K-stability and large complex structure limits
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866915183418408960 |
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| author | Stoppa, Jacopo |
| author_facet | Stoppa, Jacopo |
| contents | We discuss how, under suitable assumptions, a Kähler test configuration admits a mirror Landau-Ginzburg model, giving a corresponding expression for the Donaldson-Futaki invariant as a residue pairing. We study the general behaviour of such mirror formulae under large scaling of the Kähler form. We exploit the observation that this scaling trivially preserves $K$-stability, but takes the mirror Landau-Ginzburg model to a large complex structure limit. In certain cases the mirror formulae for the Donaldson-Futaki invariant simplify in this limit. We focus on a special type of limiting behaviour, when the Donaldson-Futaki invariant concentrates at a single critical point of the Landau-Ginzburg potential, and show that this leads to new formulae for the Donaldson-Futaki invariant in terms of theta functions on the mirror. We provide a main application, which shows that such limiting behaviour actually occurs for test configurations in several nontrivial examples, both toric and non-toric, in the case of slope (in)stability for polarised surfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_05291 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | K-stability and large complex structure limits Stoppa, Jacopo Algebraic Geometry Differential Geometry 32Q25, 14J33 (Primary) 32A27 (Secondary) We discuss how, under suitable assumptions, a Kähler test configuration admits a mirror Landau-Ginzburg model, giving a corresponding expression for the Donaldson-Futaki invariant as a residue pairing. We study the general behaviour of such mirror formulae under large scaling of the Kähler form. We exploit the observation that this scaling trivially preserves $K$-stability, but takes the mirror Landau-Ginzburg model to a large complex structure limit. In certain cases the mirror formulae for the Donaldson-Futaki invariant simplify in this limit. We focus on a special type of limiting behaviour, when the Donaldson-Futaki invariant concentrates at a single critical point of the Landau-Ginzburg potential, and show that this leads to new formulae for the Donaldson-Futaki invariant in terms of theta functions on the mirror. We provide a main application, which shows that such limiting behaviour actually occurs for test configurations in several nontrivial examples, both toric and non-toric, in the case of slope (in)stability for polarised surfaces. |
| title | K-stability and large complex structure limits |
| topic | Algebraic Geometry Differential Geometry 32Q25, 14J33 (Primary) 32A27 (Secondary) |
| url | https://arxiv.org/abs/2309.05291 |