This paper studies the following problem: suppose \((M,\omega)\) is a compact Kähler manifold with an extremal Kähler metric, and \(p\in M\). When does the blowup \(\tilde{M}\) of \(M\) at \(p\) admit an extremal metric in all Kähler classes which makes the exceptional divisor sufficiently small?
This is a challenging problem, because in general the existence of extremal Kähler metrics in a (rational) Kähler class is conjectured to be equivalent to an algebro-geometric notion of stability of the polarized manifold (known as “relative K-polystability”). The first results concerning this question are due to C. Arezzo and F. Pacard [Acta Math. 196, No. 2, 179–228 (2006; Zbl 1123.53036); Ann. Math. (2) 170, No. 2, 685–738 (2009; Zbl 1202.53069)] and C. Arezzo, F. Pacard and M. Singer [Duke Math. J. 157, No. 1, 1–51 (2011; Zbl 1221.32008)], who proved that indeed \(\tilde{M}\) admits such extremal metrics, under certain additional hypotheses. The first of theses papers also completely treated the case when \(M\) has no nonzero holomorphic vector fields, so the author assumes that this is not the case for \(M\).
The paper under review removes most of the additional hypotheses, and in fact settles the above-mentioned conjecture in the case when \((M,\omega)\) is Kähler-Einstein. This requires a refining of the gluing techniques of the above-mentioned papers, a solid understanding of the deformation problem for relatively stable points in geometric invariant theory, and some technical algebro-geometric calculations of Futaki invariants on blowups. Overall, this is a groundbreaking paper, where analysis, differential geometry and algebraic geometry come together beautifully. We expect that a further improvement of the techniques in this paper will lead to a complete solution to the existence problem of extremal Kähler metrics on blowups of extremal manifolds.