17 problems

Let $X$ be a unibranch variety, let $L$ be a line bundle on $X$, and consider the space of non-Archimedean plurisubharmonic metrics $\mathrm{PSH}(L^{\mathrm{an}})$. An increasing n…

Let $X$ be the compact complex manifold, let $\theta$ be the closed $(1,1)$-form, and let $\theta$-plurisubharmonic function $\varphi\in\operatorname{PSH}(X,\theta)$ be as above, w…

Khabibullin's conjecture. The stated integral implication holds for every such $S$, every $\rho>1$, and every integer $n\geq 2$. This is an extremal integral inequality arising in…

Full-mass conjecture. $e_n(\varphi)=1$ if and only if $u\in\mathcal{E}(\omega_0)$.

Let $D\subset\mathbb{C}^n$ be a domain, and let $\varphi$ be a strongly upper semi-continuous function on $D$, meaning that for every $x\in D$ and every subset $S\subset D$ of meas…

Let $\Omega$ be the domain under consideration, let $\mathcal{E}$ be the Cegrell class of negative plurisubharmonic functions on $\Omega$, and let $g_u$ denote the Green–Poisson re…

Let $X$ be a compact $n$-dimensional complex manifold, and let $\{\beta\}\in H^{1,1}(X,\mathbb{R})$ be nef with $\int_X\beta^n>0$. Fix points $x_1,\ldots,x_N\in X$ and positive rea…

Let $\varphi$ and $\psi$ be plurisubharmonic germs with isolated singularities at $0\in\mathbb{C}^n$. For every divisorial valuation $v$ centered at $0$, write $v(\varphi)$ for its…

Let $\varphi$ be plurisubharmonic in a bounded pseudoconvex neighborhood of $0\in\mathbb{C}^n$, locally bounded away from $0$. For $m\in\mathbb{Z}_{>0}$, let $\varphi_m$ be the ana…

Let $f:X\to Z$ be as in the setting, with fibers $X_t$, forms $\theta_t$ and $\omega_t$, and let $V$ denote the common volume appearing in the fiberwise averages. Write…

Schwarz symmetrization conjecture. One has

Modification characterization conjecture.

Demailly–Kollár jumping-number openness conjecture. If $c_x^{\mathfrak q}(\varphi)<\infty$, then $|\mathfrak q|^2\exp(-2c_x^{\mathfrak q}(\varphi)\varphi)$ is not locally integrabl…

Let $\mathcal{B}\subset\mathbb{C}^{n}$ be the ball, and consider the equation referred to in the source as equation, with parameter $\gamma$. Let $u_{\gamma}$ denote the radial sol…

Strong openness conjecture. The following equality of sheaves holds:

Modified openness conjecture. Then

Capacity lower-bound conjecture. There exists a constant $C_{a,n}>0$ such that, for every compact non-pluripolar set $E\bigsubset A$,