10 problems

Let $D$ be a divisor in a Fano manifold, and consider conical Kähler–Einstein metrics along $D$ with cone angles tending to zero. The complement of $D$ is equipped with a complete…

Let $(X,D)$ be a two-dimensional klt pair with $D=\sum_{j=1}^{n}(1-\beta_j)D_j$, and let $g$ be a weak conical Kähler–Einstein metric on $(X,D)$. Let $S=\operatorname{Sing}(X,D)$ b…

Let $(X,D)$ be a pair with curve singularity locus $C$, let $p\in C$, and let $g$ be a weak conical Kähler–Einstein metric on $(X,D)$ with prescribed cone angle parameters. The met…

Tian–Yau convergence conjecture. As $\beta\to0$, these Kähler–Einstein cone metrics converge to the complete Calabi–Yau metrics on $X\setminus D$ constructed by Tian and Yau.

Let $X$ be a smooth compact Kähler surface, let $D=\sum_i \beta_iD_i$ be an $\mathbb{R}$-divisor with $0<\beta_i<1$, where the $D_i$ are smooth irreducible divisors and $D$ has sim…

Let $X$ be a compact complex manifold, let $L\to X$ be an ample line bundle, let $D$ be a divisor on $X$, and let $\beta$ determine a cone angle $2\pi\beta$ along $D$. Write…

Let $C_{\epsilon} \subset \mathbb{CP}^2$ be the family of cubic curves defined by … where $\epsilon_j\to 0$ through positive values, and let $g_{\epsilon_j}$ be conical Kähler–Eins…

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…

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…