127 problems

Donald's conjecture. One has

Let $D$ be a strongly pseudoconvex domain with smooth boundary. Let the Cheng–Yau metric be the canonical complete Kähler–Einstein metric on $D$, and let the Bergman metric be the…

Donaldson–Sun's metric-independence conjecture. Both $W$ and $\mathcal C$ depend only on the germ of the singularity $(Z,p)$, rather than on the metric.

Kähler–Einstein conjecture. If $g$ is Kähler–Einstein, then $(U,g)$ is biholomorphically isometric to a bounded symmetric domain.

Tian's conjecture. K-stability implies the existence of a Kähler–Einstein metric on $M$.

Yau's conjecture. The existence of a Kähler–Einstein metric is related to certain algebraic notions of stability.

Donald's conjecture. There does not exist a conical Kähler–Einstein metric solving

ODP conjecture.

Let $(X,D)$ be a strongly asymptotically log Fano manifold with $D$ smooth and irreducible. Set … and suppose that there exist Kähler–Einstein edge metrics $\omega_\beta$ on…

Tangent-cone product conjecture. These spaces converge in pointed Gromov–Hausdorff topology to a product of copies of $\mathbb{R}$ and $\mathbb{C}$, compact Calabi–Yau varieties, a…

Picard number conjecture. Then $S$ has Picard number one.

Let $D$ be a divisor in a Fano manifold, and consider conical Kähler–Einstein metrics along $D$ with cone angles tending to zero. The complement of $D$ is equipped with a complete…

Let $\Omega \subset \operatorname{\mathbb{C}}^d$ with $d \geq 2$ be a bounded pseudoconvex domain whose boundary is $C^2$. Suppose that $\Omega$ admits an open quotient of finite v…

Let $(X_i,\omega_i)$ be a sequence of smooth Kähler–Einstein Del Pezzo surfaces converging in the Gromov–Hausdorff sense to a limiting metric space $X$. Tian's conjecture. The limi…

Let $(M,\omega_{KE})$ be a Kähler–Einstein manifold with positive first Chern class and $\operatorname{Ric}(\omega_{KE})=\omega_{KE}$, and let $P(M,\omega_{KE})$ be the space of no…

Tian's conjecture. Hilbert–Mumford stability of $X_f$ is sufficient for the existence of a Kähler–Einstein metric on $X_f$.

Tian's conjecture. Nonnegativity of the real part of the Futaki invariant for every special degeneration of $X$, with equality only for trivial degenerations, is sufficient for the…

Let $V$ be a Fano orbifold, meaning a compact Kähler orbifold whose Ricci-form cohomology class is represented by a positive $(1,1)$-form. In the quasi-regular setting, $V$ is the…

Let $\Sigma$ be the closed surface under consideration, let $\mathcal L^*(g)$ denote the relevant set of surface subgroups of genus $g$, and let $A(h)(\Pi)$ be the associated area…

Higher-dimensional coupled Kähler–Einstein conjecture. For every integer $n\geq 4$, there exists a toric Fano manifold of dimension $n$ that admits a two-coupled Kähler–Einstein me…

Let $D\subset\mathbb{C}^n$ be a bounded pseudoconvex domain. Its Bergman metric is the invariant Kähler metric associated with the Bergman kernel, and a domain is bounded homogeneo…

Let $M$ be a Fano manifold, meaning a compact complex manifold with positive first Chern class. A Fano manifold is K-stable when it satisfies the algebro-geometric K-stability cond…

Let $X$ be the Fano variety considered in the paper, let $e_0$ be the maximizer of the specific entropy $S(e)$, let $e_c$ be the threshold beyond which strict concavity may fail, a…

Hashimoto's conjecture. The existence of coupled Kähler–Einstein metrics should be equivalent to an equivariant version of uniform coupled Ding stability. The conjectured stability…

Hultgren–Witt Nyström's conjecture. The existence of coupled Kähler–Einstein metrics is equivalent to K-polystability or Ding polystability of…