Reverse Hölder inequalities on the space of Kähler metrics of a Fano variety and effective openness
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866914784978403328 |
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| author | Berman, Robert J. |
| author_facet | Berman, Robert J. |
| contents | A reverse Hölder inequality is established on the space of Kähler metrics in the first Chern class of a Fano manifold X endowed with Darvas L^{p}-Finsler metrics. The inequality holds under a uniform bound on a twisted Ricci potential and extends to Fano varieties with log terminal singularities. Its proof leverages a "hidden" log-concavity. An application to destabilizing geodesic rays is provided, which yields a reverse Hölder inequality for the speed of the geodesic. In the case of Aubin's continuity path on a K-unstable Fano variety, the constant in the corresponding Hölder bound is shown to only depend on p and the dimension of X. This leads to some intruiging relations to Harnack bounds and the partial C^{0}-estimate. In another direction, universal effective openness results are established for the complex singularity exponents (log canonical thresholds) of ω-plurisubharmonic functions on any Fano variety. Finally, another application to K-unstable Fano varieties is given, involving Archimedean Igusa zeta functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_16278 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Reverse Hölder inequalities on the space of Kähler metrics of a Fano variety and effective openness Berman, Robert J. Differential Geometry Algebraic Geometry Complex Variables A reverse Hölder inequality is established on the space of Kähler metrics in the first Chern class of a Fano manifold X endowed with Darvas L^{p}-Finsler metrics. The inequality holds under a uniform bound on a twisted Ricci potential and extends to Fano varieties with log terminal singularities. Its proof leverages a "hidden" log-concavity. An application to destabilizing geodesic rays is provided, which yields a reverse Hölder inequality for the speed of the geodesic. In the case of Aubin's continuity path on a K-unstable Fano variety, the constant in the corresponding Hölder bound is shown to only depend on p and the dimension of X. This leads to some intruiging relations to Harnack bounds and the partial C^{0}-estimate. In another direction, universal effective openness results are established for the complex singularity exponents (log canonical thresholds) of ω-plurisubharmonic functions on any Fano variety. Finally, another application to K-unstable Fano varieties is given, involving Archimedean Igusa zeta functions. |
| title | Reverse Hölder inequalities on the space of Kähler metrics of a Fano variety and effective openness |
| topic | Differential Geometry Algebraic Geometry Complex Variables |
| url | https://arxiv.org/abs/2309.16278 |