6 problems

Mabuchi properness conjecture. The existence of cscK metrics is equivalent to the properness of the Mabuchi functional.

Let $\Phi:X\rightarrow\Sigma_g$ be a minimal elliptic surface, and let $H(X,\Phi)$ be the group of pairs of automorphisms preserving $\Phi$: … A semi-flat cscK current has fiber vo…

Existence and uniqueness conjecture. There is a semi-flat cscK current on $X$ whose regular fibers have volume $1$ and whose scalar curvature is $-3$; moreover, if $\omega_1$ and…

Let $X$ be a nonsingular algebraic variety with semi-ample canonical line bundle $K_X$, canonical model $X_{can}$ of dimension $\kappa$ satisfying $0<\kappa<\dim X=n$, and let…

Chen's conjecture. The quantity $R([\omega_0],\chi)$ is an invariant of the Kähler class $[\chi]$. This quantity measures how far the continuity path proceeds toward the constant s…

Let $(M,[\omega],J)$ and $(M,[\omega],J')$ be polarized complex manifolds, with the latter admitting a cscK metric. Let $\chi>0$ be a form in the class $[\omega]$, and let $s$ deno…