6 problems

Let $X$ be a weakly pseudoconvex manifold, let $\mathscr{J}$ be an ideal sheaf on $X$, and let $L\longrightarrow X$ be a holomorphic line bundle that is exhaustion singular-positiv…

Positivity conjecture. If $h$ is singular positive, then $h_\natural$ should also be singular positive. The construction already gives that $h_\natural$ is singular positive on…

Let $E$ be a holomorphic vector bundle with a singular Hermitian metric $h$, and let $c9$ be the Kähler metric from Proposition cref{smoothing of tr-c9 semi-posi for constant secti…

Coherence conjecture. If $h$ is Griffiths semi-positive, then $\mathcal{E}(h)$ is coherent.

Let $X$ be a compact Kähler manifold, let $H\to X$ be a holomorphic line bundle, and let $Z\subset X$ be a smooth hypersurface. Let $h$ be a singular Hermitian metric on $H$ satisf…

Bergman-kernel asymptotics conjecture. The upper and lower local volumes coincide: