13 problems

Let $X$ be a minimal class VII surface with positive second Betti number $b_2(X)>0$. A global spherical shell is a smooth $3$-sphere with a neighborhood biholomorphic to a neighbor…

Class VII surface conjecture. All class VII surfaces with $b_2>0$ are Kato surfaces.

Let $S$ be a surface of class VII$_0^+$, meaning that $b_1(S)=1$ and $b_2(S)>0$. A degeneration of blown-up primary Hopf surfaces is a surface arising as a degeneration of surfaces…

Let $S$ be a minimal compact complex surface of class VII$_0^+$, meaning that $b_1(S)=1$ and $n:=b_2(S)>0$, and let $\mathcal S\to B$ be its versal deformation over a ball of dimen…

Let $S$ be a minimal compact complex surface of Kodaira class $VII_0$, with first Betti number $b_1(S)=1$, and suppose that $S$ contains more than $0$ rational curves, where…

Let $X$ be a minimal compact holomorphic surface in class VII, with positive second Betti number $b_2(X)>0$. Standard class VII conjecture. The surface $X$ has exactly $b_2(X)$ rat…

Let $S$ be a compact complex surface in class $VII_0$, and let $\mathcal T(S)=]a,b[$, with $-\infty\leq a$ and $b<0$, be the moduli space of locally conformally symplectic structur…

Let $S$ be a minimal compact complex surface in class $VII_0^+$, meaning $b_1(S)=1$ and $b_2(S)\geq 1$. A GSS conjecture asserts that every such surface contains a global spherical…

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…

Let $M$ be a Class VII$_0$ surface with $b_2(M)\geqslant1$, and let the Chern-Ricci flow develop its finite singular time $T$. Suppose that as $t\to T$, the metrics converge smooth…

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 $S$ be a compact complex surface with $b_1(S)=1$, $b_2(S)>0$, and Kodaira dimension $-infty$. A surface is minimal if it contains no exceptional curves of the first kind, and a…

Let $X$ be a non-Kähler compact complex surface in the setting of class VII surfaces, and let $b_{2}(X)$ denote its second Betti number. A cycle is a configuration of curves formin…