26 problems

Let $X$ be a compact Kähler Calabi-Yau manifold, and let $\alpha\in H^{1,1}_{\mathbb{R}}(X)$ be a class that is nef and big but not Kähler. A smooth $(1,1)$-form representing…

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

Tosatti's minimal-singularities conjecture. If $\eta\in\mathcal{T}_{[\alpha]}$ and

Let $M_\zeta$ be the Calabi–Yau ALE resolution in complex dimension $3$, let $g_\zeta$ be its quotient Kähler metric, and let $(\mathcal{H}^{0,1}_\alpha)^\Gamma$ denote the invaria…

Existence conjecture. The conclusions of Theorem 1 hold for $M$, depending on the sign of $c_1(\det E\otimes K_B^{\vee})$. If $d_\ell=0$ and $\det E\otimes K_B^{\vee}\to B$ is triv…

Fano-cone Ricci-flatness conjecture. On a Fano cone $X$, every Kähler-Ricci shrinker is a Ricci-flat Kähler cone metric.

Let $g$ be a complete asymptotically conical Ricci-flat Kähler metric on $\mathbb{C}^n$, asymptotic to a Kähler cone $\mathcal{C}$ whose link is smooth. The AC metric rigidity conj…

Let $X$ be a compact complex manifold of dimension $n$ with holomorphic volume form $\Omega$. Let $\omega$ be a Hermitian metric satisfying … and denote its balanced class by…

Let $Y_{[3]}^\Gamma$ be a resolved threefold with exceptional divisor $\mathcal{ED}$, and let the Ricci-flat metric and the Kähler metric on $\mathcal{ED}$ be those considered in t…

Kronheimer–Ricci-flat coincidence conjecture. The two metrics are different on $Y^\Gamma_{[3]}$, but they coincide on the exceptional divisor $\mathcal{ED}$.

Let $(M_n,\mathbf{g}_n)_n$ be a sequence of compact smooth Ricci-flat metrics converging in the Gromov–Hausdorff sense to a Ricci-flat orbifold $(M_o,\mathbf{g}_o)$, while Eguchi–H…

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 $((X,D),L)$ be a polarized dlt log Calabi–Yau pair with $\lfloor D\rfloor=\sum_iD_i$, $K_X+D\equiv0$, and $L$ ample on $X^{o}:=X\setminus\operatorname{Supp}(\lfloor D\rfloor)$.…

Let $X^{o}$ be a non-proper separated complex variety with at most Kawamata log terminal singularities, carrying a complete Ricci-flat weak Kähler metric $g$ and a base point $p$.…

Let $X$ be a $K3$ surface, let $\alpha$ be a closed real $(1,1)$-form with $0\neq[\alpha]\in\partial\mathcal{C}_X$ and $\int_X\alpha^2=0$, and let $\omega$ be a Kähler metric on…

Let $\mathfrak{X}$ be a Calabi–Yau family as in Setup II that does not satisfy the equivalent conditions in Theorem. Logarithmic diameter growth conjecture. There exist constants…

Let $X$ be a $K3$ surface and let $[\alpha]\in\partial\mathcal{C}_X$ be as in Setup I.B.2. Point-limit conjecture. The Ricci-flat manifolds $(X,\omega_t)$ converge to a point in th…

Compactification conjecture. The following hold:

In Setup I.B.1, let $f:X\to Y$ be the fiber space and let $S$ be the singular-fiber locus. Smooth convergence conjecture. The Ricci-flat Kähler metrics satisfy … in…

In Setup I.A, let $(X,\omega_t)$ be the corresponding family of Ricci-flat manifolds and let $Z$ denote its Gromov–Hausdorff limit. Gromov–Hausdorff limit conjecture. The limit $Z$…

Let $(M,g)$ be a Kähler metric on an $n$-dimensional complex manifold, and let $a_j$ denote the coefficients associated to $g$ in the Tian–Yau–Zelditch expansion. The Ricci-flat TY…

A Ricci-flat metric is a Kähler metric whose Ricci curvature vanishes, and a metric is projectively induced if it admits a Kähler immersion into some projective space…

Ricci-flat convergence conjecture. As $t\to\infty$, the flow converges in the Gromov-Hausdorff sense to the unique Ricci-flat Kähler metric in $[H]$. Smooth convergence is known wh…

Let $\mathop{\rm Al}(M)$ denote the relevant space of metrics, and let $c_1$ and $n$ be the diameter-bound parameters for a fixed Ricci-flat Kähler metric $\omega_0$. A local minim…