Arcs, stability of pairs and the Mabuchi functional
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916403791003648 |
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| author | Dervan, Ruadhaí Reboulet, Rémi |
| author_facet | Dervan, Ruadhaí Reboulet, Rémi |
| contents | We prove various results involving arcs - which generalise test configurations - within the theory of K-stability.
Our main result characterises coercivity of the Mabuchi functional on spaces of Fubini-Study metrics in terms of uniform K-polystability with respect to arcs, thereby proving a version of a conjecture of Tian. The main new tool is an arc version of a numerical criterion for Paul's theory of stability of pairs, for which we also provide a suitable generalisation applicable to pairs with nontrivial stabiliser.
We give two applications. Firstly, we give a new proof of a version of the Yau-Tian-Donaldson conjecture for Fano manifolds, along the lines originally envisaged by Tian - allowing us to reduce the general Yau-Tian-Donaldson conjecture to an analogue of the partial C^0-estimate. Secondly, for a (possibly singular) polarised variety which is uniformly K-polystable with respect to arcs, we show that the associated Cartan subgroup of its automorphism group is reductive. In particular, uniform K-stability with respect to arcs implies finiteness of the automorphism group. This generalises work of Blum-Xu for Fano varieties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_13617 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Arcs, stability of pairs and the Mabuchi functional Dervan, Ruadhaí Reboulet, Rémi Algebraic Geometry Differential Geometry We prove various results involving arcs - which generalise test configurations - within the theory of K-stability. Our main result characterises coercivity of the Mabuchi functional on spaces of Fubini-Study metrics in terms of uniform K-polystability with respect to arcs, thereby proving a version of a conjecture of Tian. The main new tool is an arc version of a numerical criterion for Paul's theory of stability of pairs, for which we also provide a suitable generalisation applicable to pairs with nontrivial stabiliser. We give two applications. Firstly, we give a new proof of a version of the Yau-Tian-Donaldson conjecture for Fano manifolds, along the lines originally envisaged by Tian - allowing us to reduce the general Yau-Tian-Donaldson conjecture to an analogue of the partial C^0-estimate. Secondly, for a (possibly singular) polarised variety which is uniformly K-polystable with respect to arcs, we show that the associated Cartan subgroup of its automorphism group is reductive. In particular, uniform K-stability with respect to arcs implies finiteness of the automorphism group. This generalises work of Blum-Xu for Fano varieties. |
| title | Arcs, stability of pairs and the Mabuchi functional |
| topic | Algebraic Geometry Differential Geometry |
| url | https://arxiv.org/abs/2409.13617 |