Duality and kernels in microlocal geometry
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915224184946688 |
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| author | Kuo, Christopher Li, Wenyuan |
| author_facet | Kuo, Christopher Li, Wenyuan |
| contents | We study the dualizability of sheaves on manifolds with isotropic singular supports $\operatorname{Sh}_Λ(M)$ and microsheaves with isotropic supports $\operatorname{μsh}_Λ(Λ)$ and obtain a classification result of colimit-preserving functors by convolutions of sheaf kernels. Moreover, for sheaves with isotropic singular supports and compact supports $\operatorname{Sh}_Λ^b(M)_0$, the standard categorical duality and Verdier duality are related by the wrap-once functor, which is the inverse Serre functor in proper objects, and we thus show that the Verdier duality extends naturally to all compact objects $\operatorname{Sh}_Λ^c(M)_0$ when the wrap-once functor is an equivalence, for instance, when $Λ$ is a full Legendrian stop or a swappable Legendrian stop. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_15211 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Duality and kernels in microlocal geometry Kuo, Christopher Li, Wenyuan Symplectic Geometry Algebraic Geometry 53Dxx (Primary) 55N30 (Secondary) We study the dualizability of sheaves on manifolds with isotropic singular supports $\operatorname{Sh}_Λ(M)$ and microsheaves with isotropic supports $\operatorname{μsh}_Λ(Λ)$ and obtain a classification result of colimit-preserving functors by convolutions of sheaf kernels. Moreover, for sheaves with isotropic singular supports and compact supports $\operatorname{Sh}_Λ^b(M)_0$, the standard categorical duality and Verdier duality are related by the wrap-once functor, which is the inverse Serre functor in proper objects, and we thus show that the Verdier duality extends naturally to all compact objects $\operatorname{Sh}_Λ^c(M)_0$ when the wrap-once functor is an equivalence, for instance, when $Λ$ is a full Legendrian stop or a swappable Legendrian stop. |
| title | Duality and kernels in microlocal geometry |
| topic | Symplectic Geometry Algebraic Geometry 53Dxx (Primary) 55N30 (Secondary) |
| url | https://arxiv.org/abs/2405.15211 |