16 problems

A Fano variety is a variety with ample anticanonical divisor. A normal test configuration is a normal test configuration of the Fano variety, while a special test configuration is…

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…

Donaldson's equality conjecture. Equality should hold in this lower bound when the supremum is taken over the space of test configurations.

Let $X$ be a Fano variety, and let $v_t$ be a geodesic ray in the metric completion of the space of Kähler potentials on $X$. A test configuration for $X$ is a suitable algebraic d…

Let $(Y,H)$ be a polarised manifold, let $c_1(H)$ be its first Chern class, let $S(\omega)$ denote the scalar curvature of a Kähler metric $\omega\in c_1(H)$, and let $\hat S$ be t…

Let $(X,L)$ be a polarised variety. It is K-semistable if its test-configuration invariants satisfy the nonnegativity condition defining K-semistability, and a test configuration f…

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 ${\mathbb{Q}}$-Fano variety, and let $\tilde{\beta}$ be the valuation functional whose infimum agrees with the infimum of the non-Archimedean ${\bf H}^{\rm NA}$-functi…

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…

Yau–Tian–Donaldson conjecture. From a variational point of view, the boundary of the space $\mathcal{H}$ should contain, as a dense subset, potentials coming from certain algebraic…

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…

Tian's normal-central-fiber conjecture. For Fano manifolds, it suffices, even with Donaldson's definition of K-stability, to consider test configurations with normal central fibers…

Let $(X,L)$ be a polarized projective manifold, let $\mathcal{T}$ be a test configuration for $(X,L)$, and let $\varphi$ be a positive metric on $L$. Phong–Sturm showed that these…