Related papers: Weights for relative motives; relation with mixed …

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…

The aim of this work is to develop a theory parallel to that of motivic complexes based on cycles and correspondences with coefficients in quadratic forms. This framework is closer to the point of view of $\mathbb{A}^1$-homotopy than the…

We describe recent achievements in the theory of weight systems, which are functions on chord diagrams satisfying so-called $4$-term relations. Our main attention is devoted to constructions of weight systems. The two main sources of these…

We construct an isomorphism of graded Frobenius algebras between the orbifold Chow ring of weighted projective spaces and graded algebras of groups of roots of the unity.

The apparatus of motivic stable homotopy theory provides a notion of Euler characteristic for smooth projective varieties, valued in the Grothendieck-Witt ring of the base field. Previous work of the first author and recent work of…

For a flat $p$-adic formal family $S$ of log points over a complete discrete valuation ring with perfect residue field of mixed characteristics $(0,p)$ and for a simple normal crossing log scheme $X$ over an exact closed log subscheme of…

We study a natural enlargement of the BGG Category O for a semisimple Lie algebra: the category of weight modules with trivial central character and finite-dimensional weight spaces supported on the root lattice. We give a geometric…

We study "canonical weight decompositions" slightly generalizing that defined by J. Wildeshaus. For an triangulated category $C$, any integer $n$, and a weight structure $w$ on $C$ a triangle $LM\to M\to RM\to LM[1]$, where $LM$ is of…

We study weighted Chebyshev polynomials on compact subsets of the complex plane with respect to a bounded weight function. We establish existence and uniqueness of weighted Chebyshev polynomials and derive weighted analogs of Kolmogorov's…

We present a detailed introduction of the theory of constructible sheaf complexes in the complex algebraic and analytic setting. All concepts are illustrated by many interesting examples and relevant applications, while some important…

The Beilinson-Bloch type conjectures predict that the low degree rational Chow groups of intersections of quadrics are one dimensional. This conjecture was proved by Otwinowska. Making use of homological projective duality and the recent…

We define and compare two different definitions of Chow motives for Deligne-Mumford stacks, associated with two definitions of Chow rings. The main result we prove is that both categories of motives are equivalent to the usual category of…

This article is about motives of quadric bundles. In the case of odd dimensional fibers and where the basis is of dimension two we give an explicit relative and absolute Chow-K\"unneth decomposition. This shows that the motive of the…

Let $X$ be a compact Riemann surface of genus $g \geq 2$ and let $D\subset X$ be a fixed finite subset. We considered the moduli spaces of parabolic Higgs bundles and of parabolic connections over $X$ with the parabolic structure over $D$.…

This article provides some basic results on weight structures, weight complex functors and homotopy categories. We prove that the full subcategories K(A)^{w < n}, K(A)^{w > n}, K(A)^- and K(A)^+ (of objects isomorphic to suitably bounded…

Motivic cohomology is powerful tool in algebraic geometry with associated realization maps giving important information about the relations between cohomological invariants of schemes and their classifying spaces. The problem of computing…

Over a perfect field k, let G be an extension of an abelian variety by the multiplicative group $\G_m$. We compute the motive of G in Voevodsky's category of etale motivic complexes with rational coefficients. The result is a decomposition…

Given a perfect field of exponential characteristic $e$ and a functor $f:\mathcal A\to\mathcal B$ between symmetric monoidal strict $V$-categories of correspondences satisfying the cancellation property such that the induced morphisms of…

We associate a weight function to pairs consisting of a smooth and proper variety X over a complete discretely valued field and a differential form on X of maximal degree. This weight function is a real-valued function on the…

In this paper, we present a general approach to establish motivic cohomology and build part of its six operations formalism. Applying this together with symplectic orientation on MW-motivic cohomology, we discuss the embedding theorem of…