36 problems

Let $(X,L)$ be a polarised smooth projective variety over $\a9c$ of complex dimension $n$. An extremal metric is a Calabi extremal Kähler metric on $(X,L)$, and $(X,L)$ is relative…

Let $(X,\omega_0)$ be a compact Kähler manifold, and let…

Relative K-stability conjecture. The existence of an extremal metric is conjectured to be equivalent to relative K-stability.

Toric Yau–Tian–Donaldson conjecture. For every $f\in\mathcal{T}(P,\Lambda)$,

Tian's conjecture. Every Kähler manifold degenerates in a suitable sense to a manifold with an extremal metric.

Existence conjecture. A solution exists if and only if $L_{A,\sigma} f\geq 0$ for every convex function $f$ having $L^1$ boundary values, with strict inequality whenever $f$ is not…

Donald's relative K-polystability conjecture. If $(Q_i,d\sigma_i)$ is relatively K-polystable, then there exists a complete extremal metric on $V_{Q_i}\setminus D_i$.

Relative K-polystability conjecture for pairs. $(V,D,L)$ is relatively K-polystable with a positive modulus of stability if and only if there exists a complete extremal metric on…

Modified K-stability conjecture. The modified K-stability condition should be equivalent to the existence of an extremal metric in the polarisation class.

Let $(M,[\omega])$ be a compact Kähler manifold with a fixed Kähler class, and let $Ca$ denote the Calabi energy. Suppose the class admits an extremal Kähler metric, whose Calabi e…

Uniqueness conjecture for the maximizing metric. The value $12\pi E(2\sqrt{2}/3)$ is uniquely achieved, up to a dilatation, by $g_0$.

Let $(X,L)$ be a polarised variety, let $T$ be a maximal torus of automorphisms of $X$, and let $F_\chi(\alpha)$ denote the modified Futaki invariant associated with the extremal v…

Let $X$ be a polarised smooth projective variety. K-stability conjecture. K-stability is equivalent to the existence of a Kähler–Einstein metric, or more generally, a Kähler metric…

Let the Kähler cone be the set of Kähler classes, and let the extremal Kähler cone be the subset consisting of classes represented by extremal Kähler metrics. Suppose the flow equa…

Let $(M,J)$ be a complex surface of positive first Chern class polarized by a Kähler class $\Omega$. The positive first Chern class convergence conjecture. There exists an initial…

Let $(X,\omega)$ be a compact Kähler manifold and let $D$ be a smooth divisor. Let $T$ be a maximal compact torus in the reduced automorphism group…

Calabi's conjecture. Extremal metrics exist precisely when these obstructions are overcome.

Let $M$ be a Kähler manifold with an extremal metric, and let $S\subset M$ be a submanifold invariant under a maximal torus in the Hamiltonian isometry group. Consider the blowup o…

Pingali's conjecture. There is a maximum value of $m$ beyond which no solution to this problem exists; equivalently, higher extremal Kähler metrics need not exist in the classes…

Let $S^n$ be the standard sphere with $n\geq2$, and let $M^q$ be a space form of dimension $q$ with curvature at most zero. A Riemannian metric on a compact manifold with positive…

Conjecture 1. The following facts hold true:

Uniform K-stability conjecture.

Let a moment polytope be split into subpolytopes, with zero boundary measure assigned to sides introduced by the splitting. Consider a subpolytope in such a splitting that is not s…

Let $P$ be a bounded convex polytope equipped with a boundary measure, and let the facets of $P$ correspond to divisors in a toric variety. Attach non-negative weights to the irred…

Asymptotic relative Chow polystability conjecture. The polarized manifold $(X,L)$ is asymptotically Chow polystable relative to $T$, and there exist integers $m_k\to\infty$ and met…