Fano Fibrations and Twisted Kähler-Einstein Metrics II: The Kähler-Ricci Flow
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918264055005184 |
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| author | Bednarek, Alexander |
| author_facet | Bednarek, Alexander |
| contents | This is the second of two papers studying both the geometric structure of Fano fibrations and the application to Kähler-Ricci flows developing a singularity in finite time. We assume that the Kähler-Ricci flow on a compact Kähler manifold has a rational initial metric and develops a singularity in finite time such that the manifold admits a Fano fibration structure. Moreover, it is assumed that the volume form of the flow collapses uniformly at the rate of $C^{-1}(T-t)^{n-m} Ω\leq ω(t)^n\leq C(T-t)^{n-m}Ω$. Under this setting, a diameter bound is obtained in any compact set away from singular fibres and the diameter of the fibres is proven to collapse at the optimal rate $\sqrt{T-t}$. Furthermore, several precise $C^0$-estimates are proven for the potential of the complex Monge-Ampere flow which involve the potentials of singular twisted Kähler-Einstein metrics on the base variety from part I. Finally, in the case of Kähler-Einstein Fano fibres, we deduce Type I scalar curvature in any compact set away from singular fibres and globally for a submersion. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_21910 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fano Fibrations and Twisted Kähler-Einstein Metrics II: The Kähler-Ricci Flow Bednarek, Alexander Differential Geometry Complex Variables 53E30, 32W50 This is the second of two papers studying both the geometric structure of Fano fibrations and the application to Kähler-Ricci flows developing a singularity in finite time. We assume that the Kähler-Ricci flow on a compact Kähler manifold has a rational initial metric and develops a singularity in finite time such that the manifold admits a Fano fibration structure. Moreover, it is assumed that the volume form of the flow collapses uniformly at the rate of $C^{-1}(T-t)^{n-m} Ω\leq ω(t)^n\leq C(T-t)^{n-m}Ω$. Under this setting, a diameter bound is obtained in any compact set away from singular fibres and the diameter of the fibres is proven to collapse at the optimal rate $\sqrt{T-t}$. Furthermore, several precise $C^0$-estimates are proven for the potential of the complex Monge-Ampere flow which involve the potentials of singular twisted Kähler-Einstein metrics on the base variety from part I. Finally, in the case of Kähler-Einstein Fano fibres, we deduce Type I scalar curvature in any compact set away from singular fibres and globally for a submersion. |
| title | Fano Fibrations and Twisted Kähler-Einstein Metrics II: The Kähler-Ricci Flow |
| topic | Differential Geometry Complex Variables 53E30, 32W50 |
| url | https://arxiv.org/abs/2512.21910 |