Sectorial descent for wrapped Fukaya categories

Ganatra, Sheel; Pardon, John; Shende, Vivek

doi:10.1090/jams/1035

Sectorial descent for wrapped Fukaya categories
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by Sheel Ganatra, John Pardon and Vivek Shende;

J. Amer. Math. Soc. 37 (2024), 499-635

DOI: https://doi.org/10.1090/jams/1035

Published electronically: October 24, 2023

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Abstract:

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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