A transcendental non-Archimedean Calabi--Yau Theorem with applications to the cscK problem
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| Format: | Preprint |
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2025
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| author | Mesquita-Piccione, Pietro Nyström, David Witt |
| author_facet | Mesquita-Piccione, Pietro Nyström, David Witt |
| contents | Let $X$ be a compact Kähler manifold and $α$ a Kähler class on $X$. We prove that if $(X,α)$ is uniformly K-stable for models, then there is a unique cscK metric in $α$. This was first proved in the algebraic case by Chi Li, and it strengthens a related result in an article of Mesquita-Piccione. K-stability for models is defined in terms of big test configurations, but we also give a valuative criterion as in the work of Boucksom--Jonsson together with an explicit formula for the associated $β$-invariant. To accomplish this we further develop the non-Archimedean pluripotential theory in the transcendental setting, as initiated in the works of Darvas--Xia--Zhang and Mesquita-Piccione. In particular we prove the continuity of envelopes and orthogonality properties, and using that, we are able to extend the non-Archimedean Calabi-Yau Theorem found in an article of Boucksom--Jonsson to the general Kähler setting. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_09442 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A transcendental non-Archimedean Calabi--Yau Theorem with applications to the cscK problem Mesquita-Piccione, Pietro Nyström, David Witt Algebraic Geometry Differential Geometry 32Q15, 32Q26, 53C55, 32P05 Let $X$ be a compact Kähler manifold and $α$ a Kähler class on $X$. We prove that if $(X,α)$ is uniformly K-stable for models, then there is a unique cscK metric in $α$. This was first proved in the algebraic case by Chi Li, and it strengthens a related result in an article of Mesquita-Piccione. K-stability for models is defined in terms of big test configurations, but we also give a valuative criterion as in the work of Boucksom--Jonsson together with an explicit formula for the associated $β$-invariant. To accomplish this we further develop the non-Archimedean pluripotential theory in the transcendental setting, as initiated in the works of Darvas--Xia--Zhang and Mesquita-Piccione. In particular we prove the continuity of envelopes and orthogonality properties, and using that, we are able to extend the non-Archimedean Calabi-Yau Theorem found in an article of Boucksom--Jonsson to the general Kähler setting. |
| title | A transcendental non-Archimedean Calabi--Yau Theorem with applications to the cscK problem |
| topic | Algebraic Geometry Differential Geometry 32Q15, 32Q26, 53C55, 32P05 |
| url | https://arxiv.org/abs/2509.09442 |