36 problems

Let $\Omega \subset \mathbb{C}^n$ be a smoothly bounded strictly pseudoconvex domain. Its Bergman kernel has an expansion … where $\psi$ is a defining function for $\Omega$, meanin…

For a symmetric convex body $K\subseteq\mathbb{R}^n$, let $T_K=\mathbb{R}^n+\sqrt{-1}K$ be its tube domain, and let $\mathcal{B}_{T_K}$ denote its Bergman kernel. Błocki's Bergman-…

Bergman-kernel growth conjecture. With this notation,

Blocki–Zwonek conjecture. The function

Let $(M,\omega)$ be a compact Kähler manifold of complex dimension $m$ such that $[\omega]=2\pi c_1(M)$, $\operatorname{Ric}(\omega)\ge R_0>0$, and…

Bergman measure equilibrium conjecture. The normalized Bergman measure

Let $M$ be a compact Kähler orbifold of dimension at least $2$ with only finite isolated singularities, and let $(L,h)\rightarrow M$ be a positive holomorphic orbifold line bundle.…

Cheng–Yau conjecture. The domain $\Omega$ is Bergman–Einstein if and only if it is biholomorphic to the unit ball of the same dimension.

Yau's conjecture. The Bergman metric of $\Omega$ is complete and Einstein if and only if $\Omega$ is biholomorphic to a bounded homogeneous domain.

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 $k\to\infty$,

For a convex body $K\subseteq\mathbb{R}^n$, let $T_K=\mathbb{R}^n+\sqrt{-1}K$ be its tube domain, let $\mathcal{B}_{T_K}$ denote its Bergman kernel, and let $\Delta_{n,0}$ be the s…

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

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…

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 $(L,h)\to M$ be a positive Hermitian holomorphic line bundle over a compact complex manifold, with Bergman kernels $K_k$ associated to the spaces of holomorphic sections…

Tian's conjecture. For such Fano manifolds, the Bergman kernel should have a uniform positive lower bound for some bounded multiple $k$.

Let $\Omega\subset\mathbb{C}^n$ be a bounded pseudoconvex domain with $\mathcal C^2$ smooth boundary. The $p$-Bergman kernel of $\Omega$ is called exhaustive when it has the exhaus…

Let $M$ be a quantizable compact Gevrey Kähler manifold, and let $s>1$ be its Gevrey order. In the Gevrey version of the Bergman-kernel expansion, the asymptotic summation is perfo…

Let $V$ be a neighborhood with a Kähler potential $\phi$ in Gevrey class $G^a(V)$, where $a>1$, and let $b_m(x,\bar{x})$ be the Bergman kernel coefficients. Write $v=(x,z)$, let…

Let $h$ be a Hermitian metric with $\partial^2\log h\in\mathcal C_M$, where $M$ satisfies the conditions of Theorem, and let $f(k)$ be given by. For points $z,w$ in the underlying…