20 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'…

Let $X$ and $Y$ be smooth projective varieties of dimension $n$, and let $Z\in CH^n(X\times Y)_{\mathbb Q}$ be a correspondence. Let $F^\bullet$ be the Bloch–Beilinson filtration a…

Let $X$ be a smooth projective variety whose algebra of holomorphic forms is generated in degree at most $2$, and let $x,y$ be closed points of $X$. Let $F_{\mathrm{BB}}^\bullet$ d…

Let $M/\mathbf{C}$ be a smooth projective variety, and let $E$ be a vector bundle on $M$ equipped with a flat connection. For each $i$, write $c_i(E)$ for its Chow-theoretic $i$-th…

Let $M$ be a projective hyper-Kähler manifold. A filtration on the Chow ring is called a Beauville–Voisin filtration when it is analogous to the filtration on the Chow group of zer…

Let $X$ be a smooth projective variety. Write $A^2_{AJ}(X)$ for the subgroup of codimension-two cycles Abel–Jacobi trivial, $A^2_{alg}(X)$ for the subgroup algebraically trivial, a…

Let $\mathcal X/\overline{\mathbb Q}$ be a quasi-projective variety. Let $CH^m(\mathcal X)_{\mathbb Q}$ denote codimension-$m$ cycles defined over $\overline{\mathbb Q}$ modulo rat…

Let $Y$ be a double EPW quartic, and let $S$ be the associated $K3$ surface. The construction as a moduli space gives a map … Let $F^\bullet$ denote the Bloch--Beilinson filtration…

Martin--Vial's conjecture.

Green–Griffiths implication. The induced second filtration step vanishes:

Let $X$ be a smooth projective variety whose algebra of holomorphic forms is generated in degrees at most $d$. A birational idempotent correspondence is an idempotent $\varpi$ acti…

Let $X$ be a smooth projective variety whose algebra of holomorphic forms is generated in degree at most $d$, and let $x_1,\ldots,x_m,y_1,\ldots,y_m$ be closed points of $X$. Let…

Let $X$ be a smooth projective variety and let $i$ be a positive integer. Assume that $\mathrm{H}^{*}_{\mathrm{tr}}(X)$ is generated by $\mathrm{H}^i_{\mathrm{tr}}(X)$. Bloch–Beili…

Rational-equivalence conjecture for points. For $x,y\in X$, one has

Generalized Bloch conjecture. The action of $\iota$ on $\operatorname{CH}_{0}(X)$ should be the identity on

Let $X$ be a smooth projective variety, let $A^i(X)=CH^i(X)_{\mathbb Q}$ denote codimension-$i$ cycles modulo rational equivalence, and let $F^iA^i(X)$ be the deepest level of the…

Let $X$ be a projective holomorphic symplectic manifold of dimension $2n$. For a point $x\in X$, let $O_x$ be its rational orbit, and for $k=1,\ldots,n$ let $S_k(X)$ be the subset…

Let $X$ be a hyper-Kähler variety of dimension $2n$. For an integer $i$, let $C_i(X)\subset \operatorname{CH}_i(X)=\operatorname{CH}^{2n-i}(X)$ be the $\mathbb{Q}$-vector space gen…

Let $X$ be a projective hyperkähler manifold, and let $z\in \operatorname{CH}(X)_{\mathbf Q}$ be a polynomial with $\mathbf Q$-coefficients in the first Chern classes of line bundl…

Let $Y$ be a smooth projective variety and let $F$ be a Bloch–Beilinson filtration with rational coefficients on its Chow groups. Assume that $F$ also induces a filtration on Chow…