The sign of scalar curvature on Kähler blowups

Abstract

We show that if $(M,\omega )$ is any compact Kähler manifold, then the blowup of $M$ at any point furnishes a Kähler 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 Kähler 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 Kähler 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).

21 pages, no figures, all comments are welcome. Edited to emphasize the most general version of the results, extended discussion in section 7