on motives over algebraically closed fields hold, then over any 'reasonable' $S$ there exists a motivic $t$-structure for the category of Voevodsky's $S$-motives (as constructed by Cisinski and Deglise). If $S$ is 'very reasonable' (for example, of finite type over a field) then the heart of this $t$-structure (the category of mixed motivic sheaves over $S$) is endowed with a weight filtration with semi-simple factors. We also prove a certain 'motivic decomposition theorem' (assuming the conjectures mentioned) and characteri…
In earlier work (arXiv:0801.0261), we gave a definition of an abelian category of motivic (constructible) sheaves over a base in characteristic zero using Nori's method. This category has Hodge and etale realizations, and is stable under inverse and direct images for projective and constant maps. The goal of this paper is to give a simpler definition of (a slight modification of) the category, and to give an easier proof of the direct image theorem. This paper relies on the previous article for some technical results, but is otherwise self contained.
With representation-theoretic applications in mind, we construct a formalism of reduced motives with integral coefficients. These are motivic sheaves from which the higher motivic cohomology of the base scheme has been removed. We show that reduced stratified Tate motives satisfy favorable properties including weight and t-structures. We also prove that reduced motives on cellular (ind-)schemes unify various approaches to mixed sheaves in representation theory, such as Soergel-Wendt's semisimplified Hodge motives, Achar-Riche's complexes of parity sheaves, as well as Ho-Li's recent category of…
We introduce the notion of a quasi DG category, generalizing that of a DG category. To a quasi DG category satisfying certain additional conditions, we associate another quasi DG category, the quasi DG category of $C$-diagrams. We then show the homotopy category of the quasi DG category of $C$-diagrams has the structure of a triangulated category. This procedure is then applied to produce a triangulated category of mixed motives over a base variety.
These are the notes accompanying three lectures given by the second author at the Motivic Geometry program at CAS, which aim to give an introduction and an overview of some recent developments in the field of reciprocity sheaves.
Let $alb_X: X \rightarrow A$ be the Albanese map of a smooth projective variety and $f: X \rightarrow Y$ the fibration from the Stein factorization of $alb_X$. For a positive integer $m$, if $f$ and $m$ satisfy the assumptions AS(1,2), then the translates through the origin of all components of cohomological locus $V^0(ω_X^m, alb_X)$ generates $I^*Pic^0(S)$ where $I: X \rightarrow S$ denotes the Iitaka fibration. This result applies to studying pluricanonical maps. We also considered the problem about whether a fibration is isotrivial and isogenous to a product.
We give a new geometric characterization of the motivic ramification filtration of reciprocity sheaves, by imitating a method used by Abbes and (Takeshi) Saito to study the ramification of torsors under finite étale groups. This new characterization is used to define characteristic forms for reciprocity sheaves. We obtain applications on pseudo-rational singularities and on questions regarding the representability of certain cohomology groups of reciprocity sheaves in the triangulated category of motives with modulus introduced by Kahn-Miyazaki-Saito-Yamazaki.
