40 problems

Chen's properness conjecture. A constant scalar curvature Kähler metric exists if and only if the $K$-energy is proper with respect to geodesic distance.

Mabuchi–Tian conjecture. The Mabuchi K-energy functional is proper if and only if there exists a CSCK metric in the class $[\omega]$.

Let $(X,\omega)$ be a compact Kähler manifold, let $(\mathcal E^1,d_1)$ be the metric completion of the space of Kähler potentials, and let the extended Mabuchi K-energy be a map…

Let $(M,\omega,\chi,J)$ be the initial twisted Kähler data, and suppose that the twisted Calabi flows have global existence. The transformed data along the flow may be written…

Let $M$ be a compact Kähler manifold, let $[\omega]$ be a Kähler class on $M$, and consider a global $C^{1,1}$ minimizer of the $K$-energy in $[\omega]$. Chen's conjecture. Every s…

Let $(X,L)$ be a polarized smooth projective variety with $\mathbb{T}$, $\mathfrak{t}_+$, $\mathcal{EH}$, $\mathcal{Y}_{\mathrm{sup}}$, and the three conditions of Proposition as i…

K-polystability conjecture. $X$ admits a constant scalar curvature Kähler metric if and only if $X$ is K-polystable.

Let $X$ be a compact Kähler manifold, let $\omega$ be a Kähler form, and let … be the space of Kähler potentials, where the Mabuchi functional $\mathcal{M}$ on $\mathcal{H}$ has cs…

Variational Yau–Tian–Donaldson conjecture. The Mabuchi functional is coercive if, and only if, $(X,\alpha)$ is uniformly K-stable.

Rubinstein's Ricci iteration convergence conjecture. The iteration exists for all $i\in\mathbb{N}$ and converges in an appropriate sense to a constant scalar curvature metric. The…

Let $X$ be a polarized variety with polarization $L_X$, and let $D$ be a divisor. A metric on $X\setminus D$ is said to be of Poincaré type when it has the prescribed Poincaré asym…

Let $X$ be a polarized variety with polarization $L_X$, and let $D$ be a divisor. Suppose that $D$ admits a constant-scalar-curvature Kähler metric with nonpositive scalar curvatur…

Moduli-space conjecture. There exists a Hausdorff complex space $\mathfrak{M}_{\gamma}(X,[\omega^{\mathbb{C}}])$ serving as a moduli space of Kähler manifolds with a complexified p…

Yau–Tian–Donaldson conjecture with B-field. The cscK equation with B-field is solvable in the complexified Kähler class $[d4d8^]$, with coupling constant $|d6fe|=\sin(d707)$, if an…

Let $(X,L)$ be a polarized projective manifold. A constant scalar curvature Kähler (cscK) metric in the Kähler class $c_1(L)$ is a metric whose scalar curvature is constant, and K-…

Let $(M,\omega_0)$ be a compact Kähler manifold with semi-ample canonical bundle $K_M$. Let $c_1(M)=[\operatorname{Ric}(\omega_0)]$ and consider any sequence of cscK metrics in the…

Let $(X,L)$ be a polarized manifold and let $\mathbb G$ be a reductive subgroup of ${\rm Aut}(X,L)_0$ containing a maximal torus of ${\rm Aut}(X,L)_0$. The three notions of…

Let $\Phi$ be a maximal geodesic ray, with non-Archimedean limit $\Phi_{\rm NA}$, and let ${\bf H}'^\infty(\Phi)$ and ${\bf H}^{\rm NA}(\Phi_{\rm NA})$ denote respectively the asym…

Let $(X,L)$ be a polarized manifold, let ${\bf H}'^\infty(\Phi)$ denote the asymptotic slope of the entropy along a maximal geodesic ray $\Phi$, and let ${\bf H}^{\rm NA}$ be the n…

Let $(X,H)\to(B,L)$ be a polarised fibration, with each fibre equipped with its fibrewise cscK geometry. An optimal symplectic connection is a fibrewise constant scalar curvature K…

Let a polarized projective manifold be equipped with its constant scalar curvature Kähler problem, and let K-stability denote the stability condition introduced for this problem. U…

Let $X$ be an $n$-dimensional Kähler manifold with semi-ample or nef canonical bundle $K_X$. Let $[\omega]$ be a Kähler class, let $\epsilon_j\to0$, and let…

Chen's invariant conjecture. The quantity $R([\omega_0],\chi)$ is an invariant of the Kähler class $[\chi]$.

The cscK cone metric and log-K-polystability conjecture. The following three conditions are equivalent: (i) $X$ admits a cscK cone metric along $\mathcal{D}$ in any ample class

Log Donaldson–Tian–Yau conjecture. $((X,D);L)$ is log $K$-polystable with cone angle $2\pi\beta$ if and only if $X$ admits a cscK metric in $c_1(L)$ with cone singularities along…