Borel–Moore motivic homology and weight structure on mixed motives

Jin, Fengnian

doi:10.1007/s00209-016-1636-7

Cited by 15 publications

(6 citation statements)

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“…Property (7) follows from the fact that over a perfect field their construction reduces to Voevodsky's original construction [Voe00] (see [CD09,11.1.14]), where this is known by [Voe00, 2.1.4] (after composing with the duality on CHM ef f (k), DM (k)). Alternatively, Chow motives can be identified as the heart of Bondarko's weight structure due to [Fan16] giving us (7). Property (8) can be deduced from the corresponding computation in Voevodsky's category, see [MVW06, 3.5, 4.2, 19.3] (this is also present in [CD09, 11.2.3]).…”

Section: And An Essentially Stein Factorizationmentioning

confidence: 99%

On weight truncations in the motivic setting

Vaish

2018

Advances in Mathematics

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Section: And An Essentially Stein Factorizationmentioning

confidence: 99%

On weight truncations in the motivic setting

Vaish

2018

Advances in Mathematics

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“…Fix a subfield R of R and a scheme X of finite type over Q: according to [Héb11], the category DM B,c (X) R (see 1.4.3) is equipped with a canonical weight structure, the motivic weight structure, whose heart (the subcategory of weight 0 objects) is denoted by CHM (X) R and called the (R-linear version of the) category of Chow motives over X. If X = S K as in subsection 1.4.3, then, after [Fan16], this category is equivalent to the homonymous category introduced in that subsection.…”

Section: The Degeneration Of the Canonical Construction At The Boundarymentioning

confidence: 99%

On the boundary and intersection motives of genus 2 Hilbert-Siegel varieties

Cavicchi¹

2018

Preprint

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We study genus 2 Hilbert-Siegel varieties, i.e. Shimura varieties S K corresponding to the group GSp 4,F over a totally real field F , along with the relative Chow motives λ V of abelian type over S K obtained from irreducible representations V λ of GSp 4,F . We analyse the weight filtration on the degeneration of such motives at the boundary of the Baily-Borel compactification and we find a criterion on the highest weight λ, potentially generalisable to other families of Shimura varieties, which characterises the absence of the middle weights 0 and 1 in the corresponding degeneration. Thanks to Wildeshaus' theory, the absence of these weights allows us to construct Hecke-equivariant Chow motives over Q, whose realizations equal interior (or intersection) cohomology of S K with V λ -coefficients. We give applications to the construction of motives associated to automorphic representations.

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“…where SmP roj denotes the category of smooth projective varieties over Spec k, CHM (k) denotes the category of Chow motives over the field k, and the last functor is fully faithful. This can be extended to give a functor: h k : (Sch/k) op → DM (Spec k) More generally, due to work of Corti-Hanamura [CH00], there is a category of Chow motives CHM (S) over any regular base S. Due to work of Hebert [Héb11] and Bondarko [Bon14] there is a weight structure (DM w≤0 (S), DM w>0 (S)) on DM (S) and due to work of Fangzhou [Fan16], it's heart can be identified with the category of Chow motives. Therefore we have functors…”

Section: Motivic Sheavesmentioning

confidence: 99%

“…Recall that in [Vai17a,4.2.7] one could define a canonical intersection motive for an arbitrary threefold X as an object in the triangulated category of mixed motivic sheaves, IM X ∈ DM (X, Q) (see 2.2.1). One of the key limitations there was that we could not show that IM X satisfies the intrinsic characterization of an intersection motive as defined in Wildeshaus [Wil12a], or, even more elementarily, show that it is a relative Chow motive (that is of weight 0 in the sense of Bondarko [Bon14] or Hebert [Héb11], or, equivalently due to Fangzhou [Fan16] lives in the full subcategory of relative Chow motives CHM (X) of Corti-Hanamura [CH00]).…”

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confidence: 95%

Motivic Intersection Complex of Certain Shimura varieties

Vaish¹

2018

Preprint

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Using a version of weight conservativity we demonstrate that for certain Shimura varieties (including all Shimura three-folds, most Shimura four-folds and the Siegel sixfold) the construction of the motivic intersection complex due to Wildeshaus compares with a motivic weight truncation in the sense of S. Morel. In particular it is defined up to a unique isomorphism, and satisfies the intrinsic characterization for an intermediate extension due to Wildeshaus.This general result yields to more specific results for Shimura varieties (see 3.3.1 for precise definitions):Corollary (See 3.3.4). Let X be any arbitrary Shimura threefold, or the Siegel sixfold (defined over it's reflex field k) and let Y := X * denote it's Baily-Borel compactification. Then, the intersection motive IM Y exists in DM (Y ) in the sense of [Wil12a] (see 2.4.3).Note that, the methods here also work for most Shimura fourfolds, see 3.3.6. As stated earlier, for the Siegel sixfold, we can even work with certain local systems:Corollary (See 3.3.7). Let X be the Siegel sixfold (defined over it's reflex field k) and let Y := X * denote it's Baily-Borel compactification. Let π : A → X denote the universal abelian scheme over X (which exists, for appropriate choice of arithmetic subgroups). Then, the intersection motive IM Y (N ) := j ! * N exists in DM (Y ) in the sense of [Wil12a] (see 2.4.3) where N is any summand of π * 1 A which realizes to a local system (upto shifts).1.4. The main method in this article is motivated by [Wil15b] of Wildeshaus -we use conservativity of the realization functors restricted to the triangulated category generated by motives of Abelian varieties (and arbitrary Tate twists) to calculate weights. However, for our purpose, it is not sufficient to work with weights in the sense of Bondarko [Bon14] or Hebert [Héb11] (which are motivic version of weights in the sense of Deligne), but we need to work with the motivic analogue of constructions due to S. Morel [Mor08, §3] (or more precisely, the mild generalization in [NV15, §3]). We briefly motivate the method below.

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Borel–Moore motivic homology and weight structure on mixed motives

Abstract: ABSTRACT. By defining and studying functorial properties of the Borel-Moore motivic homology, we identify the heart of Bondarko-Hébert's weight structure on Beilinson motives with Corti-Hanamura's category of Chow motives over a base, therefore answering a question of Bondarko.

Cited by 15 publications

References 20 publications

On weight truncations in the motivic setting

On weight truncations in the motivic setting

On the boundary and intersection motives of genus 2 Hilbert-Siegel varieties

Motivic Intersection Complex of Certain Shimura varieties

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