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Generating the Fukaya categories of Hamiltonian $ G$-manifolds


Authors: Jonathan David Evans and Yankı Lekili
Journal: J. Amer. Math. Soc. 32 (2019), 119-162
MSC (2010): Primary 53D40
DOI: https://doi.org/10.1090/jams/909
Published electronically: September 27, 2018
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Abstract: Let $ G$ be a compact Lie group, and let $ k$ be a field of characteristic $ p \geq 0$ such that $ H^*(G)$ has no $ p$-torsion if $ p>0$. We show that a free Lagrangian orbit of a Hamiltonian $ G$-action on a compact, monotone, symplectic manifold $ X$ split-generates an idempotent summand of the monotone Fukaya category $ \mathcal {F}(X; k)$ if and only if it represents a nonzero object of that summand (slightly more general results are also provided). Our result is based on an explicit understanding of the wrapped Fukaya category $ \mathcal {W}(T^*G; k)$ through Koszul twisted complexes involving the zero-section and a cotangent fibre and on a functor $ D^b \mathcal {W}(T^*G; k) \to D^b\mathcal {F}(X^{-} \times X; k)$ canonically associated to the Hamiltonian $ G$-action on $ X$. We explore several examples which can be studied in a uniform manner, including toric Fano varieties and certain Grassmannians.


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Additional Information

Jonathan David Evans
Affiliation: Department of Mathematics, University College London, London, United Kingdom

Yankı Lekili
Affiliation: Department of Mathematical Sciences, King’s College London, London, United Kingdom

DOI: https://doi.org/10.1090/jams/909
Received by editor(s): July 30, 2015
Received by editor(s) in revised form: February 24, 2018
Published electronically: September 27, 2018
Additional Notes: The second author is partially supported by the Royal Society and NSF Grant No. DMS-1509141.
Article copyright: © Copyright 2018 Jonathan David Evans and Yankı Lekili

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