Related papers: Perfect correspondences and Chow motives

In this article we introduce a new class of non-commutative projective curves and show that in certain cases the derived category of coherent sheaves on them has a tilting complex. In particular, we prove that the right bounded derived…

Let X and Y be complex smooth projective varieties, and D^b(X) and D^b(Y) the associated bounded derived categories of coherent sheaves. Assume the existence of a triangulated category T which is admissible both in D^b(X) as in D^b(Y).…

We extend some of the results of Bondal-Orlov on the equivalence of derived categories to the case of orbifolds by using the category of coherent orbifold sheaves.

To any finite group G in SL_2(C), and each `t' in the center of the group algebra of G, we associate a category, Coh_t. It is defined as a suitable quotient of the category of graded modules over (a graded version of) the deformed…

We describe certain criteria for a motif $M$ to be $r$-effective, i.e., to belong to the $r$th Tate twist $Obj DM^{eff}_{gm,R}(r)=Obj DM^{eff}_{gm,R} \otimes L^{\otimes r}$ of effective Voevodsky motives (for $r\ge 1$; $R$ is the…

We show that the deformation theory of a perfect complex and that of its determinant are related by the trace map, in a general setting of sheaves on a site. The key technical step, in passing from the setting of modules over a ring where…

Orlov's famous representability theorem asserts that any fully faithful exact functor between the bounded derived categories of coherent sheaves on smooth projective varieties is a Fourier-Mukai functor. In this paper we show that this…

We prove analogs of Whitehead's theorem (from algebraic topology) for both the Chow groups and for the Grothendieck group of coherent sheaves: a morphism between smooth projective varieties whose pushforward is an isomorphism on the Chow…

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…

We survey some recent results concerning the so called Categorical Torelli problem. This is to say how one can reconstruct a smooth projective variety up to isomorphism, by using the homological properties of special admissible…

In this paper, we establish a structure theorem for a smooth projective variety $X$ with semi-positive holomorphic sectional curvature. Our structure theorem contains the solution for Yau's conjecture and it can be regarded as a natural…

Let X be a quasi-compact scheme, equipped with an open covering by affine schemes. A quasi-coherent sheaf on X gives rise, by taking sections over the covering sets, to a diagram of modules over the various coordinate rings. The resulting…

For any smooth projective moduli space $M$ of Gieseker stable sheaves on a complex projective K3 surface (or an abelian surface) S, we prove that the Chow motive $\mathfrak{h}(M)$ becomes a direct summand of a motive $\bigoplus…

Given a coherent sheaf E on a scheme of finite type X over a perfect field, we introduce a category of complexes of \'etale sheaves on X with logarithmic conductors bounded by E and study its compatibilities with finite push-forward.

Suppose $F\colon \mathcal{D}(X)\to \mathcal{T}$ is an exact functor from the bounded derived category of coherent sheaves on a smooth projective variety $X$ to a triangulated category $\mathcal{T}$. If $F$ possesses left and right adjoints,…

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.…

In this paper, we prove that the bounded derived category $D^b_{coh}(Y)$ of coherent sheaves on a separated scheme $Y$ of finite type over a field $\mathrm{k}$ of characteristic zero is homotopically finitely presented. This confirms a…

In topology, the Steenrod problem asks whether every singular homology class is the pushforward of the fundamental class of a closed oriented manifold. Here, we introduce an analogous question in algebraic geometry: is every element on the…

In this paper certain Chow weight structures on the "big" triangulated motivic categories $DM_R^{eff}\subset DM_R$ are defined in terms of motives of all smooth varieties over the base field. This definition allows studying basic properties…

Let $k$ be a perfect field of characteristic $p>0$. In this paper, without assuming resolution of singularities, we prove that the triangulated category of motives with modulus with rational coefficients is equivalent to Voevodsky's…