3 problems

Let $M$ be a blown-up ruled surface of genus at least $2$, and let $[\omega]$ be an admissible Kähler class, meaning that $c_1\cup[\omega]=0$. Associate to $[\omega]$ its correspon…

Let $M$ be a ruled compact complex surface, and let $\widetilde{M}$ be its blow-up at sufficiently many points. Blow-up conjecture. The surface $\widetilde{M}$ admits a scalar-flat…

Classification conjecture. Any scalar-flat Kähler toric metric on $X$ is a Donaldson generalized Taub–NUT metric corresponding to an admissible parameter $\nu\in\mathbb{R}^2$.