9 problems

Let $X$ be a Calabi–Yau threefold with $pi_1(X)=0$. For $c\in H^4(X,\mathbf{Z})$, call $c$ a DRY class if there exist an ample class $H\in H^2(X,\mathbf{R})$ and an integer…

Minimal degree conjecture. When $C$ is generic,

Douglas–Reinbacher–Yau strengthened Bogomolov conjecture. The Chern classes of $E$ satisfy

Existence conjecture. There exists a stable vector bundle $V$ on $D$ such that

For a smooth rational surface $X$, let $\mathfrak{M}_H(\mathbf{v})$ denote a moduli space of slope-stable bundles with respect to a polarization $H$. Costa–Miró-Roig's rationality…

Pullback-image conjecture. The image of this morphism consists precisely of $H$-stable vertical vector bundles on $X$ whose restriction to the general fiber is trivial.

Extremal metric decomposition conjecture. The projective bundle $(M,J)=P(E)$ admits an extremal Kähler metric in some Kähler class if and only if $E$ decomposes as a direct sum of…

Let $S$ be the completed and suitably localized Grothendieck ring of motives over $\mathbb{C}$, and let $\Gamma$ denote the set of characters $\alpha=(n,d)$. For each…

Virtual Hodge polynomial conjecture. One has