Abstract:
The interaction between curvature and algebraic geometry is a guiding principle in Kähler geometry. For example, it is a result of Yau that a Kähler manifold with positive average scalar curvature has negative Kodaira dimension, and that minimal surfaces in this class admit pscK metrics. It was conjectured by Lebrun that all negative Kodaira dimension surfaces should admit pscK metrics. In this talk, I will explain a recent resolution to this conjecture by showing one can actually make the scalar curvature on the blowup of any Kähler manifold C^0 close to that of any starting Kähler metric on the base.