A YTD correspondence for constant scalar curvature metrics
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909883928936448 |
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| author | Darvas, Tamás Zhang, Kewei |
| author_facet | Darvas, Tamás Zhang, Kewei |
| contents | Given a compact Kähler manifold, to better understand Mabuchi's $K$ energy we introduce a family of $K^β$ energies, whose favorable properties are similar to those of the Ding energy from the Fano case. The construction uses Berman's transcendental quantization, and we show that the slope of the $K^β$ energies along test configurations can be computed using intersection theory. With these ingredients in place we provide a uniform Yau-Tian-Donaldson correspondence that characterizes the existence of a unique constant scalar curvature Kähler metric using test configurations. Combining our techniques with the non-Archimedean approach to $K$-stability pioneered by Boucksom--Jonsson, we show that the properness of the classical $K$ energy can be tested by checking its slope along a distinguished subclass of Chi Li-type models, called log discrepancy models, thus yielding another $G$-uniform Yau--Tian--Donaldson correspondence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_15173 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A YTD correspondence for constant scalar curvature metrics Darvas, Tamás Zhang, Kewei Differential Geometry Complex Variables Given a compact Kähler manifold, to better understand Mabuchi's $K$ energy we introduce a family of $K^β$ energies, whose favorable properties are similar to those of the Ding energy from the Fano case. The construction uses Berman's transcendental quantization, and we show that the slope of the $K^β$ energies along test configurations can be computed using intersection theory. With these ingredients in place we provide a uniform Yau-Tian-Donaldson correspondence that characterizes the existence of a unique constant scalar curvature Kähler metric using test configurations. Combining our techniques with the non-Archimedean approach to $K$-stability pioneered by Boucksom--Jonsson, we show that the properness of the classical $K$ energy can be tested by checking its slope along a distinguished subclass of Chi Li-type models, called log discrepancy models, thus yielding another $G$-uniform Yau--Tian--Donaldson correspondence. |
| title | A YTD correspondence for constant scalar curvature metrics |
| topic | Differential Geometry Complex Variables |
| url | https://arxiv.org/abs/2509.15173 |