6 problems

Let $G$ be a connected semisimple group, let $P$ be a parabolic subgroup of $G$, and let $U_P$ be its unipotent radical. Let $B$ be a Borel subgroup and set … as a homogeneous vect…

Let $G$ be a reductive linear algebraic group and let $U$ be a maximal unipotent subgroup. The affine closure $\overline{T^*(G/U)}$ is the variety whose coordinate ring is the ring…

Irregular conical Calabi–Yau structure conjecture. For each $k\geqslant 5$, the horospherical cone constructed from $\mathcal{G}_k$ has an irregular conical Calabi–Yau structure.

The orbit classification conjecture. Two $G$-orbits $\mathcal{O}$ and $\mathcal{O}'$ lie in the same $\mathrm{Aut}(X)^0$-orbit if and only if

Let $X$ be a horospherical complexity-zero variety, let $\tau$ be a face of its cone $\sigma^\vee$, and let $O_\tau$ be the corresponding $G$-orbit. Let $p_1,\ldots,p_k$ be the pri…

Let $X$ be a Fano manifold with Picard number one. Odaka–Odaka conjecture. The manifold $X$ should be K-semistable. The conjecture is refuted: Fujita provided two counterexamples,…