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We develop a set of tools for doing computations in and of (partially) wrapped Fukaya categories. In particular, we prove (1) a descent (cosheaf) property for the wrapped Fukaya category with respect to so-called Weinstein sectorial coverings and (2) that the partially wrapped Fukaya category of a Weinstein manifold with respect to a mostly Legendrian stop is generated by the cocores of the critical handles and the linking disks to the stop. We also prove (3) a ‘stop removal equals localization’ result, and (4) that the Fukaya–Seidel category of a Lefschetz fibration with Liouville fiber is generated by the Lefschetz thimbles. These results are derived from three main ingredients, also of independent use: (5) a Künneth formula (6) an exact triangle in the Fukaya category associated to wrapping a Lagrangian through a Legendrian stop at infinity and (7) a geometric criterion for when a pushforward functor between wrapped Fukaya categories of Liouville sectors is fully faithful.
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This content will become publicly available on March 1, 2026
Duality and Kernels in Microlocal Geometry
Abstract We study the dualizability of sheaves on manifolds with isotropic singular supports $$\operatorname{Sh}_\Lambda (M)$$ and microsheaves with isotropic supports $$\operatorname{\mu sh}_\Lambda (\Lambda )$$ and obtain a classification result of colimit-preserving functors by convolutions of sheaf kernels. Moreover, for sheaves with isotropic singular supports and compact supports $$\operatorname{Sh}_\Lambda ^{b}(M)_{0}$$, 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 $$\operatorname{Sh}_\Lambda ^{c}(M)_{0}$$ when the wrap-once functor is an equivalence, for instance, when $$\Lambda $$ is a full Legendrian stop or a swappable Legendrian stop.
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- Award ID(s):
- 1928930
- PAR ID:
- 10600030
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- Volume:
- 2025
- Issue:
- 6
- ISSN:
- 1073-7928
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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