Questions tagged [motives]
for questions about motives in algebraic geometry, including constructions of categories of motives and motivic sheaves, and aspects of the standard conjectures.
477 questions
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Annihilator of the Lefschetz motive in Grothendieck ring of varieties
Does there exist a paper in which the annihilator of the Lefschetz motive in the Grothendieck ring of varieties was calculated explicitly (using K-theory of assemblers)?
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Do odd-weight Galois representations have positive and negative eigenspaces for the infinite Frobenius?
Suppose $V$ is a geometric $\ell$-adic Galois representation of $G_{\mathbb{Q}} = \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, pure of weight $2n+1$ for $n \in \mathbb{Z}$. This means that $V$ is ...
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Do odd-weight motives have positive and negative eigenspaces for the infinite Frobenius?
Let's think of a motive (over $\mathbb{Q}$ with $\mathbb{Q}$-coefficients) as a system of realizations $M_B, M_{dR}, M_{\ell}$ coming from a piece of the cohomology of an algebraic variety. (Formally, ...
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Does coarse moduli morphism induce isomorphisms on rational $\mathbb{A}^1$ homology sheaves?
Let $S$ be a Noetherian scheme and $\mathrm{Sm}_S$ the category of smooth schemes over $S$ of finite type. A motivic space over $S$ is an $\mathbb{A}^1$-local Nisnevich $\infty$-sheaf on $\mathrm{Sm}...
- 611
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Rational homology Whitehead theorem for motivic infinite loop spaces
Let $S$ be a scheme and $\mathcal{H}(S)$ be the category of motivic spaces over $S$, which is a reflective subcategory of Nisnevich $\infty$-sheaves $\mathrm{Shv}(\mathrm{Sm}_S)$. Let $L_{\mathbb{A}^1}...
- 611
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Does $S^1$-loop space preserves motivic equivalence?
Let $S$ be a scheme. Let $\mathrm{sPsh}(\mathrm{Sm}_S)$ be the category of simplicial presheaves on the category of smooth schemes over $S$ of finite type. By Nisnevich topology it is endowed with a ...
- 611
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What is the inertia-Deligne group?
In Peter Scholze’s “Geometrization of Local Langlands, Motivically”, he discusses an object coined as the “inertia-Deligne group”, which is an object that lies in an extension of the inertia group by ...
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$\infty$-Categorically enhanced filtered phi realization of lisse motives
Is there an $\infty$-categorically enhanced filtered $\phi$ realization functor with expected properties (e.g. symmetric monoidal) from the stable $\infty$-category of lisse mixed motives over a mixed ...
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Are there new developments on the Schanuel Conjecture from a Motivic Perspective?
In September 2025, a paper appeared on arXiv(https://arxiv.org/pdf/2509.08700) claiming significant progress on the motivic interpretation of Schanuel’s Conjecture.
The author (C. Bertolin) argues ...
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Grothendieck site vs simplicial site: the correct definition of category of unstable motives
Let $X$ be a topological space and $\mathrm{Sing}(X)$ be its singular simplicial set. Then we have a canonical equivalence of $\infty$-topoi
$$
\mathcal{S}/\mathrm{Sing}(X) \simeq \mathrm{PSh}_\infty(\...
- 611
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Rigidity for the property being an Abelian motive
Let $K$ be a field embeddable into $\mathbb{C}$, and $\text{Mot}(K)$ the category of motives for Absolute Hodge cycles. If the base change $h_\mathbb{C}$ of an object $h\in\text{Mot}(K)$ is in the ...
- 151
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If the motive of an abelian variety is defined over a subfield, is the abelian variety defined over that field?
Let $L / k$ be a finite extension of fields, let $M$ be a pure motive over $k$ (with rational coefficients), and suppose that the base change $M_L$ of $M$ to $L$ is isomorphic to $\mathfrak{h}^1(A)$. ...
- 1,146
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Does the second Severi-Brauer variety satisfy the Hasse principle?
Let $k$ be the rational numbers and assume $A$ is a central simple algebra over $k$ with ind$(A) > 2$.
We denote by $X$ the variety of all right ideals in $A$ of reduced dimension $2$. This variety ...
- 1,509
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Idempotents in Chow ring
In his introductory paper on motives, James Milne gives a "first attempt" on the construction of the category of motives and he writes that in this attempt, images on idempotents are ...
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First cohomology of a variety and of its Albanese variety
Let $X$ be a smooth proper variety over an algebraically closed field $k$, $A_X$ its Albanese variety and $\alpha:X\to A_X$ the "canonical" map. It is known that for both étale and ...
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