8 problems

Let $X$ be a complex hyper-Kähler variety, and let $A^*(X)=\bigoplus_i A^i(X)$ be its Chow ring with $\mathbb{Q}$-coefficients. A subvariety $Z\subset X$ is a constant cycle subvar…

Let $X=F_r(Y)$ with $r\geq2$, let $\Psi:X\dashrightarrow X$ be the Voisin map, and let $F\subset X$ be the closure of its fixed locus. Fixed-locus constant-cycle conjecture. The va…

Let $X$ be a hyper-Kähler manifold of dimension $2n$, and let $L,L'\subset X$ be $n$-dimensional constant-cycle subvarieties, meaning that all their points are rationally equivalen…

Let $X=F_r(Y)$ be as above, with Voisin rational self-map $\Psi:X\dashrightarrow X$, and let $F\subset X$ be the closure of its fixed locus. A closed subvariety is constant-cycle w…

Let $X$ be a K3 surface and ${\mathsf M}$ a moduli space of stable sheaves on $X$ of dimension $2n$. Let $V\subset {\mathsf M}$ be a constant-cycle subvariety, meaning that all poi…

Let $X$ be a K3 surface and ${\mathsf M}$ a moduli space of stable sheaves on $X$ with special zero-cycle $c_{\mathsf M}$. A subvariety $V\subset {\mathsf M}$ is Lagrangian constan…

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 hyper-Kähler variety of dimension $2n$. For each $i$, let $C_{2n-i}(X)\subset \operatorname{CH}^i(X)$ be the $\mathbb{Q}$-vector space generated by codimension-$i$ sub…