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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Generating the Fukaya categories of Hamiltonian $G$-manifolds
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by Jonathan David Evans and Yankı Lekili HTML | PDF
J. Amer. Math. Soc. 32 (2019), 119-162

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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Additional 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