12 problems

Let $(\mathcal{X},\omega)$ be a closed Kähler manifold, and let $[\omega]$ be a Kähler class. A Kähler metric in $[\omega]$ is constant scalar curvature (cscK) when its scalar curv…

A canonical metric is a metric associated naturally with a geometric structure, while stability refers to a suitable notion of stability in geometric invariant theory. Yau's conjec…

Let $L$ be an ample line bundle on a projective variety, and let $c_{1}(L)$ denote its first Chern class. A projective variety with polarization $L$ is K-stable in the sense of alg…

Let $(X,\alpha)$ be a Kähler manifold with Kähler class $\alpha$. Z-stability conjecture. The manifold $(X,\alpha)$ admits a $Z$-critical Kähler metric in $\alpha$ if and only if…

Let $X$ be a polarized manifold with discrete automorphism group. A cscK metric is a Kähler metric on $X$ with constant scalar curvature, and uniform K-stability is the strengthene…

Let $(X,L)$ be a polarized variety. Denote by $\mathcal{H}^{\mathbb{R}}_{\mathrm{NA}}(X,L)$ the space of non-archimedean metrics and by $\mathcal{H}(X,L)$ the space of Kähler metri…

Let $\pi:X\to B$ be a holomorphic submersion, let $\alpha$ be a relatively Kähler class, and let $\omega_X\in\alpha$ be a relatively Kähler metric whose restriction to each fibre i…

In a fixed Kähler class, consider extremal Kähler metrics and the action of biholomorphisms preserving that class. Mabuchi's uniqueness conjecture. Extremal Kähler metrics are uniq…

A Kähler-Ricci soliton on a Fano manifold $M$ is a pair $(\omega,X)$ consisting of a Kähler form $\omega\in 2\pi c_1(M)$ and a holomorphic vector field $X$ satisfying … The modifie…

Let $(M_\Omega(\mu),g(\mu))$ be a Cartan–Hartogs domain, and let $a_2$ denote the coefficient of the Engliš expansion of the Rawnsley $\varepsilon$-function associated to $g(\mu)$.…

Let … be the punctured disk, and let $h_{\mathrm{can},\Delta^{*}}$ be the canonical singular hermitian metric on $K_{\Delta^{*}}$ constructed from pluricanonical forms. Punctured-d…

Let $f:X\to S$ be a surjective proper Kähler morphism with connected fibers between connected complex manifolds, let $S^{\circ}$ be the complement of the discriminant locus, and le…