5 problems

Zero-angle limit conjecture. As $\beta$ tends to zero, $(S,C,\omega_\beta)$ converges in an appropriate sense to a generalized Kähler–Einstein metric $\omega_\infty$ on…

Asymptotic logarithmic Kähler–Einstein conjecture. The surface $S$ admits conical Kähler–Einstein metrics with angle $\beta$ along $C$ for all sufficiently small $\beta$ if and onl…

Tian's conjecture. Then $(M,J)$ admits a conical Kähler–Einstein metric with angle $2\pi\beta_j$ along $D_j$, unique in the negative case and unique in each Kähler class in the zer…

Donald's conjecture. For all $0<\beta<R(M)$ there exists a cone-singularity solution to this equation, and there is no solution for $R(M)<\beta<1$. Equivalently,

Let $\pi: ({\mathcal{X}}, {\mathcal{D}})\rightarrow \mathbb C$ be a special degeneration for $(X,D)$. Suppose the central fiber $({\mathcal{X}}_0,{\mathcal{D}}_0)$ admits a singula…