28 problems

Let $X$ be a smooth projective variety admitting a Chow–Künneth decomposition as in Murre's Conjecture (A), and let $F^\bullet$ be the induced filtration on its Chow groups. Murre'…

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 $X$ be a hyper-Kähler manifold, and let $h(X)$ denote its Chow motive with rational coefficients. Write $\mathcal{M}_{rat}(\mathbf{C})$ for the category of Chow motives over…

Let $X$ and $Y$ be hyper-Kähler varieties of dimension $2n$, let $\mathcal E$ be the twisted vector bundle used in the source, and let $\widetilde{\mathfrak F}$ and…

Let $X$ and $Y$ be hyper-Kähler varieties of dimension $2n$, and let $\mathfrak{F}$ and $\mathfrak{F}^{-1}$ be the Fourier transforms and their inverses considered in the source, w…

Let $X$ and $X'$ be derived equivalent hyper-Kähler varieties, and let $h_{\mathrm{hom}}(X)$ denote the homological motive of $X$, equipped with its cup-product. The homological mu…

Let $X$ and $X'$ be derived equivalent hyper-Kähler varieties. The Chow motive $h(X)$ is equipped with its cup-product structure, written $(h(X),\cup)$. Fu–Vial's multiplicative Or…

Let $X$ and $B$ be smooth connected varieties, and let $f:X\to B$ be a smooth projective morphism such that a very general fiber of $f$ admits motivic $0$-multiplicativity. Assume…

Let $X$ be a smooth connected projective variety of dimension $n$ with CK projectors $\{\pi_i^X\}_{0\leq i\leq 2n}$, and let $\Delta_I^X\in\operatorname{CH}^{mn}(X^{m+1})$ be the…

Let $X$ be a smooth connected projective variety of dimension $n$ equipped with a CK decomposition. For an integer $m\geq 2$, an $m$-fold $r$-multiplicative CK decomposition is one…

Let $M_{n,d}\to B$ be the Hitchin system, and let $h(M_{n,d})=\bigoplus_i h_i(M_{n,d})$ be its motivic decomposition in the category of relative Chow motives over $B$. A motivic de…

Let $\pi:\overline{J}_C\to B$ be as in the motivic Beauville decomposition conjecture, and let $\operatorname{Corr}^k_B(\overline{J}_C,\overline{J}_C)$ denote the degree-$k$ relati…

Let $\pi:\overline{J}_C\to B$ be the compactified Jacobian fibration associated with a flat family of integral projective locally planar curves of arithmetic genus $g$ over a nonsi…

Griffiths-group rank prediction. One expects

Let $(A,\sigma)$ be a central simple algebra with orthogonal involution and trivial discriminant, and let $G=\operatorname{PGO}^+(A,\sigma)$ be the connected component of its autom…

Isogenous hyper-Kähler varieties conjecture. If $X$ and $Y$ are isogenous hyper-Kähler varieties, then there exists an isomorphism of rational Chow motives

Let $k$ be an algebraically closed field of characteristic $p\geq 0$, let $X$ be a smooth projective surface over $k$, and let $t_{2}(X)$ denote its transcendental Chow motive with…

Dimension bound conjecture. Then

A birational Chow–Künneth decomposition of a smooth projective variety is co-multiplicative when the induced grading on its zero-cycles is compatible with the diagonal coproduct. C…

Shen–Vial conjecture. The motive of any hyper-Kähler variety admits a multiplicative Chow–Künneth decomposition.

Chin-Lung Wang's conjecture. If $X$ and $Y$ are K-equivalent smooth projective complex varieties, then they have isomorphic integral Chow motives.

Let $M$ be a smooth projective holomorphic symplectic variety endowed with a faithful symplectic action of a finite group $G$. Let $X:=M/G$ be the quotient, and suppose that $X$ ha…

Nilpotence conjecture. If $\Gamma\in \operatorname{End}_{{\mathcal M}_{k,R}}(M)$ is homologically trivial, then there exists an integer $n\gg0$ such that

Let $X$ be a holomorphic symplectic variety of dimension $2n$. Then the motivic splitting property. there is a canonical self-dual multiplicative Chow–Künneth decomposition … of Bl…

Let $M$ be a smooth projective holomorphic symplectic variety equipped with a faithful action of a finite group $G$ by symplectic automorphisms of $M$. If $Y$ is a symplectic resol…