11 problems

Let $X_1$ and $X_2$ be generalized-dual smooth Gushel–Mukai varieties, and let $\mathcal A_{X_1}$ and $\mathcal A_{X_2}$ denote their K3 categories. Kuznetsov's conjecture. One sho…

Conjecture. The morphism $\phi_{\mathcal{L}}$ is generically finite of degree two over the Plücker variety.

Generic projected-push-forward conjecture. For every $[E]\in U$, the object $\operatorname{pr}_X(j_*E)$ is $\sigma_X$-semistable.

Generic hyperplane-section stability conjecture. Theorem holds generically for every GM fourfold and cubic fourfold.

Let $X$ and $X'$ be GM varieties of the same dimension. The Kuznetsov–Perry conjecture. If there exists an equivalence … then $X$ and $X'$ are birationally equivalent. This conject…

Let $X_1$ and $X_2$ be Gushel–Mukai varieties of the same dimension, and let $\mathcal{K}\!u(X_i)$ denote their Kuznetsov components. Birational categorical Torelli conjecture. If…

Kuznetsov's inverse duality conjecture. If $X$ and $X'$ are GM varieties of the same dimension and their Kuznetsov components are equivalent, then $X$ and $X'$ are birationally equ…

Explicit principally polarized abelian variety conjecture. The principally polarized abelian variety $({\mathbb J},\theta)$ is isomorphic to the explicit 10-dimensional principally…

Let $X$ be a GM fourfold. An associated K3 surface of $X$ is a K3 surface arising from a period point of $X$ lying on a hypersurface $\mathcal{D}_d$, with a nef algebraic class $h$…

Let $X$ be a Gushel–Mukai fivefold or sixfold, and let $X^\vee$ be a generalized dual GM variety with $\dim(X^\vee)\leq\dim(X)-2$. A generalized duality prediction. Then … Here…

Let $X$ be a rational Gushel–Mukai fourfold, and let $\mathcal{A}_X$ denote its GM category. A rationality conjecture. Then … for some K3 surface $S$. This is the GM analogue of th…