17 problems

Rigidity conjecture. Then $\Phi$ is either a biholomorphism or an anti-biholomorphism, and moreover

Let $(M, g; L, h)$ be a polarized manifold, and let $S$ be the circle bundle of $(L^*, h^*)$. The manifold $(M,g)$ is locally homogeneous if its local biholomorphic isometries act…

Let $D$ be a planar regular region with finitely many boundary components that are analytic Jordan curves. Let $c_{\beta}(z)$ denote the logarithmic capacity and let $B(z)$ denote…

Let $n\geq 1$, let $\Gamma\subset\operatorname{Sp}_n(\mathbf Z)$ be a congruence subgroup of level $N$, and let $S^{(n)}_k(\Gamma)$ and $\mathbb B_{k,\Gamma}(Z)$ be as above. Level…

Weight-aspect Bergman-kernel size conjecture. As $k\to\infty$,

Saitoh's conjecture. If $D$ is not simply connected, then

Let $\Omega$ be an open Riemann surface admitting a nontrivial Green function $G_\Omega$, let $u$ be harmonic on $\Omega$, and set $\rho=e^{-2u}$. Define $B_{\Omega,\rho}$ from an…

Let $M$ be a Stein manifold of complex dimension $n$, let $S$ be a closed complex submanifold, and let $G(,S)$ be the upper envelope of the polar functions associated with $S$. Let…

Bergman-kernel growth conjecture. With this notation,

Let $n\in\mathbb{N}$, $c,d>0$, and $\eta\in(0,1)$. Let $(M,\omega,\mathcal{L},h)$ be a polarized Kähler manifold with $|\operatorname{Ric}|\leq 1$ and…

Let $G\subset \mathbb{C}^n$ with $n\geq 2$ be a smoothly bounded pseudoconvex domain, and let $K$ be the Bergman kernel of $G$. The minimal-degree conjecture. If $K$ is algebraic,…

Let $\{(M_i,g^i)\}$ be a sequence of $n$-dimensional Kähler–Einstein manifolds with constant scalar curvature $n$. For each $M_i$, let $\rho_l(M_i,g^i)$ denote the Bergman kernel o…

Tian's partial $C^{0}$-estimate. Given $t_0\in(0,1]$, there exist $k>0$ and $\delta>0$ such that $kL$ is very ample and, for every $\omega\in\mathcal{K}(X,t_0)$,

For $n,m\in\mathbb N$, let $r_{n,m}$ denote the largest real root of the polynomial associated with the Bergman kernel of the Fock-Bargmann-Hartogs domain $D_{n,m}$. Limit conjectu…

For $n\in\mathbb N$, let $m_0(n)$ be the least positive integer such that the associated Fock-Bargmann-Hartogs domain $D_{n,m}$ is a Lu Qi-Keng domain, and let $[x]$ denote the nea…

For $n\in\mathbb N$, let $m_0(n)$ be the least positive integer such that the associated Fock-Bargmann-Hartogs domain $D_{n,m}$ is a Lu Qi-Keng domain. Strict monotonicity conjectu…

Bergman-kernel asymptotics conjecture. The upper and lower local volumes coincide: