9 problems

Let $(M,L)$ be a polarized manifold and let $\hat K$ be a compact subgroup of $\operatorname{Aut}(M)$. Suppose that $L$ is $\hat K$-linearized. Let $H_{\hat K}^0(M,L^m)$ be the max…

Jiang's conjecture. $\alpha(X)<\frac{1}{n}$ if and only if $X\cong \mathbb{P}^n$.

Let $S$ be a smooth del Pezzo surface and let $A$ be an ample divisor on $S$ of one of the specified $\mathbb{P}^2$-, $\mathbb{F}_1$-, or $\mathbb{P}^1\times\mathbb{P}^1$-types. Le…

Let $X$ be a smooth variety and let $L$ be an ample Cartier divisor on $X$. For a positive integer $n$, define … with $\alpha_n(X,L)=+\infty$ if $|nL|$ is empty, and define … Assum…

Higher-eigenvalue control conjecture. For each natural number $k$ such that $2\leq k\leq n$, the $k$-th eigenvalue of $K_M^{-m}$ is controlled.

Ambro's conjecture. Fix $0<\epsilon<1$ and an integer $n>0$. Then there exists a number $\mu(n,\epsilon)>0$ depending only on $n$ and $\epsilon$ such that, for every $\epsilon$-klt…

Alpha-invariant conjecture. If $X_d$ does not contain star points, then

Higher-dimensional uniformization conjecture. (i) There exist no Kähler–Einstein edge metrics with small $\beta$ if $\kappa=\infty$. (ii) If $(K_X+D)^{\dim X}=0$, then there exist…

Tian's stabilization conjecture. When $m$ is sufficiently large, the numbers $\alpha_{m,1}(L)$ stabilize to $\alpha(M)$. This conjecture concerns the asymptotic relationship betwee…