7 problems

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…

Exhaustion conjecture. The above list of examples essentially exhausts all possible real-analytic Nijenhuis operators on closed two-dimensional surfaces.

Rational-cycle conjecture. If every such $T$ belongs to $L^2_{-1}(X)$, but not every such $T$ satisfies $I(T)=0$, then $X$ admits a cycle of rational curves.

Let $S$ be an analytic surface in $\mathbb R^3$. Suppose that through each point of $S$ there are a line segment and a parabolic arc with vertical axis, both fully contained in $S$…

Let $S$ be an analytic surface in $\mathbb R^3$. Suppose that through each point of $S$ there are two transversal arcs of isotropic circles, lying on one isotropic sphere, fully co…

Let $C_2$ be a curve of genus $2$, and let $f\colon S\longrightarrow \operatorname{Sym^2}(C_2)$ be a degree-$4$ generic cover branched over the diagonal, arising from the construct…

Let $(S,D)$ be a logarithmic del Pezzo surface of rank $1$ with shrinkable boundary, and let $\bar S$ be its associated surface. Classification conjecture. Then … where $G$ is a fi…