12 problems

Form-type Calabi–Yau equation solvability conjecture. There exist a constant $b$ and a function $\varphi$ satisfying these equations. The equation is introduced as the form-type Ca…

Let $M$ be a compact complex manifold with vanishing Bott–Chern first Chern class $c_1^{BC}=0$ and a balanced metric $\omega_0$. A Hermitian metric is balanced if…

Let $X$ be a compact complex manifold of dimension $3$ with holomorphic volume form $\Omega$. Let $\omega$ be a Hermitian metric satisfying … and denote its balanced class by…

Let $X$ be a compact complex manifold of dimension $n$ with holomorphic volume form $\Omega$. Let $\omega$ be a Hermitian metric satisfying … and denote its balanced class by…

Alesker–Verbitsky conjecture. Every HKT $\mathrm{SL}(n,\mathbb{H})$-manifold admits a balanced HKT metric, which need not coincide with its initial HKT metric.

Let $\mathcal{E}$ be a Gieseker-stable vector bundle, and let $(P^E_m)_{m\gg 0}$ be its sequence of balanced metrics. Suppose the strong operator topology limit … exists. Balanced-…

Let $M$ be a compact complex manifold. A Hermitian metric is balanced if its fundamental form $\omega$ satisfies $d(\omega^{n-1})=0$, and it is pluriclosed if $dd^c\omega=0$. The b…

Prescribed-volume conjecture. On any compact complex manifold with a fixed balanced metric, one can always find a balanced metric in the same suitable cohomology class whose normal…

Let $X_1={\mathbb P}(T_X)$, let $f$ be the parabolic curve under consideration, and let $T[f']$ be the associated lifted Nevanlinna current. Let $\omega$ be a fixed reference metri…

The balanced Calabi–Yau metric conjecture. There is a balanced metric $\tilde{\omega}$ with

Form-type Calabi–Yau equation conjecture. For any $c>(\int_X\alpha^n)^{-1}$, equation (eon) has solutions in the balanced class $[\alpha^{n-1}]$.

Critical-point and convergence conjecture. If $\operatorname{Aut}_{0}(X)$ is trivial, then, for all sufficiently large $k$, $\mathcal{F}_{k}$ admits a critical point…