Gromov–Hausdorff limits of Kähler manifolds and algebraic geometry

s. donaldson and s. sun unrealistic, numbers.This is connected to certain definite and tractable mathematical questions which we take up briefly again at the very end of the paper, where we formulate a conjectural sharper version of Theorem 1.1 (Conjecture 5.15).We finish this introduction with some words about the origins of this paper.While the question that we answer in Theorem 1.1 is a central one in the field of Kähler-Einstein geometry, it is not something that the authors have focused on until recently.The main construction in this paper emerged as an off-shoot of a joint project by the first-named author and Xiuxiong Chen, studying the slightly different problem of Kähler-Einstein metrics with cone singularities along a divisor.A companion article by the first-named author and Chen, developing this related theory, will appear shortly.Both authors are very grateful to Chen for discussions of these matters, extending over many years.We are also very grateful to the referees for many helpful comments that improved the exposition of this paper.