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Sheaf quantization of Legendrian isotopy

Part of: Symplectic geometry, contact geometry

Published online by Cambridge University Press:  17 February 2023

Peng Zhou
Peng Zhou*
Affiliation:
University of California, Berkeley, 970 Evans Hall, Berkeley, CA 94720, USA pzhou.math@gmail.com
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Abstract

Let $\{\Lambda ^\infty _t\}$ be an isotopy of Legendrians (possibly singular) in a unit cosphere bundle $S^*M$ that arise as slices of a singular Legendrian $\Lambda _I^\infty \subset S^*M \times T^*I$. Let $\mathcal {C}_t = Sh(M, \Lambda ^\infty _t)$ be the differential graded derived category of constructible sheaves on $M$ with singular support at infinity contained in $\Lambda ^\infty _t$. We prove that if the isotopy of Legendrians embeds into an isotopy of Liouville hypersurfaces, then the family of categories $\{\mathcal {C}_t\}$ is constant in $t$.

Keywords

nearby cyclesingular support estimate

MSC classification

Primary: 53D25: Geodesic flows
Secondary: 53D37: Mirror symmetry, symplectic aspects; homological mirror symmetry; Fukaya category
Type
Research Article
Information
Compositio Mathematica , Volume 159 , Issue 2 , February 2023 , pp. 419 - 435
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Copyright
© 2023 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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Footnotes

This work was supported by an IHES Simons Postdoctoral Fellowship as part of the Simons Collaboration on HMS.

In memory of Steve

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