We define a motivic conductor for any presheaf with transfers $F$ using the categorical framework developed for the theory of motives with modulus by Kahn-Miyazaki-Saito-Yamazaki. If $F$ is a reciprocity sheaf this conductor yields an increasing and exhaustive filtration on $F(L)$, where $L$ is any henselian discrete valuation field of geometric type over the perfect ground field. We show if $F$ is a smooth group scheme, then the motivic conductor extends the Rosenlicht-Serre conductor; if $F$ assigns to $X$ the group of finite characters on the abelianized étale fundamental group of $X$, then…
We construct a motivic Eilenberg-MacLane spectrum with a highly structured multiplication over smooth schemes over Dedekind domains which represents Levine's motivic cohomology. The latter is defined via Bloch's cycle complexes. Our method is by gluing p-completed and rational parts along an arithmetic square. Hereby the finite coefficient spectra are obtained by truncated étale sheaves (relying on the now proven Bloch-Kato conjecture) and a variant of Geisser's version of syntomic cohomology, and the rational spectra are the ones which represent Beilinson motivic cohomology. As an application…
We propose a construction of a tensor exact category F_X^m of Artin-Tate motivic sheaves with finite coefficients Z/m over an algebraic variety X (over a field K of characteristic prime to m) in terms of etale sheaves of Z/m-modules over X. Among the objects of F_X^m, in addition to the Tate motives Z/m(j), there are the cohomological relative motives with compact support M_cc^m(Y/X) of varieties Y quasi-finite over X. Exact functors of inverse image with respect to morphisms of algebraic varieties and direct image with compact supports with respect to quasi-finite morphisms of varieties Y\to …
The category of framed correspondences $Fr_*(k)$ and framed sheaves were invented by Voevodsky in his unpublished notes [V2]. Based on the theory, framed motives are introduced and studied in [GP1]. These are Nisnivich sheaves of $S^1$-spectra and the major computational tool of [GP1]. The aim of this paper is to show the following result which is essential in proving the main theorem of [GP1]: given an infinite perfect base field $k$, any $k$-smooth scheme $X$ and any $n\geq 1$, the map of simplicial pointed Nisnevich sheaves $(-,\mathbb{A}^1//\mathbb G_m)^{\wedge n}_+\to T^n$ induces a Nisne…
We introduce exponential complexes of sheaves on manifolds. They are resolutions of the (Tate twisted) constant sheaves of the rational numbers, generalising the short exact exponential sequence. There are canonical maps from the exponential complexes to the de Rham complex. Using this, we introduce new complexes calculating the rational Deligne cohomology. We call them exponential Deligne complexes. Their advantage is that, at least at the generic point of a complex variety, one can define Beilinson's regulator map to the rational Deligne cohomology on the level of complexes. Namely, we defin…
We apply the machinery developed by the first-named author to the K-theory of coherent G-sheaves on a finite type G-scheme X over a field, where G is a finite group. This leads to a definition of G-equivariant higher Chow groups (different from the Chow groups of classifying spaces constructed by Totaro and generalized to arbitrary X by Edidin-Graham) and an Atiyah-Hirzebruch spectral sequence from the G-equivariant higher Chow groups to the higher K-theory of coherent G-sheaves on X. This spectral sequence generalizes the spectral sequence from motivic cohomology to K-theory constructed by Bl…
In this paper, we construct a refined, relative version of the étale realization functor of motivic spaces, first studied by Isaksen and Schmidt. Their functor goes from the $\infty$-category of motivic spaces over a base scheme $S$ to the $\infty$-category of $p$-profinite spaces, where $p$ is a prime which is invertible in all residue fields of $S$. In the first part of this paper, we refine the target of this functor to an $\infty$-category where $p$-profinite spaces is a further completion. Roughly speaking, this $\infty$-category is generated under cofiltered limits by those spaces whose …
We give an algebro-geometric interpretation of $C_2$-equivariant stable homotopy theory by means of the $b$-topology introduced by Claus Scheiderer in his study of $2$-torsion phenomena in étale cohomology. To accomplish this, we first revisit and extend work of Scheiderer on equivariant topos theory by functorially associating to a $\infty$-topos $\mathscr{X}$ with $G$-action a presentable stable $\infty$-category $\mathrm{Sp}^G(\mathscr{X})$, which recovers the $\infty$-category $\mathrm{Sp}^G$ of genuine $G$-spectra when $\mathscr{X}$ is the terminal $G$-$\infty$-topos. Given a scheme $X$ w…
Rutgers, The State University Of New Jersey（新泽西州立罗格斯大学）；The University of Tokyo（东京大学）；Simion Stoilow Institute of Mathematics（罗马尼亚科学院西米恩·斯托伊洛数学研究所）；Université de Paris（巴黎大学）
