Related papers: Berkovich Motives
This text is devoted to the systematic study of relative properties in the context of Berkovich analytic spaces. We first develop a theory of flatness in this setting. After having shown through a counter-example that naive flatness cannot…
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…
We review the shape theory of $\infty$-topoi, and relate it with the usual cohomology of locally constant sheaves. Additionally, a new localization of profinite spaces is defined which allows us to extend the \'etale realization functor of…
For a reductive group $G$, we prove that complex irreducible rigid $G$-local systems with quasi-unipotent monodromies and finite order abelianization on a smooth curve are motivic, generalizing a theorem of Katz for $GL_n$. We do so by…
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…
We analyze stabilization with respect to ${\mathbb P}^1$ in the Morel--Voevodsky unstable motivic homotopy theory. We introduce a refined notion of cellularity (a.k.a., biconnectivity) in various motivic homotopy categories taking into…
For a variety with a Whitney stratification by affine spaces, we study categories of motivic sheaves which are constant mixed Tate along the strata. We are particularly interested in those cases where the category of mixed Tate motives over…
We discuss relations between the motives of two varieties with equivalent derived categories of coherent sheaves.
We fill a gap in the study of contextuality by extending the sheaf-theoretic framework for non-locality by Abramsky and Brandenburger to deal with operational scenarios in the presence of arbitrary definite causal orders.
In this short note we show how results of Orlov and To\"en imply that any equivalence between the derived categories of coherent sheaves on two varieties lifts to an equivalence at the level of dg-categories. This establishes the link…
We introduce coefficient systems of pro-\'etale motives and pro-\'etale motivic spectra with coefficients in any condensed ring spectrum and show that they afford the six operations. Over locally \'etale bounded schemes, \'etale motivic…
These notes, written version of a Bourbaki talk, survey Morel-Voevodsky's motivic homotopy theory over a field, with a focus on computations of motivic homotopy sheaves, both stable and unstable. We also describe Isaksen-Wang-Xu's…
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…
We study Medvedev reducibility in the context of set theory -- specifically, forcing and large cardinal hypotheses. Answering a question of Hamkins and Li \cite{HaLi}, we show that the Medvedev degrees of countable ordinals are far from…
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.
A twisted rational map over a non-archimedean field $K$ is the composition of a rational function over $K$ and a continuous automorphism of $K$. We explore the dynamics of some twisted rational maps on the Berkovich projective line.
We apply Wildeshaus's theory of motivic intermediate extensions to the motivic decomposition conjecture, formulated by Deninger-Murre and Corti-Hanamura. We first obtain a general motivic decomposition for the Chow motive of an arbitrary…
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…
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…
A mixed Weil cohomology with values in an abelian rigid tensor category is a cohomological functor on Voevodsky's category of motives which is satisfying K\"unneth formula and such that its restriction to Chow motives is a Weil cohomology.…