Twisted Kähler-Einstein metrics in big classes

Abstract

We prove existence of twisted Kähler-Einstein metrics in big cohomology classes, using a divisorial stability condition. In particular, when $-K_X$ is big, we obtain a uniform Yau-Tian-Donaldson existence theorem for Kähler-Einstein metrics. To achieve this, we build up from scratch the theory of Fujita-Odaka type delta invariants in the transcendental big setting, using pluripotential theory. We do not use the K-energy in our arguments, and our techniques provide a simple roadmap to prove Yau-Tian-Donaldson existence theorems for Kähler-Einstein type metrics, that only needs convexity of the appropriate Ding energy. As an application, we give a simplified proof of Li-Tian-Wang's existence theorem in the log Fano setting.

v1. Comments welcome! v2. Some imprecisions in presentation fixed. References updated v3. Substantial rewrite of presentation, further context added v.4 final version