11 problems

Let ${X}_{mer}$ be an algebraic $n$-dimensional Calabi–Yau manifold over the field of meromorphic germs, with a non-vanishing volume form, and let $(X_t,\Omega_t)$ be the associate…

Let ${{\cal X}}_{mer}$ be a maximally degenerate Calabi–Yau family at $q=0$, let $X_q^{new}$ be the rescaled family, and let $n$ be the complex dimension. Maximal-degeneration limi…

Tosatti's diameter-collapse conjecture. For every sequence $[\alpha_i]\in\mathcal{C}$ with $[\alpha_i]\to[\alpha]$,

Folklore conjecture generalizing Kontsevich–Soibelman's conjecture. The collapsing boundary densely parametrizes singular tropical metric spaces homeomorphic to essential skeleta…

Let $M$ be a projective manifold with a semiample line bundle $L$ inducing a holomorphic map $f:M\to X$, and let $X^\circ\subset X$ be a Zariski open subset over which $f$ is a hol…

Let $(X_j,\omega_j,p_j)$ be a sequence of $n$-dimensional Kähler–Einstein manifolds with $\operatorname{Ric}(\omega_j)=c_j\omega_j$, $|c_j|\leq 1$, and…

Let ${\mathbb F}_2$ be the second Hirzebruch surface, equipped with the family of Kähler–Einstein edge metrics $\omega_{\beta_1,\beta_2}$ described in the source, with…

Let $(\mathcal X^*,\mathcal L^*)\to\Delta^*$ be a degeneration satisfying the paper's special maximal-degeneration expectation, and let $(S,M)$ be a fixed pointed polarized $m$-dim…

In a polarized flat punctured holomorphic family $\pi^*:(\mathcal X^*,\mathcal L^*)\to\Delta^*$ of $n$-dimensional polarized klt projective varieties with continuous Kähler metrics…

Sharp asymptotic bounds conjecture. For any $n,c,V$ and $\eta<1$, there is a number $k_{0}(n,c,V,\eta)$ such that if $k\geq k_{0}$ then for any $X$ in ${\mathcal K}(n,c,V)$ the dis…

Anderson–Cheeger–Colding–Tian conjecture. The metric space $(M,d)$ is itself a manifold.