11 problems

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 $X$ be a hyper-Kähler manifold of dimension $2n$, and let $\pi:X\to B$ be a Lagrangian fibration. Normal-base conjecture. If the base $B$ is normal, then $B\cong\mathbb P^n$. T…

Let $\pi:X\longrightarrow B$ be a nontrivial and non-birational fibration of a hyper-Kähler manifold $X$ of dimension $2n$. Base conjecture. The base $B$ is isomorphic to…

Let $X$ be a projective hyper-Kähler manifold and let $f:X\to B$ be a Lagrangian fibration. For an integer $i$, let $c_I\in\operatorname{CH}^{2i}(X)$ be a monomial of weighted degr…

Let $X$ be a projective hyper-Kähler manifold of dimension $2n$, let $q$ be its Beauville–Bogomolov form, let $L$ be a line bundle on $X$ with $q(L)=0$, let $i$ and $j$ be integers…

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

Base conjecture for Lagrangian fibrations. The base $B$ of any Lagrangian fibration is smooth, and in fact

Hyper-Kähler SYZ conjecture. The line bundle $L$ is semi-ample.

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

Let $X$ be a hyper-Kähler manifold with holomorphic and Kähler classes $[\Omega]\in H^2(X,\mathbb{C})$ and $[\omega]\in H^2(X,\mathbb{R})$. Suppose a MUM condition gives a vanishin…

Let $X$ be a general cubic fourfold, let $B=(\mathbb{P}^5)^\vee$ parametrize its hyperplane sections, and let $U=B\setminus X^\vee$ be the locus of smooth hyperplane sections. Writ…