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Constructible sheaves and the Fukaya category


Authors: David Nadler and Eric Zaslow
Journal: J. Amer. Math. Soc. 22 (2009), 233-286
MSC (2000): Primary 53D40, 32S60
DOI: https://doi.org/10.1090/S0894-0347-08-00612-7
Published electronically: September 3, 2008
MathSciNet review: 2449059
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Abstract: Let $ X$ be a compact real analytic manifold, and let $ T^*X$ be its cotangent bundle. Let $ Sh(X)$ be the triangulated dg category of bounded, constructible complexes of sheaves on $ X$. In this paper, we develop a Fukaya $ A_\infty$-category $ Fuk(T^*X)$ whose objects are exact, not necessarily compact Lagrangian branes in the cotangent bundle. We write $ Tw Fuk(T^*X)$ for the $ A_\infty$-triangulated envelope of $ Fuk(T^*X)$ consisting of twisted complexes of Lagrangian branes. Our main result is that $ Sh(X)$ quasi-embeds into $ Tw Fuk(T^*X)$ as an $ A_\infty$-category. Taking cohomology gives an embedding of the corresponding derived categories.


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Additional Information

David Nadler
Affiliation: Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, Illinois 60208
Email: nadler@math.northwestern.edu

Eric Zaslow
Affiliation: Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, Illinois 60208
Email: zaslow@math.northwestern.edu

DOI: https://doi.org/10.1090/S0894-0347-08-00612-7
Received by editor(s): October 5, 2006
Published electronically: September 3, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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