Advertisement

Selecta Mathematica

, Volume 24, Issue 5, pp 4105–4140 | Cite as

Arboreal singularities in Weinstein skeleta

  • Laura Starkston
Article
  • 9 Downloads

Abstract

We study the singularities of the isotropic skeleton of a Weinstein manifold in relation to Nadler’s program of arboreal singularities. By deforming the skeleton via homotopies of the Weinstein structure, we produce a Morse–Bott* representative of the Weinstein homotopy class whose stratified skeleton determines its symplectic neighborhood. We then study the singularities of the skeleta in this class and show that after a certain type of generic perturbation either (1) these singularities fall into the class of (signed Lagrangian versions of) Nadler’s arboreal singularities which are combinatorially classified into finitely many types in a given dimension or (2) there are singularities of tangency in associated front projections. We then turn to the singularities of tangency to try to reduce them also to collections of arboreal singularities. We give a general localization procedure to isolate the Liouville flow to a neighborhood of these non-arboreal singularities, and then show how to replace the simplest singularities of tangency (those of Thom-Boardman type \(\Sigma ^{1,0}\)) by arboreal singularities.

Mathematics Subject Classification

53D05 57R17 57R45 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

This work has greatly benefited from many discussions with David Nadler and Yasha Eliashberg. I am grateful for David’s invaluable intuition on arborealization which confirmed throughout when things were on track and corrected them when they were not. I have learned an enormous amount from Yasha and every discussion we have had has taught me a new way of thinking about Lagrangians, symplectic manifolds, and singularities. I have tried to incorporate some of these perspectives into my definitions and proofs, which I believe has significantly advanced the clarity and scope of these results. I am also grateful for advice, interest, shared knowledge, and suggestions from Daniel Álvarez-Gavela, Roger Casals, Kai Cieleibak, Josh Sabloff, Vivek Shende, and Alex Zorn. During the course of this work, I have been supported by an NSF Postdoctoral Fellowship Grant No. 1501728.

References

  1. 1.
    Austin, D.M., Braam, P.J.: Morse–Bott theory and equivariant cohomology. In: Hofer, H., Taubes, C.H., Weinstein, A., Zehnder, E. (eds.) The Floer Memorial Volume, volume 133 of Progress in Mathematics, pp. 123–183. Birkhäuser, Basel (1995)CrossRefGoogle Scholar
  2. 2.
    Cieliebak, K., Eliashberg, Y.: From Stein to Weinstein and Back, volume 59 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI (2012). (Symplectic geometry of affine complex manifolds)zbMATHGoogle Scholar
  3. 3.
    Eliashberg, Y., Gromov, M.: Convex symplectic manifolds. In: Krantz, S., Bedford, E., D’Angelo, J., Greene, R. (eds.) Several Complex Variables and Complex Geometry, Part 2 (Santa Cruz, CA, 1989), volume 52 of Proceedings Symposium Pure Mathematics, pp. 135–162. American Mathematical Society, Providence, RI (1991)Google Scholar
  4. 4.
    Eliashberg, Y.: Topological characterization of Stein manifolds of dimension \(>\) 2. Int. J. Math. 1(1), 29–46 (1990)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Gavela, D.Á.: The simplification of singularities of Lagrangian and Legendrian fronts. arXiv:1605.07259 [math.sg]
  6. 6.
    Kontsevich, M.: Symplectic geometry of homological algebra. https://www.ihes.fr/~maxim/TEXTS/Symplectic_AT2009.pdf
  7. 7.
    Nadler, D.: Non-characteristic expansion of Legendrian singularities. arXiv:1507.01513
  8. 8.
    Nadler, D.: Microlocal branes are constructible sheaves. Sel. Math. (N.S.) 15(4), 563–619 (2009)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Nadler, D.: Arboreal singularities. Geom. Topol. 21(2), 1231–1274 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Nadler, D., Zaslow, E.: Constructible sheaves and the Fukaya category. J. Am. Math. Soc. 22(1), 233–286 (2009)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Weinstein, A.: Contact surgery and symplectic handlebodies. Hokkaido Math. J. 20(2), 241–251 (1991)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.DavisUSA

Personalised recommendations