3 problems

Bounded-sectional-curvature conjecture. The existence of such a Kähler metric imposes strong restrictions on the complex geometry of the pair $(X,D)$.

Log Donaldson–Tian–Yau conjecture. $((X,D);L)$ is log $K$-polystable with cone angle $2\pi\beta$ if and only if $X$ admits a cscK metric in $c_1(L)$ with cone singularities along…

The cone-angle existence conjecture. There is a cone-singularity solution $\omega_\beta$ to (27) for every parameter $\beta\in(0,R(X))$. If $R(X)<1$, there is no solution for param…