Twisted Kähler-Einstein metrics in big classes
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866912800992919552 |
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| author | Darvas, Tamás Zhang, Kewei |
| author_facet | Darvas, Tamás Zhang, Kewei |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_08324 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Twisted Kähler-Einstein metrics in big classes Darvas, Tamás Zhang, Kewei Differential Geometry Algebraic Geometry Complex Variables 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. |
| title | Twisted Kähler-Einstein metrics in big classes |
| topic | Differential Geometry Algebraic Geometry Complex Variables |
| url | https://arxiv.org/abs/2208.08324 |