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Sheaves via augmentations of Legendrian surfaces

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Abstract

Given an augmentation for a Legendrian surface in a 1-jet space, \(\Lambda \subset J^1(M)\), we explicitly construct an object, \(\mathcal {F} \in \mathbf {Sh}^\bullet _{\Lambda }(M\times \mathbb {R}, \mathbb {K})\), of the (derived) category from Shende, Treumann and Zaslow (Invent Math 207(3), 1031–1133 (2017)) of constructible sheaves on \(M\times \mathbb {R}\) with singular support determined by \(\Lambda \). In the construction, we introduce a simplicial Legendrian DGA (differential graded algebra) for Legendrian submanifolds in 1-jet spaces that, based on Rutherford and Sullivan (Cellular Legendrian contact homology for surfaces, Part I, arXiv:1608.02984.) Rutherford and Sullivan (Internat J Math 30(7):135, 2019) Rutherford and Sullivan (Internat J Math 30(7):111, 2019), is equivalent to the Legendrian contact homology DGA in the case of Legendrian surfaces. In addition, we extend the approach of Shende, Treumann and Zaslow (Invent Math 207(3), 1031–1133 (2017)) for 1-dimensional Legendrian knots to obtain a combinatorial model for sheaves in \(\mathbf {Sh}^\bullet _{\Lambda }(M\times \mathbb {R}, \mathbb {K})\) in the 2-dimensional case.

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Change history

  • 30 October 2021

    The original article has been revised to correct the first and last citations in the Abstract section.

Notes

  1. Here and elsewhere in the article we refer to objects of \(\mathbf {Sh}^\bullet _{\Lambda }(M\times \mathbb {R}, \mathbb {K})\) as “sheaves” even though they are in fact cochain complexes of sheaves.

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Acknowledgements

The first author is partially supported by grant 429536 from the Simons Foundation. The second author is partially supported by grant 317469 from the Simons Foundation. He thanks the Centre de Recherches Mathematiques for hosting him while some of this work was done. The authors also thank Baptiste Chantraine, Honghao Gao, Stephane Guillermou, Vivek Shende, David Treumann, Eric Zaslow, and Mahmoud Zeinalian for educational conversations and e-mail correspondence, as well as the anonymous referee for useful comments.

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  1. Ball State Unversity, Muncie, IN, USA

    Dan Rutherford

  2. University of Massachusetts Amherst, Amherst, USA

    Michael Sullivan

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  1. Dan Rutherford
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  2. Michael Sullivan
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Correspondence to Michael Sullivan.

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Communicated by Scott Wilson.

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Rutherford, D., Sullivan, M. Sheaves via augmentations of Legendrian surfaces. J. Homotopy Relat. Struct. 16, 703–752 (2021). https://doi.org/10.1007/s40062-021-00292-6

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