33 problems

Let $\Omega\subset\mathbb{C}^n$ be the strictly pseudoconvex domain and let $u_0$ be the initial datum considered in Theorem. Replace the assumption … by the condition that $u_0$ h…

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 $(X,\omega)$ 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…

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 $D=B_1$ be the unit ball in ${\mathbb C}^n$, and let $u$ be a plurisubharmonic function on $B_1$ that is locally bounded outside the origin. The Monge–Ampère measure…

Let $f$ be an analytic function, and consider its graph and pluripolar hull. A finely analytic extension of $f$ may be possibly multi-valued; its maximal such extension is understo…

Let $(X,L)$ be a polarized variety. Every $d_2$-bounded weakly convergent sequence in $\mathcal{E}_{\mathrm{NA}}^2(X,L)$ should be $d_1$-convergent. This conjecture would provide t…

Let $(X,L)$ be a polarized variety. For any $\varphi\in\mathcal{E}_{\mathrm{NA}}^1(X,L)$, there exists a sequence…

Let $(X,L)$ be a polarized manifold, let $\mathcal{R}^1$ be the space of test curves, and let $\mathcal{E}^{1,\mathrm{NA}}(L)$ be the subspace consisting of those…

Let $X$ be a Fano manifold and let $L=-K_X$. Assume that $\delta<1$, and let $G$ be a pluricomplex Green function obtained as an $L^1$ limit of normalized potentials along Aubin's c…

Let $(X,L)$ be a polarized manifold, and let $\delta$ and $\delta_{\mathrm{pp}}$ denote the stability threshold and pluripotential-theoretic stability threshold, respectively. A plur…

Valuative invariance conjecture. If $\varphi$ and $\psi$ are valuatively equivalent, then their $n$-th Lelong numbers at $0$ are equal:

Demailly's conjecture. The $n$-th Lelong numbers converge:

Let $X^n$ be a Calabi–Yau manifold and let $\alpha$ be a closed real $(1,1)$-form such that $[\alpha]\in\partial\mathcal{C}_X$ is nef. Bounded-potential conjecture. There is a boun…

Let $\widetilde{\Omega} \subset \mathbb{C}^n \times \Delta$ be a pseudoconvex domain. For each $t \in \Delta$, define the fiber … Fix $w\in\mathbb{C}^n$, and assume that every…

Let $P\subset\mathbb{R}^{n}$ be a convex body containing $0$ in its interior, and let $R_P\in(0,1]$ be the invariant of $P$ defined in the source. For fixed $\beta<0$, let…

Let $dV$ be a probability measure on $\mathbb{C}^{n}$ satisfying the assumptions stated before the conjecture, and let $E(dV)$ be its non-weighted pluricomplex energy. Energy Green…