436 problems

Let $X$ and $Y$ be smooth projective varieties. Orlov's motivic conjecture predicts … where $h(X)_{\mathbb Q}$ denotes the rational Chow motive. A Fourier-Mukai equivalence produce…

Let $M$ be a $\mathbb{Q}$-motive with coefficients in a number field $F$ and odd weight. Let $H_B^+(M)$ be the $+1$-eigenspace of complex conjugation, let $\alpha_M^+$ be the compa…

Orlov's motivic conjecture. The rational Chow motives of $X$ and $Y$ are isomorphic.

Let $X$ be a smooth projective variety, and let $z$ be an algebraic cycle on $X$. Two cycles are numerically equivalent if they have the same intersection numbers with all cycles o…

A Chow motive is a motive in the category of Chow motives. Kimura–O'Sullivan conjecture. Every Chow motive is Kimura-finite. Kimura finiteness is a structural finiteness property o…

Let $G$ be a reductive group and let a local system on a smooth curve be irreducible and rigid, with quasi-unipotent local monodromies and finite-order abelianization. Simpson's co…

Grothendieck's period conjecture. The period map

Let $({\mathbb V}^{\sigma})_{\sigma}$ be a de Rham motivic variation of Hodge structure on a smooth connected complex quasi-projective variety $S$. Let ${\mathbf G}$ be the group d…

Let $F$ be a number field, $E$ a number field, $M$ a polarized motive in the pseudo-Abelian category of Chow motives over $F$ with coefficients in $E$, and let $\lbd$ be a finite p…

Rost nilpotence principle. For every $\Gamma\in \operatorname{End}_k(X)$ such that $\Gamma_E=0$ for some field extension $E/k$, $\Gamma$ is nilpotent as a correspondence.

Let $k$ be a base field and consider the diagram of categories of motives and realization functors described in the paper, with rational coefficients. A morphism of motives is test…

Corti–Hanamura's motivic decomposition conjecture. Every such morphism admits a motivic decomposition.

Let $M$ be a motive over a number field $F$ with Betti realization $M_B$, Mumford–Tate group $G_{MT}$, and adelic Galois representation … Assume that the Betti realization $M_B$ is…

Let $X$ be a smooth projective variety over a field $k$ of pure dimension $d$. A Chow–Künneth decomposition of $X$ consists of pairwise orthogonal projectors…

A deformation quantization is a quantized deformation of an algebraic or geometric structure, and the motivic Galois group is the symmetry group arising from the theory of motives.…

Let $M$ be a motive and let $\operatorname{MS}_{\sim}(M)$ be its motivic Serre group. Motivic Sato–Tate conjecture. For any prime number $l$, … The paper obtains this formulation a…

Let $M$ be a motive in the chosen motivic category, let $G_{l,K}^{\operatorname{alg}}$ be the algebraic group attached to its $l$-adic realization, and let…

Let $\Pi$ be an algebraic cuspidal automorphic representation of $\operatorname{GL}_n(\mathbb A)$, and let $\mathbb Q(\Pi)$ be its rationality field. Clozel's conjecture. There exi…

Let $A$ be an abelian variety over $\overline{\mathbb{Q}}$. Write $\langle\mathrm{H}^1_{\mathrm{dRB}}(A,\mathbb{Q})\rangle^{\otimes}$ for the tensor category generated by its first…

Fontaine–Perrin-Riou's conjecture. The category

Let $A$ be an abelian variety over $\mathbb{Q}^{\mathrm{al}}$ with good reduction at $w$, and let $\gamma$ be a Hodge class on $A$. Call $\gamma$ $w$-Lefschetz if its specializatio…

Let $F$ be a field. Beilinson–Deligne conjecture. For every field $F$, there should exist an abelian category of mixed Tate motives. This conjecture would provide the expected cate…

Let $X$ be a smooth projective curve of genus $g\geq 2$, let $L$ be a line bundle on $X$ of degree $d$, and let $\mathrm{M}_X(r,L)$ be the moduli space of semistable vector bundles…

Let $M$ be a motive in the relevant motivic category, with rational polarized Hodge realization $(V,\psi)$. Let $\operatorname{MMT}_{\sim}(M)$ be its motivic Mumford–Tate group. Se…

Let $k$ be a base field and let $\pi'$ be the quotient functor from homological motives to numerical motives in the diagram of motive categories. Homological-equals-numerical conje…