7 problems

Let $(X,L)$ be a smooth polarized variety, and let $\mathbb{G}\subset \operatorname{Aut}^\circ(X,L)$ be a reductive complex Lie group. Let $(\mathcal{X},\mathcal{L})$ be a dominati…

Let $v_*$ be a minimizer of the valuation functional $\tilde{\beta}$ for a ${\mathbb{Q}}$-Fano variety $X$, and let $\mathcal{F}_{v_*}$ be the filtration induced by $v_*$. Minimizi…

Let $X$ be a Fano manifold with automorphism group $\operatorname{Aut}(X)$. A Loewy filtration is the canonical filtration of the section ring constructed from the action of…

Let $f(z_0,z_1,z_2,z_3)$ be an isolated hypersurface singularity. The extremal monomials are the monomials selected from the four sets $I$, $II$, $III$, and $IV$ displayed in the s…

Birational functoriality conjecture. There is a K-unstable pair $(X”,L”)$ destabilized by a test configuration $(\mathcal X”,\mathcal L”)$, a birational map…

Torus-relative stability conjecture. If

Let $M$ be a Fano manifold and consider weak conical Kähler–Ricci solitons on varying pairs $(W,\Delta)$, satisfying … A test-configuration for $(M,K_M^{-1})$ is a $\mathbf{C}^*$-e…