Generating the Fukaya categories of Hamiltonian $G$-manifolds
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- by Jonathan David Evans and Yankı Lekili
- J. Amer. Math. Soc. 32 (2019), 119-162
- 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.References
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Bibliographic Information
- Jonathan David Evans
- Affiliation: Department of Mathematics, University College London, London, United Kingdom
- MR Author ID: 897392
- Yankı Lekili
- Affiliation: Department of Mathematical Sciences, King’s College London, London, United Kingdom
- MR Author ID: 858151
- 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.
- © Copyright 2018 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
- MathSciNet review: 3868001