The sign of scalar curvature on K\"ahler blowups

Abstract

We show that if $(M,\omega)$ is any compact K\"ahler manifold, then the blowup of $M$ at any point furnishes a K\"ahler metric with scalar curvature globally and arbitrarily $C^0$-close to the scalar curvature of $\omega$. It follows that if $M$ admits a positive scalar curvature K\"ahler metric, then so do all of its blowups. This special case extends a result of N. Hitchin to surfaces and answers a conjecture of C. LeBrun in the affirmative, consequently completing the classification of positive scalar curvature K\"ahler surfaces as being precisely those of negative Kodaira dimension (i.e. blowups of either the projective plane or a holomorphic bundle of projective lines over a Riemann surface).