Blowing up extremal K\"ahler manifolds II

TL;DR

This paper proves a conjecture relating blowups of extremal K"ahler manifolds to their K-stability, showing that under certain conditions, blowups admit constant scalar curvature K"ahler metrics if and only if they are K-stable.

Contribution

It confirms a conjecture connecting blowups of extremal K"ahler manifolds with K-stability and extends the understanding of cscK metrics on blown-up manifolds.

Findings

Proved the conjecture on blowups of extremal K"ahler manifolds.

Established the equivalence between existence of cscK metrics and K-stability for blowups.

Connected blowup geometry with stability conditions in K"ahler geometry.

Abstract

This is a continuation of the work of Arezzo-Pacard-Singer and the author on blowups of extremal K\"ahler manifolds. We prove the conjecture stated in [32], and we relate this result to the K-stability of blown up manifolds. As an application we prove that if a K\"ahler manifold M of dimension greater than 2 admits a cscK metric, then the blowup of M at a point admits a cscK metric if and only if it is K-stable, as long as the exceptional divisor is sufficiently small.