Cyclic A-infinity Algebras and Calabi--Yau Structures in the Analytic Setting
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866914875517698048 |
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| author | van Garderen, Okke |
| author_facet | van Garderen, Okke |
| contents | This paper considers $A_\infty$-algebras whose higher products satisfy an analytic bound with respect to a fixed norm. We define a notion of right Calabi--Yau structures on such $A_\infty$-algebras and show that these give rise to cyclic minimal models satisfying the same analytic bound. This strengthens a theorem of Kontsevich--Soibelman, and yields a flexible method for obtaining analytic potentials of Hua-Keller.
We apply these results to the endomorphism DGAs of polystable sheaves considered by Toda, for which we construct a family of such right CY structures obtained from analytic germs of holomorphic volume forms on a projective variety. As a result, we can define a canonical cyclic analytic $A_\infty$-structure on the Ext-algebra of a polystable sheaf, which depends only on the analytic-local geometry of its support. This shows that the results of Toda can be extended to the quasi-projective setting, and yields a new method for comparing cyclic $A_\infty$-structures of sheaves on different Calabi--Yau varieties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_00771 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Cyclic A-infinity Algebras and Calabi--Yau Structures in the Analytic Setting van Garderen, Okke Algebraic Geometry Algebraic Topology 18G70, 14F08 (Primary) 32G13, 14J32, 14N35 (Secondary) This paper considers $A_\infty$-algebras whose higher products satisfy an analytic bound with respect to a fixed norm. We define a notion of right Calabi--Yau structures on such $A_\infty$-algebras and show that these give rise to cyclic minimal models satisfying the same analytic bound. This strengthens a theorem of Kontsevich--Soibelman, and yields a flexible method for obtaining analytic potentials of Hua-Keller. We apply these results to the endomorphism DGAs of polystable sheaves considered by Toda, for which we construct a family of such right CY structures obtained from analytic germs of holomorphic volume forms on a projective variety. As a result, we can define a canonical cyclic analytic $A_\infty$-structure on the Ext-algebra of a polystable sheaf, which depends only on the analytic-local geometry of its support. This shows that the results of Toda can be extended to the quasi-projective setting, and yields a new method for comparing cyclic $A_\infty$-structures of sheaves on different Calabi--Yau varieties. |
| title | Cyclic A-infinity Algebras and Calabi--Yau Structures in the Analytic Setting |
| topic | Algebraic Geometry Algebraic Topology 18G70, 14F08 (Primary) 32G13, 14J32, 14N35 (Secondary) |
| url | https://arxiv.org/abs/2306.00771 |