7 problems

Let ${\mathcal M}_g$ be the moduli space of Riemann surfaces of genus $g\geq 2$, and let the Weil–Petersson metric be the Kähler metric on ${\mathcal M}_g$. Its Ricci curvature is…

A polarized Calabi–Yau manifold is a pair $(X,\omega)$ consisting of a compact algebraic manifold $X$ with zero first Chern class and a Kähler form $\omega\in H^2(X,\mathbb Z)$. Le…

Let $S$ be the surface under consideration, let $d(S)$ denote its complexity parameter, and let $\operatorname{Teich}(S)$ be its Teichmüller space equipped with the Weil–Petersson…

Let the moduli space parametrize canonically polarized projective varieties with klt singularities, and let its compactification be the compactified KSB moduli space. A Weil–Peters…

Weil–Petersson extension and completion conjecture. The metric $\omega_{WP}$ extends uniquely to a non-negative closed $(1,1)$-current on $O{S}$ with continuous local potentials, a…

Wang's conjecture. The converse is also true: if the Weil–Petersson distance between general fibers and the central fiber is finite, then the central fiber $X_0$ is a Calabi-Yau va…

Wang's conjecture. $X$ has finite Weil–Petersson distance if and only if