43 problems

An extremal transition is a process connecting topologically distinct Calabi–Yau manifolds through a singular Calabi–Yau variety and its smoothing or resolution. Candelas–de la Oss…

Loi–Salis–Zuddas conjecture. A Ricci-flat metric on $M$ such that $a_{n+1}=0$ should be flat.

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…

Let $X$ be a Hopf surface, and call a Hermitian metric $\omega$ Levi-Civita Ricci-flat when its Levi-Civita Ricci curvature satisfies $\mathfrak{Ric}(\omega)=0$. Existence conjectu…

Let $M$ be a manifold and let ${\cal R}$ be a set of Ricci-flat metrics on $M$. Denote by $\hat{T}_{\cal R}$ the map assigning to each metric in ${\cal R}$ its Lazzeri Jacobian. Lo…

Odaka's conjecture. The volume growth of geodesic balls in $X$ is bounded below by the dimension of the essential skeleton:

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 $[\alpha_i]\to[\alpha]$,

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…

Calabi–Yau conjecture. There exists a unique smooth function $\phi:X\to\mathbb{R}$, up to addition of a constant, such that the hermitian metric $g_{CY}$ associated with…

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…

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

Kronheimer–Ricci-flat coincidence conjecture. The Kronheimer Kähler metric and the Ricci-flat Kähler metric coincide on the exceptional divisor of $Y^\Gamma$.

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…

A Ricci-flat metric is a metric with vanishing Ricci curvature; the asymptotic model is the quotient cone $\mathbb{R}^4/\mathbb{Z}_2$, and the Eguchi–Hanson metric is a Ricci-flat…

Let $(M_o^4,\mathbf{g}_o)$ be a Ricci-flat orbifold, and suppose it can be desingularized in the Gromov–Hausdorff sense by Einstein metrics $(M^4,\mathbf{g}_n)_{n\in\mathbb{N}^*}$…

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 $X^{o}$ be a complex manifold carrying a complete Ricci-flat Kähler metric. A compactification should consist of a compact Kähler manifold $X$ and an anticanonical divisor $D$…