The sign of scalar curvature on K\"ahler blowups

TL;DR

This paper proves that blowing up a compact K"ahler manifold allows the construction of new K"ahler metrics with scalar curvature arbitrarily close to the original, extending known results and confirming conjectures about scalar curvature on K"ahler surfaces.

Contribution

It demonstrates that blowups of K"ahler manifolds can admit metrics with scalar curvature close to the original, confirming conjectures and extending previous results to surfaces.

Findings

Blowups admit K"ahler metrics with scalar curvature close to the original.

Positive scalar curvature metrics are preserved under blowups.

Complete classification of positive scalar curvature K"ahler surfaces achieved.

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).