4 problems

Parabolic stability necessity conjecture. If $\widehat M$ admits a scalar-flat Kähler metric, then $E$ must be parabolically stable. This is a necessary-condition conjecture for th…

LeBrun–Simanca conjecture. The Kähler class $[\omega]$ contains a scalar-flat Kähler representative if and only if the parabolic bundle $E$ is stable. This conjectures that the qua…

Let $\Gamma$ be any group acting on $\mathbb{C}^2$ so that the corresponding quotient singularity admits a versal deformation family. Consider complex structures in this versal fam…

Let $\Gamma \subset {\rm{U}}(2)$ be a finite subgroup containing no complex reflections, and let $\widetilde{X}$ be the minimal resolution of $\mathbb{C}^2/\Gamma$. A small deforma…