Canonical Metrics on Singular Fano Varieties
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| Natura: | Recurso digital |
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2025
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| author | SÉRGIO DE ANDRADE, PAULO |
| author_facet | SÉRGIO DE ANDRADE, PAULO |
| contents | This paper provides a comprehensive overview of the theory of canonical metrics on singular Fano varieties, focusing on the existence and properties of Kähler-Einstein metrics. Fano varieties, characterized by their ample anticanonical bundle, are central objects in algebraic geometry. In the smooth setting, the celebrated Yau-Tian-Donaldson conjecture posits that the existence of a Kähler-Einstein metric is equivalent to an algebro-geometric stability condition known as K-polystability. This conjecture was resolved by Chen, Donaldson, and Sun. This work explores the extension of this correspondence to the singular setting, which introduces significant analytic and geometric challenges. We review the foundational concepts of pluripotential theory required to define metrics on singular spaces, the appropriate formulation of the complex Monge-Ampère equation, and the algebraic notion of K-stability for singular Q-Fano varieties. The paper surveys the key results that establish the equivalence between the existence of singular Kähler-Einstein metrics and K-polystability, highlighting the crucial role of valuative criteria and the continuity method. Furthermore, we discuss the regularity of these metrics and their profound implications for the construction and study of moduli spaces of Fano varieties. The discussion also addresses the limitations of current theories and outlines major open problems in the field, including the study of other canonical metrics and the extension to broader classes of singularities. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17677695 |
| institution | Zenodo |
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| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Canonical Metrics on Singular Fano Varieties SÉRGIO DE ANDRADE, PAULO This paper provides a comprehensive overview of the theory of canonical metrics on singular Fano varieties, focusing on the existence and properties of Kähler-Einstein metrics. Fano varieties, characterized by their ample anticanonical bundle, are central objects in algebraic geometry. In the smooth setting, the celebrated Yau-Tian-Donaldson conjecture posits that the existence of a Kähler-Einstein metric is equivalent to an algebro-geometric stability condition known as K-polystability. This conjecture was resolved by Chen, Donaldson, and Sun. This work explores the extension of this correspondence to the singular setting, which introduces significant analytic and geometric challenges. We review the foundational concepts of pluripotential theory required to define metrics on singular spaces, the appropriate formulation of the complex Monge-Ampère equation, and the algebraic notion of K-stability for singular Q-Fano varieties. The paper surveys the key results that establish the equivalence between the existence of singular Kähler-Einstein metrics and K-polystability, highlighting the crucial role of valuative criteria and the continuity method. Furthermore, we discuss the regularity of these metrics and their profound implications for the construction and study of moduli spaces of Fano varieties. The discussion also addresses the limitations of current theories and outlines major open problems in the field, including the study of other canonical metrics and the extension to broader classes of singularities. |
| title | Canonical Metrics on Singular Fano Varieties |
| url | https://doi.org/10.5281/zenodo.17677695 |