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arXiv · 2405.15211

Duality and kernels in microlocal geometry

Christopher Kuo·Wenyuan Li
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Abstract

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.

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Christopher Kuo, Wenyuan Li. 2025-04-03. Duality and kernels in microlocal geometry. https://doi.org/10.1093/imrn%2Frnaf070

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