5 problems

Let $Y$ be the variety in the preceding construction and let $X=F_r(Y)$, with $r\geq 2$, where $\Psi:X\dashrightarrow X$ is the Voisin rational map. For $z\in CH_0(X)_{hom}$, the s…

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 $X$ be a projective hyper-Kähler manifold, and let $C^*$ be the subalgebra of $CH^*(X)$ generated by divisors and Chern classes. Voisin's conjecture. The cycle class map is inj…

Numerical Riemann–Roch conjecture. The Huybrechts–Riemann–Roch polynomial of $X$ satisfies

Hyper-Kähler SYZ deformation conjecture. Any hyper-Kähler manifold can be deformed into a hyper-Kähler manifold with a Lagrangian fibration.