Kähler-Einstein metrics arising from micro-canonical measures and Hamiltonian dynamics
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866929759793971200 |
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| author | Berman, Robert J. |
| author_facet | Berman, Robert J. |
| contents | We introduce new probabilistic and variational constructions of (twisted) Kähler-Einstein metrics on complex projective algebraic varieties, drawing inspiration from Onsager's statistical mechanical model of turbulence in two-dimensional incompressible fluids. The probabilistic construction involves microcanonical measures associated with the level sets of the pluricomplex energy, which give rise to maximum entropy principles. These, in turn, yield novel characterizations of Kähler-Einstein metrics, as well of Fano varieties admitting such metrics. Additionally, connections to Hamiltonian dynamics are uncovered, resulting in a new evolution equation, that generalizes both the 2D incompressible Euler equation and the 2D semi-geostrophic equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_11157 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Kähler-Einstein metrics arising from micro-canonical measures and Hamiltonian dynamics Berman, Robert J. Differential Geometry Mathematical Physics Complex Variables We introduce new probabilistic and variational constructions of (twisted) Kähler-Einstein metrics on complex projective algebraic varieties, drawing inspiration from Onsager's statistical mechanical model of turbulence in two-dimensional incompressible fluids. The probabilistic construction involves microcanonical measures associated with the level sets of the pluricomplex energy, which give rise to maximum entropy principles. These, in turn, yield novel characterizations of Kähler-Einstein metrics, as well of Fano varieties admitting such metrics. Additionally, connections to Hamiltonian dynamics are uncovered, resulting in a new evolution equation, that generalizes both the 2D incompressible Euler equation and the 2D semi-geostrophic equation. |
| title | Kähler-Einstein metrics arising from micro-canonical measures and Hamiltonian dynamics |
| topic | Differential Geometry Mathematical Physics Complex Variables |
| url | https://arxiv.org/abs/2503.11157 |