71 problems

Let $f$ be an analytic function, and consider its graph and pluripolar hull. The Levenberg–Martin–Poletsky conjecture. The pluripolar hull of the graph of $f$ is equal to the graph…

Blocki's integrability conjecture. In the flat case, every $m$-subharmonic function belongs to $L^p$ for every

Let $E$ be a non-symmetric convex body in ${\bf R}^d$, let $\lambda(x)$ denote the density of its equilibrium measure in the interior, and let $\widetilde{G}(E,x)$ be the convex hu…

Fekete-array Monge–Ampère conjecture. The measures $\mu_n$ converge weak- to $\mu_K$.

Bergman measure equilibrium conjecture. The normalized Bergman measure

Levenberg–Poletsky–Merker conjecture. The closure of the analytic extension of $A$ coincides with $A^*_D$.

Let $k$ and $\ell$ be positive integers. Vanishing-power conjecture. There exists a smooth nonconstant function $v$ on $\mathbb{P}^k$ such that … if and only if … The proposed thre…

Let $\Omega$ be a convex domain in $\mathbb C^n$, let $a_1,\ldots,a_N\in\Omega$ be poles, and let all poles have the same positive weight $\nu$. Define the pluricomplex Green funct…

Let $(X,\omega)$ be a compact Kähler manifold, let $\theta$ be a smooth closed real $(1,1)$-form representing a big class, and let $P_\theta[u]$ denote the singularity envelope of…

Let $(X,\omega)$ be a compact Kähler manifold, and let $\theta$ be a smooth closed real $(1,1)$-form representing a big class. For $u\in\operatorname{PSH}(X,\theta)$, let…

Conjectural support and laminarity description. Point (Q:support) holds: the support of $T$ is homeomorphic to the support of $dd^c f$. Moreover, the difference…

Let $d\geq 1$, let $\mathscr Q:\mathbb C^d\to\mathbb R$ be $C^2$ and strictly plurisubharmonic, and suppose that its droplet $S_{\mathscr Q}$ has smooth boundary. For each…

Let $d\geq 1$, let $\mathscr Q:\mathbb C^d\to\mathbb R$ be $C^2$ and strictly plurisubharmonic, and let $S_{\mathscr Q}$ be its droplet with smooth boundary. For…

Strict rescaling quotient conjecture. One has

Integral characterization conjecture. The range of $MA$ on these classes should yield an integral characterization of the measures that are Monge–Ampère measures of bounded plurisu…

Let $X\to\mathbb{D}^\times$ be a projective meromorphic degeneration with relatively ample meromorphic line bundle $L$, and let $\phi_t\in\operatorname{CPSH}(L_t)$ be the continuou…

Monge–Ampère origin conjecture. The family

Let $E\subset\partial\Omega$ be a b-pluripolar set, and define its propagating singularity set by … where … Define the b-pluripolar hull by … Since $\hat E\subset E^{\ast}$ and…

Let $\Omega$ be the domain under consideration, let $\varphi$ be the prescribed singularity type, and let $\mathcal{E}(\Omega)\subseteq\mathcal{D}(\Omega)$ denote the relevant clas…

Strong Bergman asymptotics. The measures satisfy

Let $u\in {\mathcal P}(B_1)$ be a plurisubharmonic function whose alternating part has finite frequencies, meaning that near the origin, in a proper complex Hopf-coordinate, it can…

Vanishing-Lelong-number and boundedness conjecture.

Canonical limit current conjecture. There is a quasi-plurisubharmonic function $\varphi_\infty$ such that

Let $(X,\omega)$ be a compact Kähler space, and let $\varphi$ be an $\omega$-psh function on $X$. Smooth regularization conjecture. The function $\varphi$ can be written as the poi…

Let $X$ be a smooth, irreducible projective Berkovich space over a non-Archimedean field $k$ of characteristic $0$, let $\rho$ be a PL metric on $K_X$, and let…