Higher regularity for singular Kähler-Einstein metrics
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866909359209971712 |
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| author | Chiu, Shih-Kai Székelyhidi, Gábor |
| author_facet | Chiu, Shih-Kai Székelyhidi, Gábor |
| contents | We study singular Kähler-Einstein metrics that are obtained as non-collapsed limits of polarized Kähler-Einstein manifolds. Our main result is that if the metric tangent cone at a point is locally isomorphic to the germ of the singularity, then the metric converges to the metric on its tangent cone at a polynomial rate on the level of Kähler potentials. When the tangent cone at the point has a smooth cross section, then the result implies polynomial convergence of the metric in the usual sense, generalizing a result due to Hein-Sun. We show that a similar result holds even in certain cases where the tangent cone is not locally isomorphic to the germ of the singularity. Finally we prove a rigidity result for complete $\partial\bar\partial$-exact Calabi-Yau metrics with maximal volume growth. This generalizes a result of Conlon-Hein, which applies to the case of asymptotically conical manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_11083 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Higher regularity for singular Kähler-Einstein metrics Chiu, Shih-Kai Székelyhidi, Gábor Differential Geometry We study singular Kähler-Einstein metrics that are obtained as non-collapsed limits of polarized Kähler-Einstein manifolds. Our main result is that if the metric tangent cone at a point is locally isomorphic to the germ of the singularity, then the metric converges to the metric on its tangent cone at a polynomial rate on the level of Kähler potentials. When the tangent cone at the point has a smooth cross section, then the result implies polynomial convergence of the metric in the usual sense, generalizing a result due to Hein-Sun. We show that a similar result holds even in certain cases where the tangent cone is not locally isomorphic to the germ of the singularity. Finally we prove a rigidity result for complete $\partial\bar\partial$-exact Calabi-Yau metrics with maximal volume growth. This generalizes a result of Conlon-Hein, which applies to the case of asymptotically conical manifolds. |
| title | Higher regularity for singular Kähler-Einstein metrics |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2202.11083 |