scholarly journals Weight filtration on the cohomology of complex analytic spaces

2014 ◽  
Author(s):  
Joana Cirici ◽  
Francisco Guillén
Keyword(s):  
Weight Filtration

scholarly journals The Plectic Weight Filtration on Cohomology of Shimura Varieties and Partial Frobenius

2021 ◽  
Vol 9 ◽  
Author(s):  
Zhiyou Wu
Keyword(s):  
Shimura Varieties ◽  
Weight Filtration ◽  
Hilbert Modular ◽  
Hilbert Modular Varieties ◽  
Modular Varieties

Abstract We prove that there is a natural plectic weight filtration on the cohomology of Hilbert modular varieties in the spirit of Nekovář and Scholl. This is achieved with the help of Morel’s work on weight t-structures and a detailed study of partial Frobenius. We prove in particular that the partial Frobenius extends to toroidal and minimal compactifications.

scholarly journals Mixed motivic sheaves (and weights for them) exist if ‘ordinary’ mixed motives do

2015 ◽  
Vol 151 (5) ◽  
pp. 917-956 ◽  
Author(s):  
Mikhail V. Bondarko
Keyword(s):  
Finite Type ◽  
Main Tool ◽  
Decomposition Theorem ◽  
Weight Filtration ◽  
Closed Fields ◽  
Weight Structure ◽  
Residue Fields ◽  
Algebraically Closed Fields ◽  
Mixed Motives ◽  
Weight Structures

The goal of this paper is to prove that if certain ‘standard’ conjectures on motives over algebraically closed fields hold, then over any ‘reasonable’ scheme $S$ there exists a motivic$t$-structure for the category $\text{DM}_{c}(S)$ of relative Voevodsky’s motives (to be more precise, for the Beilinson motives described by Cisinski and Deglise). If $S$ is 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 semisimple factors. We also prove a certain ‘motivic decomposition theorem’ (assuming the conjectures mentioned) and characterize semisimple motivic sheaves over $S$ in terms of those over its residue fields. Our main tool is the theory of weight structures. We actually prove somewhat more than the existence of a weight filtration for mixed motivic sheaves: we prove that the motivic $t$-structure is transversal to the Chow weight structure for $\text{DM}_{c}(S)$ (that was introduced previously by Hébert and the author). We also deduce several properties of mixed motivic sheaves from this fact. Our reasoning relies on the degeneration of Chow weight spectral sequences for ‘perverse étale homology’ (which we prove unconditionally); this statement also yields the existence of the Chow weight filtration for such (co)homology that is strictly restricted by (‘motivic’) morphisms.

scholarly journals On finite dimensionality of mixed Tate motives

2008 ◽  
Vol 4 (1) ◽  
pp. 145-161 ◽  
Author(s):  
Shahram Biglari
Keyword(s):  
Grothendieck Group ◽  
Ring Structure ◽  
Triangulated Category ◽  
Cohomology Groups ◽  
Weight Filtration ◽  
Finite Dimensional ◽  
Finite Dimensionality ◽  
Tate Motives

AbstractWe prove a few results concerning the notion of finite dimensionality of mixed Tate motives in the sense of Kimura and O'Sullivan. It is shown that being oddly or evenly finite dimensional is equivalent to vanishing of certain cohomology groups defined by means of the Levine weight filtration. We then explain the relation to the Grothendieck group of the triangulated category D of mixed Tate motives. This naturally gives rise to a λ–ring structure on K0(D).

scholarly journals Differential graded motives: weight complex, weight filtrations and spectral sequences for realizations; Voevodsky versus Hanamura

2008 ◽  
Vol 8 (1) ◽  
pp. 39-97 ◽  
Author(s):  
M. V. Bondarko
Keyword(s):  
Spectral Sequence ◽  
Finite Fields ◽  
Maximal Length ◽  
Triangulated Category ◽  
Spectral Sequences ◽  
Motivic Cohomology ◽  
Weight Filtration ◽  
Singular Cohomology ◽  
Tate Motives

AbstractWe describe explicitly the Voevodsky's triangulated category of motives $\operatorname{DM}^{\mathrm{eff}}_{\mathrm{gm}}$ (and give a ‘differential graded enhancement’ of it). This enables us to able to verify that DMgm ℚ is (anti)isomorphic to Hanamura's $\mathcal{D}$(k).We obtain a description of all subcategories (including those of Tate motives) and of all localizations of $\operatorname{DM}^{\mathrm{eff}}_{\mathrm{gm}}$. We construct a conservative weight complex functor $t:\smash{\operatorname{DM}^{\mathrm{eff}}_{\mathrm{gm}}}\to\smash{K^{\mathrm{b}}(\operatorname{Chow}^{\mathrm{eff}})}$; t gives an isomorphism $K_0(\smash{\operatorname{DM}^{\mathrm{eff}}_{\mathrm{gm}}})\to\smash{K_0(\operatorname{Chow}^{\mathrm{eff}})}$. A motif is mixed Tate whenever its weight complex is. Over finite fields the Beilinson–Parshin conjecture holds if and only if tℚ is an equivalence.For a realization D of $\operatorname{DM}^{\mathrm{eff}}_{\mathrm{gm}}$ we construct a spectral sequence S (the spectral sequence of motivic descent) converging to the cohomology of an arbitrary motif X. S is ‘motivically functorial’; it gives a canonical functorial weight filtration on the cohomology of D(X). For the ‘standard’ realizations this filtration coincides with the usual one (up to a shift of indices). For the motivic cohomology this weight filtration is non-trivial and appears to be quite new.We define the (rational) length of a motif M; modulo certain ‘standard’ conjectures this length coincides with the maximal length of the weight filtration of the singular cohomology of M.

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