21 problems

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…

Streets–Tian conjecture. In analogy with the Kähler–Ricci flow, the solution of the pluriclosed flow with initial metric $g_0$ exists on the maximal interval $[0,\tau(g_0))$.

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 $M=X(K,U)$ be an Oeljeklaus–Toma manifold of type $(s,s)$, let $\omega_0$ be a pluriclosed metric, and let $d(K,U)$ denote the canonical flat metric on the torus $\mathbb T^s$…

Pluriclosed-flow nefness conjecture. The pluriclosed flow on $(M,J)$ exists for all time if and only if $-c_1^A(M,J)$ is nef.

Let $(M,J,g)$ be a compact Hermitian manifold, with Bismut connection $\nabla^B$ and curvature tensor $R^B$. The connection is Kähler-like if $\nabla^B$ satisfies the first Bianchi…

Let $M$ be a minimal class VII surface with $b_2(M)>0$, and consider the pluriclosed flow, which evolves a starting pluriclosed metric by … Here a rational curve means a curve para…

Pluriclosed flow regularity conjecture. The flow should exist until either the volume collapses or it becomes singular on an effective divisor with negative self-intersection.

Attractiveness conjecture. SKT metrics with vanishing Bismut Ricci form should have a similar behaviour to Bismut flat metrics under the pluriclosed flow: they should be attractive…

Commuting generalized Kähler-flow regularity conjecture. The solution to generalized Kähler-Ricci flow with initial condition $g_0$ exists on $[0,\tau^*(g_0))$. This is presented a…

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 a primary Hopf surface, and let $\omega_0$ be a pluriclosed metric on $M$. Primary Hopf surface convergence conjecture. The solution $\omega_t$ to pluriclosed flow…

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 Kähler surface with $\operatorname{kod}(M)=0$, and let $\omega_0$ be a pluriclosed metric on $M$. Calabi–Yau convergence conjecture. The pluriclosed flow…

Let $\pi:M\to\Sigma$ be a minimal elliptic surface with $\operatorname{kod}(M)=1$, and let $\omega_0$ be a pluriclosed metric. Elliptic surface convergence conjecture. The solution…

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 $(M^{2n},g_0,J_{\pm})$ be a generalized Kähler manifold with $[J_+,J_-]=0$. Let $\tau^*$ be the supremum … where $\mathcal P$ is the positive cone of formally generalized Kähle…

Let $M$ be a class $VII^+$ complex surface, and consider the pluriclosed flow introduced by Streets and Tian. Streets–Tian topology conjecture. Their pluriclosed flow can be used t…

Let $(M^4,J)$ be a compact complex non-Kähler surface with pluriclosed metric, and let $\omega(t)$ be the solution to pluriclosed flow with initial condition $g_0$. Suppose that…