10 problems

Nagai's conjecture. The monodromy operators satisfy

Fix a prime $p$, let $K$ be a complete discretely valued field of mixed characteristic $(0,p)$ with perfect residue field, and let $X$ be a hyper-Kähler variety over $K$ of dimensi…

Fix $u\in B'$ and let $\mathcal{X}:\mathcal{M}'_u\times\mathbb{C}^{\times}\to\mathcal{T}'_u$ be the map satisfying the prescribed wall-crossing jumps, reality condition, and semifl…

Polynomial relation conjecture. Any Lefschetz cycle satisfies

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

SYZ conjecture. There exist a hyper-Kähler manifold $X'$ birational to $X$ and a Lagrangian fibration $X'\rightarrow B$ such that, writing $l'$ for the Lagrangian class of the fibr…

Let $X$ be a very general polarized hyper-Kähler variety with primitive curve class $\beta$. Let $M_{1,0}(X,\beta)$ denote the moduli space of stable maps from non-singular genus-o…

Let $\mathscr{X}$ be a maximally unipotent good minimal dlt degeneration of compact hyper-Kähler manifolds over $\mathbf{C}((t))$. Let $\mathcal{D}(\mathscr{X}_0)$ denote the dual…

Let $\mathcal{X}$ be a smooth algebraic orbifold with coarse moduli space $|\mathcal{X}|$, and suppose there is a crepant resolution $Y\to |\mathcal{X}|$ such that $Y$ is hyper-Käh…

Genericity conjecture. Every sufficiently general elliptic product hyper-Kähler fourfold satisfies these genericity conditions.