225 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 $X$ be a supersingular irreducible symplectic variety over a field of positive characteristic. The conjecture predicts that $X$ satisfies the supersingular Tate conjecture: the…

Hyper-Kähler motivic conjecture. If $\operatorname{D}^b(X)$ and $\operatorname{D}^b(Y)$ are exactly equivalent, then $\mathfrak{h}(X)$ and $\mathfrak{h}(Y)$ are isomorphic as Frobe…

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…

Murre's conjecture. There exists a Chow–Künneth decomposition $\{\pi^0,\ldots,\pi^{2d}\}$, and the induced filtration satisfies $F^0\operatorname{CH}^i(X)=\operatorname{CH}^i(X)$,…

Hodge standard conjecture. The quadratic form $q_Z$ is positive definite. This is Grothendieck’s Hodge-type standard conjecture; it is known in characteristic zero through Hodge–Ri…

Let $C$ be a curve over a number field $K$ with an action by a finite group $G$, let $\tau$ be an irreducible representation of $G$, and let $C^\tau$ be the corresponding motivic p…

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…

Kimura's conjecture. All smooth projective varieties have finite-dimensional motive.

Let $M=[u:\mathbb{Z}\to G^n]$ be the 1-motive with $u(1)=(R_1,\dots,R_n)$ defined by the points in the source. Let $k$ be the endomorphism field and let $\mathrm{LieBracket}$ and…

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 ${\mathcal M}_k$ be the category of motives over the field $k$. A motive $M\in{\mathcal M}_k$ is finite-dimensional in the sense of Kimura and O'Sullivan if it admits the corre…

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…

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…

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.

Full faithfulness conjecture. The motivic realization functor

Let $F$ be a number field and let $X/F$ be a curve. Let $A_1,A_2,A_3$ be three simple isogeny factors of $\operatorname{Jac}_X$. Assume that the $L$-function … has holomorphic cont…

Let $X$ be a variety over $\overline{\mathbb{Q}}$ with good reduction $X_p$ at a prime $p$. Let $Z_0$ be the $\mathbb{Q}$-vector space of algebraic classes on $X$, let $Z_p$ be the…

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…

Let $X_{{\mathbb Q}}$ be a smooth projective variety over ${{\mathbb Q}}$, and let $X_{{\mathbb R}}:=X_{{\mathbb Q}}\times_{{\mathbb Q}}{{\mathbb R}}$. Let…

Derived Hodge-invariance conjecture. Derived equivalent nice varieties defined over the complex numbers have the same Hodge numbers.

Let $K'$ be a finite extension of $K$, let $\varepsilon$ be an idempotent of $A[\mathcal{G}_{K'}]$, and let $\mathcal{R}$ be a $\mathbb{Z}_p$-order in…

Let $K$ be a number field and let $c>0$. Fix a type $\Phi=(w,(h^r)_{r\in\mathbb{Z}})$ of motives, with $h^r=\dim \operatorname{gr}_{dR}^r$, and choose $a,b$ such that $h^r=0$ unles…

Let $F$ be a number field, let $X$ be a smooth projective variety over $F$, and let $M=h^{i-1}(X)(n)$. Write $の\operatorname{Ext}^1_{\mathcal{O}_F}(\mathbb{1},M)$ for the subspace…