10 problems

Let $X$ be a compact complex manifold. If $X$ admits a Hermitian metric $h$ whose Chern holomorphic sectional curvature satisfies $H_h(v)\le 0$ for every nonzero $v\in T^{1,0}_xX$…

Let $Y$ be a projective Calabi–Yau variety with only ordinary double points, and let $X\to Y$ be a crepant resolution or let $Y$ occur as the central fiber of a smoothing. Then the…

Let $f\colon X\to \Delta$ be a proper Kähler family over a disk, with central fiber $X_0$ a simple-normal-crossing fiber, and suppose that the relative canonical bundle…

Let $X$ be a compact Kähler manifold of complex dimension $n$, and let $\alpha,\beta$ be Kähler classes on $X$. Set $\mu=\frac{n\,\alpha^{n-1}\cdot\beta}{\alpha^n}$. The Datar–Mete…

For fixed singular or big-cohomology data for which twisted Kähler–Einstein currents are defined, if $T_1$ and $T_2$ are two twisted Kähler–Einstein currents satisfying the same tw…

Let $X$ be a compact Kähler manifold, and let $\omega(t)$ solve the unnormalized Kähler–Ricci flow $\partial_t\omega(t)=-\operatorname{Ric}(\omega(t))$ on $[0,T)$, where $T<\infty$…

For every complete, noncompact Kähler manifold $(M^n,g)$ with positive holomorphic bisectional curvature, i.e. $R(X,\overline{X},Y,\overline{Y})>0$ for all points $p\in M$ and all…

For every integer $n\ge 1$, if a Reinhardt domain $\Omega\subset\mathbb{C}^n$ has smooth boundary and its Bergman metric is complete and Einstein, then $\Omega$ is biholomorphic to…

Let $(X,x)$ be an $n$-dimensional isolated log terminal singularity. If $\alpha_x^{\mathrm{an}}(X)$ and $\alpha_x^{\mathrm{alg}}(X)$ denote the two local alpha invariants introduce…

For every polarized smooth projective variety $(X,L)$ over $\mathbb{C}$, the polarization $L$ admits a constant-scalar-curvature Kähler metric in the class $c_1(L)$ if and only if…