36 problems

Let $\phi$ be a germ of a plurisubharmonic function at a point $x$ on a complex manifold, and let $c_x(\phi)$ denote its complex singularity exponent. Demailly–Kollár openness conj…

Demailly's strong openness conjecture. If

Zero mass conjecture. The residual Monge–Ampère mass of $u$ at the origin must vanish whenever $u$ has an isolated singularity and zero Lelong number there. This conjecture concern…

Let $(X^n,\omega)$ be a compact Hermitian manifold and let $\alpha$ be a closed real $(1,1)$-form whose Bott–Chern class $[\alpha]$ is nef. Fix points $x_1,\ldots,x_m\in X$ and pos…

Let $D\subset\mathbb{C}^n$ be a pseudoconvex domain containing the origin $o$, and let $\psi$ be a plurisubharmonic function on $D$. Write $c_o(\psi)$ for the complex singularity e…

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 $(X^{4n},I,J,K)$ be a hypercomplex manifold. A continuous function $h:X\to\mathbb{R}$ is quaternionic plurisubharmonic if $\partial_J\partial h$ is a non-negative generalized s…

Conjecture on the slicing constant. The inequality above is true with the constant $c_n'=n$.

Pali's potential conjecture. There exists a unique function $f\in Psh(X,J)\cap L^1_{loc}(X)$ whose associated distribution coincides with the distribution $u$.

Pali's conjecture. The function $u$ is plurisubharmonic if and only if the $(1,1)$-current

A plurisubharmonic function with sub-logarithmic growth on a complete noncompact Kähler manifold is understood as a plurisubharmonic function satisfying the stated sub-logarithmic…

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…

Zero mass conjecture. The zero mass conjecture claims that $\nu_n(u)=0$ when $\nu_1(u)=0$.

Let $\Omega$ be a smoothly bounded strictly pseudoconvex Reinhardt domain in $\mathbb C^n$, let $M=\{z\in\mathbb C^n\mid z_1\cdots z_n=0\}$, and let $\phi$ be a weak geodesic solvi…

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…

Core obstruction conjecture. For certain classes of domains in $\mathbb{C}^d$, the existence of a nontrivial core is the only obstruction to bounded representations.

Let $(X_k,\omega_k)$ be fibres in a family and let $(u_k)_k$ be a sequence of fibrewise plurisubharmonic functions converging to $u_0$ in the family sense. Continuity conjecture. T…

Let $\Omega$ be a domain in $\mathbb C^n$, let $E\subseteq\Omega$ be a closed subset with locally finite $(2n-1)$-dimensional Hausdorff measure, and let $u$ be a subharmonic functi…

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 $f:X\to Z$ 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.

General boundary-kernel conjecture. The inclusion

Ohsawa's matrix strong openness conjecture. If

Let $\varphi_1\leq\varphi_2\leq\cdots$ be plurisubharmonic functions on $\mathbb C^n$ which are locally uniformly bounded above, and let … If $f$ is an entire function such that ……