5 problems

Let $X$ be a globally $F$-split variety. Write $D_X^{(e)}$ for the sheaf of $O_X^{p^e}$-linear endomorphisms of $\mathcal O_X$, so that…

Let $X$ be a smooth projective variety with nontrivial first-order deformations, meaning $h^1(T_X)>0$, where $T_X$ is the tangent sheaf. The absolute Frobenius pushforward is denot…

Linear-obstruction conjecture. $(F_\ell)_*\mathcal O_X(D)$ is a classical generator of $D^b\operatorname{Coh}(X)$ for some $\ell\in\mathbb N$ if and only if

Multiplicity equality conjecture. Is it exactly equal, that is, does

Generalization conjecture. Do the results of Theorem hold for any semisimple adjoint group $G$?