17 problems

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 Gromov–Hausdorff limit conjecture. Even if $[\alpha]$ is not semiample, the Gromov–Hausdorff limit $(Z,d)$ should exist, be homeomorphic to a normal compact complex analy…

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…

Aspherical limit conjecture. Then $X$ is a differentiable manifold in the Ricci-curvature setting, or a topological manifold in the sectional-curvature setting, and is aspherical.

Let $(M^n_j,g_j,v_j,p_j)$ be a sequence of pointed Riemannian manifolds with a uniform lower Ricci-curvature bound whose measured Gromov–Hausdorff limit is a metric-measure space…

The paper considers Kähler–Einstein manifolds and the projective space $\mathbb CP^N$ equipped with its canonical Fubini–Study metric $\omega_{FS}$. A standard holomorphic isometri…

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…

Let $\mathcal{X}\to\mathbb{D}$ be a maximally unipotent degeneration over a disc, with $X=\mathcal{X}_\epsilon$ for some small $\epsilon\ll1$, and let $g_t$ be a suitably normalize…

Codimension 4 conjecture. The singular set $\mathcal{S}$ should have codimension $4$.

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…

Let $X$ be a limit space with a lower Ricci-curvature bound, let $\gamma$ be a geodesic in $X$, and consider tangent cones obtained from the same sequence of rescalings along…

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