Related papers: Torsion motives

Given two smooth projective varieties X and Y over a field, we say that X motivates Y if the (suitably defined) motive of Y is contained in the category generated from X by taking sums, summands and products. This notion has appeared…

To a smooth projective variety $X$ whose Chow group of $0$-cycles is $\mathbf Q$-universally trivial one can associate its torsion index $\mathrm{Tor}(X)$, the smallest multiple of the diagonal appearing in a cycle-theoretic decomposition…

In the integral cohomology ring of the classifying space of the projective linear group $PGL_n$ (over $\mathbb{C}$), we find a collection of $p$-torsions $y_{p,k}$ of degree $2(p^{k+1}+1)$ for any odd prime divisor $p$ of $n$, and $k\geq…

We introduce the notion of the boundary motive of a scheme X over a perfect field. By definition, it measures the difference between the motive X and the motive with compact support of X. We develop three tools to compute the boundary…

We determine the mod p cohomological invariants for several affine group schemes G in chararacteristic p. These are invariants of G-torsors with values in etale motivic cohomology, or equivalently in Kato's version of Galois cohomology…

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…

Let $k$ be a perfect field and $X$ be a smooth projective surface over $k$ with a rational point, we discuss the condition of splitting off the top cell for the motivic stable homotopy type of $X$. We also study some outlying examples, such…

Let $X$ be a variety. In this survey, we study (decompositions of) the motivic class, in the Grothendieck ring of stacks, of the stack $\mathscr{C}oh^n(X)$ of $0$-dimensional coherent sheaves of length $n$ on $X$. To do so, we review the…

Let $f: X \rightarrow C$ be a genus 1 fibration from a smooth projective surface, i.e. its generic fiber is a regular genus 1 curve. Let $j: J \rightarrow C$ be the Jacobian fibration of $f$. In this paper, we prove that the Chow motives of…

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…

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,…

Let $G$ be a reductive group over $\mathbb{F}_{p}$ with associated finite group of Lie type $G^{F}$. Let $T$ be a maximal torus contained inside a Borel $B$ of $G$. We relate the (rational) Tate motives of $\text{B}G^{F}$ with the…

In this paper, we show that the infinitesimal Torelli theorem implies the existence of deformations of automorphisms. In the first part, we use Hodge theory and deformation theory to study the deformations of automorphisms of complex…

Motivic Serre invariants defined by Loeser and Sebag are elements of the Grothendieck ring of varities modulo $\mathbb{L}-1$. In this paper, we show that we can lift these invariants to modulo the square of $\mathbb{L}-1$ after tensoring…

We construct elements in the motivic cohomology of certain rank 4 weight 3 Calabi--Yau motives, and write down explicit expressions for the regulators of these elements in the context of conjectures on $L$-values such as those of Beilinson…

The orientability problem in real Gromov-Witten theory is one of the fundamental hurdles to enumerating real curves. In this paper, we describe topological conditions on the target manifold which ensure that the uncompactified moduli spaces…

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.

Making use of noncommutative motives we relate exceptional collections (and more generally semi-orthogonal decompositions) to motivic decompositions. On one hand we prove that the Chow motive M(X) of every smooth proper Deligne-Mumford…

Smooth projective $\mathbb{G}_m$-varieties with isolated rational fixed points admit Tate Milnor-Witt motives. Over Euclidean fields, we give a splitting formula of such motives, which reduces the computation of their Chow-Witt groups to…

In this article, we construct a Hecke-equivariant Chow motive whose realizations equal intersection cohomology of Siegel threefolds with regular algebraic coefficients. As a consequence, we are able to define Grothendieck motives for Siegel…