61 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…

Let $X$ be a complex manifold endowed with a Riemannian metric, and let $f\colon X\to X$ be an automorphism acting hyperbolically on an invariant compact subset $K\subset X$. For…

D'Angelo's degree conjecture. The degree bound can be taken to be

Globevnik–Stout conjecture. The result of Nagel and Rudin should generalize to the case of two arbitrary enclosed domains $D^{\prime} \subset D \subset \mathbb C^n$.

Let $M\subset \mathbb{C}^n$ be a pseudoconvex real hypersurface with $n\geq 3$. For $1\leq s\leq n-2$ and $p\in M$, write $t^{(s)}(M,p)$, $c^{(s)}(M,p)$ and $a^{(s)}(M,p)$ for the…

Fix an integer $n\geq 2$. For each $k\in\mathbb N^+$ such that $k(k+1)/2<n$, define … The gap phenomenon concerns rational proper maps from the unit ball $\mathbb B^n$ to…

Let $D^1$ denote the unit disc, let $k(z)=(1+alpha)(1-|z|^2)^\alpha$ with $\alpha>4$, and let $A^2(\mathbb{C},D^1,kd\mu)$ be the corresponding weighted Bergman space. The ARS chara…

Assign weights $\alpha_j=2m_j$, with $m_j\in{\Bbb N}$, to the variables $\tilde z=(z_2,\dots,z_n)$. For a multi-index $K=(k_2,\dots,k_n)$, set…

Conjecture. The domain $D$ is biholomorphically equivalent to some $\Omega_{\alpha}$, where $\alpha$ need not be an integer. This conjecture extends the classification suggested by…

Let $D\subset\mathbb{C}^n$ be a topologically contractible, strictly pseudoconvex domain, and let $\phi:D\rightarrow D$ be a holomorphic self-map. Its fixed-point set is the set of…

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…

Let $\mathbb{B}^2$ be the unit ball in $\mathbb{C}^2$. Let $X$ be a relatively closed complex manifold of dimension one in $\mathbb{B}^2$, smooth up to the boundary and transversal…

Let $D_1,D_2\subset\mathbb{C}^n$ be smoothly bounded domains, and let a biholomorphism between them induce a map between their boundaries. Fefferman's boundary regularity conjectur…

Let $D_j\subset\mathbb{C}^{n_j}$ be domains, let $A_j\subset D_j$ be locally pluriregular Borel sets for $j=1,\dots,N$, and let … Let $U\subset D_1\times\dots\times D_N$ be a conne…

Let $\mathcal D$ be a tube type symmetric irreducible Siegel domain of type II, with Hua system $\text{\bf H}$. An $\text{\bf H}$-harmonic function is a function annihilated by thi…

A symmetric irreducible Siegel domain of type II is a complex domain equipped with the Hua system $\text{\bf H}$, and an $\text{\bf H}$-harmonic function is a function annihilated…

Strong Ramadanov conjecture. If $\rho \in C^{n+2}(\overline{D})$, then $\partial D$ is locally spherical.

Let $M$ be a pseudoconvex hypersurface in $\mathbb{C}^n$, and let $q$-dimensional complex manifolds and $q$-dimensional subbundles of the $(1,0)$ tangent space of $M$ be considered…

Several-variable Brück conjecture. The function $\alpha(z)$ is constant, say $\alpha(z)=c$, and

Let $M\subset \mathbb{C}^n$ be a pseudoconvex real hypersurface with $n\geq 3$, let $1\leq s\leq n-2$, let $p\in M$, and let $B$ be a smooth complex vector subbundle of…

Let $M\subset \mathbb{C}^n$ be a pseudoconvex real hypersurface, let $p\in M$, and let $L$ be a complex vector field in the relevant $(1,0)$ tangent directions. Denote by…

Let $\phi$ be a rational inner function on the polydisk $\mathbb{D}^d$, let $\sigma_{\alpha}$ be its Clark measure associated to $\alpha$, and let $\mathcal{C}_{\alpha}(\phi)$ deno…

Let $\Omega$ be a bounded strongly pseudoconvex domain in $\mathbb{C}^n$, where $n\geq 2$, and let $F$ be a holomorphic self-map of $\Omega$ with an interior fixed point. Huang's c…

A strongly pseudoconvex CR hypersurface is a CR hypersurface with positive Levi form; it is obstruction flat when its obstruction function vanishes identically, and spherical when…

Let $\Omega\subset\mathbb{C}^n$ be a bounded pseudoconvex domain with smooth boundary of finite type, and let $f=1$ be the trivial multiplier generated by Kohn's procedures. The or…