Intermediate extension of Chow motives of Abelian type

Wildeshaus, Jörg

doi:10.1016/j.aim.2016.09.032

Cited by 18 publications

(57 citation statements)

References 46 publications

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“…In section 4.1 we demonstrate that the truncations w ≤Id and w ≤Id +1 factor through birational motives and use them to construct several invariants, including EM Id X , EM Id +1 X . We also demonstrate the expected relationship of these objects with Wildeshaus' intersection complex [Wil12a]. Finally, in section 4.2 we construct the motivic IC X for any 3-fold X, and show that the motivic IC X as well as motives EM Id X , EM Id +1 X have the expected realizations.…”

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confidence: 71%

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On weight truncations in the motivic setting

Vaish

2018

Advances in Mathematics

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confidence: 71%

“…The relationship holds whenever IM X exists (e.g. for X the Baily-Borel compactification of a locally symmetric Hermitian space by [Wil12a]).…”

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confidence: 99%

On weight truncations in the motivic setting

Vaish

2018

Advances in Mathematics

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“…* F is a simple perverse sheaf (cf. As we have recalled, Wildeshaus constructs [1·] * between intersection motives; see also the construction leading up to Corollary 8.8 in [27]. Thus IM(S K , F W ) is a direct factor of IM(S K ′ , F W ), modulo homological equivalence.…”

Section: Connected Componentsmentioning

confidence: 88%

“…* μ (W ) under the structure morphism m : S K −→ Spec k, one gets the intersection motive IM(S K , W ) whose Betti realization is canonically isomorphic to the intersection cohomology IH * (S K (C), W ). 3 Also constructed in [27] (Theorem 0.5) is an endomorphism KgK of IM(S K , W ), for each double coset KgK ∈ K \ G(A f )/K, whose Betti realization coincides with the usual action of the Hecke operator for the coset on the intersection cohomology. The construction uses the compatibility of the motivic middle extension and the "change of level" maps [h· ] * , see Theorem 0.4 of [27].…”

Section: Review Of Motivic Constructionsmentioning

confidence: 99%

“…A possible generalization is the motive representing the intersection cohomology of the minimal compactification of S K . Such a motive is not known to exist for general varieties (though we certainly expect that it does), but in the case of minimal compactifications of Shimura varieties it has been constructed by Wildeshaus, even in the category of Chow motives over E (cf [27] Theorems 0.1 and 0.2). We then have the following generalization of the previous theorem : Theorem 1.4 Let (G, X , h) be simple PEL Shimura data, and assume that G satisfies condition (C´).…”

Section: Introductionmentioning

confidence: 99%

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The standard sign conjecture on algebraic cycles: The case of Shimura varieties

Morel

Suh

2016

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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For the purpose of modifying the commutativity constraints and getting the Tannakian category, one needs somewhat less, and the necessary weakening is called the standard "sign" conjecture (terminology proposed by Jannsen; cf.[3] 5.1.3 for the formulation and see [3] 6.1.2.1 for obtaining the Tannakian category):is concentrated in even (resp. odd) degrees.Equivalently, for every smooth projective X/k, the sum p + X (resp. p − X ) of the even (resp. odd) Künneth projectors on H * (X) is given by an algebraic cycle. Note that such a decomposition is necessarily unique (up to unique isomorphism).This conjecture has, in addition to the consequences in terms of the category of homological motives and the algebraicity of (Hodge or Tate) cohomology classes, also the following interesting consequence, due to André and Kahn. Recall that the numerical and the homological equivalences on algebraic cycles on projective smooth varieties are conjectured to be the same. Theorem 1.2 ([3] 9.3.3.3) Let M be an additive ⊗-subcategory of M hom (k) F and let M num be its image in M num (k) F . If the sign conjecture is true for every object of M , then the functor M −→ M num admits a section compatible with ⊗, unique up to ⊗-isomorphism.Next we turn to the main geometric objects of this paper, Shimura varieties. In this paper, we will take for k a subfield of C, and for H * the cohomology theory that sends a smooth projective variety X over k to the Betti cohomology of X(C) with coefficients in number fields F .

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Perverse Motives and Graded Derived Category

Soergel¹,

Wendt²

2016

J. Inst. Math. Jussieu

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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 a point is equivalent to the category of finitedimensional bigraded vector spaces. Examples of such situations include rational motives on varieties over finite fields and modules over the spectrum representing the semisimplification of de Rham cohomology for varieties over the complex numbers. We show that our categories of stratified mixed Tate motives have a natural weight structure. Under an additional assumption of pointwise purity for objects of the heart, tilting gives an equivalence between stratified mixed Tate sheaves and the bounded homotopy category of the heart of the weight structure. Specializing to the case of flag varieties, we find natural geometric interpretations of graded category O and Koszul duality.

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Intermediate extension of Chow motives of Abelian type

Cited by 18 publications

References 46 publications

On weight truncations in the motivic setting

On weight truncations in the motivic setting

The standard sign conjecture on algebraic cycles: The case of Shimura varieties

Perverse Motives and Graded Derived Category

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