29 problems

Let $(X,L)$ be a smooth polarized manifold, where $c_1(L)$ denotes the first Chern class of $L$, and let K-polystability mean that every test configuration has nonnegative Donaldso…

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…

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…

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

Chen's convexity conjecture. The function

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.

Donaldson's conjecture. The following statements are equivalent: (1) there is no constant scalar curvature Kähler metric in $\mathcal H$; (2) there is a potential…

The Yau–Tian–Donaldson conjecture for maps. The class $c_1(L)$ admits a $p^*\alpha$-twisted cscK metric if and only if the map $p:(X,L)\to(Y,T)$ is uniformly K-stable. This is the…

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 $(X,\omega)$ be a Kähler manifold, let $\mathcal H$ be the normalized space of Kähler potentials cohomologous to $\omega$, let $G=\operatorname{Aut}_0(X,J)$ act on $\mathcal H$…

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…

Donaldson–Tian–Yau conjecture. $(X,L)$ admits a cscK metric in $c_1(L)$ if and only if it is $K$-polystable.

Let $\mu_k$ be the closed positive twisting data used to define the functional $J_{\mu_k}$, and let … be the corresponding twisted K-energy functional. A twisted cscK metric is a s…

Let $(\Omega,\chi,t)$ be twisted Kähler data with $t<1$, and let the corresponding twisted continuity-path equation be the equation whose solvability at parameter $t$ defines a twi…

Let $(M,[\omega])$ be a polarized Kähler manifold, and let $E$ denote the K-energy functional. The maximal invariant, totally geodesic subspace induced by the automorphism group is…

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 $\varphi$ be a smooth initial Kähler potential, and let the twisted Calabi flow be the evolution equation introduced in the paper for the corresponding Kähler metrics and twist…

Let $\chi>0$ be a closed positive $(1,1)$-form, and define $R(\chi)$ to be the supremum of times $t\in[0,1]$ for which the twisted continuity-path equation is solvable. Chen's conj…

Let $M$ be a compact Kähler manifold with Kähler class $[\omega]$, let $\chi$ be a closed positive $(1,1)$-form, and let $t\in[0,1]$. A metric solving the corresponding twisted con…

Let $(X,\omega)$ be a Kähler manifold, and let the Mabuchi functional be the functional on the space of Kähler metrics in the fixed Kähler class represented by $\omega$. Tian's con…

Mabuchi–Tian three-way conjecture. The following are equivalent: