9 problems

Fino–Vezzoni flow conjecture. The pluriclosed flow exists for all time and converges smoothly to a Kähler metric. In particular, $M$ is Kähler.

Let $X(K,U)$ be an Oeljeklaus–Toma manifold. For a pluriclosed metric, let $R^{H,\psi}$ denote the weighted scalar curvature determined by the associated generalized Ricci-flow data…

Let $(M^{2n},g_0,J)$ be a compact complex manifold with pluriclosed metric. Define … Here $\omega_0$ is the fundamental form of $g_0$, $c_1$ is the first Chern class in Aeppli coho…

Let $(M^4,J)$ be a compact Class $\mathrm{VII}_+$ surface, let $\omega_0$ be a pluriclosed metric on $M$, and let $p\in M$ be a generic point as described in the surrounding discus…

Let $(M^4,J)$ be an Inoue surface, and let $\omega_0$ be a pluriclosed metric on $M$. Inoue surface geometrization conjecture. The solution $\omega_t$ to pluriclosed flow with this…

Let $(M^4,J)$ be a properly elliptic surface with $\operatorname{kod}(M)=1$ and odd first Betti number, and let $\omega$ be a pluriclosed metric. Properly elliptic geometrization c…

Let $(M^4,J)$ be a compact Fano surface admitting a Kähler–Ricci soliton $(\omega_{KRS},X)$. Let $\omega_0\in c_1$ be a Kähler metric invariant under the one-parameter subgroup…

Let $(M,J)$ be a complex surface of general type, and let $\omega_0$ be a pluriclosed metric. General type convergence conjecture. The solution to pluriclosed flow with initial con…

Let $(M^{2n},g_0,J)$ be a compact complex manifold with pluriclosed metric, and let $\tau^*(\omega_0)$ be the formal cohomological existence time defined by the condition…