11 problems

Let $\Omega$ be an open Riemann surface admitting a nontrivial Green function $G_{\Omega}$. For $z_0\in\Omega$, let $c_{\beta}(z_0)$ denote its logarithmic capacity and let…

Let $\Omega$ be a bounded strongly pseudoconvex domain of $\mathbb{C}^n$. Its Bergman kernel has a logarithmic term in its boundary expansion. Ramadanov's conjecture. If the log te…

Bergman density rigidity conjecture. Then $\Phi$ is either holomorphic and

Let $M$ be a Kähler manifold with Kähler form $\omega$, and let $L$ be a holomorphic line bundle with Hermitian metric whose curvature is $-i\omega$. Let $K_k(x,y)$ be the Bergman…

Let $m>n$ be positive integers with $\gcd(m,n)=1$, and let $Q_{m,n}$ be the degree $2m-2n$ palindromic polynomial associated with this pair. Write…

Let $X$ be a compact complex manifold and let $L$ be a quasi-positive line bundle on $X$. For each $k\in\mathbb N$, write $\lambda_1(X,L^k)$ for the smallest nonzero eigenvalue of…

Multidimensional Suita conjecture. For every $z\in\Omega$,

Unnecessity conjecture. Assumption (A) is completely unnecessary for the results of this paper. This concerns the construction and analysis of the Bergman projection and Bergman ke…

Let $\beta(\Omega)$ denote the Bergman kernel integrability exponent of a domain $b\Omega\subset b\mathbb C^n$. Convex-domain conjecture. If $b\Omega\subset b\mathbb C^n$ is convex…

Monotonicity conjecture. If $Omega$ is pseudoconvex in $mathbb C^n$ then this function is non-decreasing on $(-infty,0]$. The conjecture is proved in dimension one; in arbitrary di…

Let $n$ be the complex dimension and consider the free energy expansion in powers of $k$, with terms indexed by $j$. A term is an exact one-cocycle if it is an exact one-cocycle in…