Geometry & Topology

Arboreal singularities

David Nadler

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Abstract

We introduce a class of combinatorial singularities of Lagrangian skeleta of symplectic manifolds. The link of each singularity is a finite regular cell complex homotopy equivalent to a bouquet of spheres. It is determined by its face poset, which is naturally constructed starting from a tree (nonempty finite acyclic graph). The choice of a root vertex of the tree leads to a natural front projection of the singularity along with an orientation of the edges of the tree. Microlocal sheaves along the singularity, calculated via the front projection, are equivalent to modules over the quiver given by the directed tree.

Article information

Source
Geom. Topol., Volume 21, Number 2 (2017), 1231-1274.

Dates
Received: 30 October 2015
Revised: 6 March 2016
Accepted: 23 April 2016
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1510859178

Digital Object Identifier
doi:10.2140/gt.2017.21.1231

Mathematical Reviews number (MathSciNet)
MR3626601

Zentralblatt MATH identifier
06701806

Subjects
Primary: 32S05: Local singularities [See also 14J17] 53D37: Mirror symmetry, symplectic aspects; homological mirror symmetry; Fukaya category [See also 14J33]

Keywords
Lagrangian singularities microlocal sheaves

Citation

Nadler, David. Arboreal singularities. Geom. Topol. 21 (2017), no. 2, 1231--1274. doi:10.2140/gt.2017.21.1231. https://projecteuclid.org/euclid.gt/1510859178


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References

  • F Ardila, M Develin, Tropical hyperplane arrangements and oriented matroids, Math. Z. 262 (2009) 795–816
  • I N Bernstein, I M Gelfand, V A Ponomarev, Coxeter functors and Gabriel's theorem, Uspehi Mat. Nauk 28 (1973) 19–33 In Russian; translated in Russian Math. Surveys 28 (1973) 17–32
  • J Bernstein, Algebraic theory of $\cD$–modules, lecture notes Available at \setbox0\makeatletter\@url http://www.math.uchicago.edu/~mitya/langlands.html {\unhbox0
  • L J Billera, A Björner, Face numbers of polytopes and complexes, from “Handbook of discrete and computational geometry” (J E Goodman, J O'Rourke, editors), CRC, Boca Raton, FL (1997) 291–310
  • A Björner, Posets, regular CW complexes and Bruhat order, European J. Combin. 5 (1984) 7–16
  • M Develin, B Sturmfels, Tropical convexity, Doc. Math. 9 (2004) 1–27
  • T Dyckerhoff, M Kapranov, Higher Segal spaces, I, preprint (2012)
  • T Dyckerhoff, M Kapranov, Triangulated surfaces in triangulated categories, preprint (2013)
  • M Kashiwara, P Schapira, Sheaves on manifolds, corrected reprint of 1st edition, Grundl. Math. Wissen. 292, Springer (1994)
  • B Keller, On differential graded categories, from “International Congress of Mathematicians, II” (M Sanz-Solé, J Soria, J L Varona, J Verdera, editors), Eur. Math. Soc., Zürich (2006) 151–190
  • M Kontsevich, Symplectic geometry of homological algebra, lecture notes (2009) Available at \setbox0\makeatletter\@url http://www.ihes.fr/~maxim/TEXTS/Symplectic_AT2009.pdf {\unhbox0
  • J-L Loday, The multiple facets of the associahedron, preprint (2005) Available at \setbox0\makeatletter\@url http://www-irma.u-strasbg.fr/~loday/PAPERS/MultFAsENG2.pdf {\unhbox0
  • C McCrory, Cone complexes and PL transversality, Trans. Amer. Math. Soc. 207 (1975) 269–291
  • D Nadler, Cyclic symmetries of $A_n$–quiver representations, Adv. Math. 269 (2015) 346–363
  • D Nadler, Non-characteristic expansions of Legendrian singularities, preprint (2015)
  • D Nadler, Mirror symmetry for the Landau–Ginzburg A-model $M=\mathbb C^n$, $ W=z_1 \cdots z_n$, preprint (2016)
  • D Nadler, Wrapped microlocal sheaves on pairs of pants, preprint (2016)
  • D Nadler, A combinatorial calculation of the Landau–Ginzburg model $M=\mathbb{C}^3$, $W=z_1 z_2 z_3$, Selecta Math. 23 (2017) 519–532
  • L Ng, D Rutherford, V Shende, S Sivek, E Zaslow, Augmentations are sheaves, preprint (2015)
  • P Seidel, Fukaya categories and Picard–Lefschetz theory, European Mathematical Society, Zürich (2008)
  • V Shende, D Treumann, E Zaslow, Legendrian knots and constructible sheaves, preprint (2014)
  • D Speyer, B Sturmfels, Tropical mathematics, lecture notes (2004)
  • J D Stasheff, Homotopy associativity of $H$–spaces, I, Trans. Amer. Math. Soc. 108 (1963) 275–292
  • J D Stasheff, Homotopy associativity of $H$–spaces, II, Trans. Amer. Math. Soc. 108 (1963) 293–312
  • M L Wachs, Poset topology: tools and applications, from “Geometric combinatorics” (E Miller, V Reiner, B Sturmfels, editors), IAS/Park City Math. Ser. 13, Amer. Math. Soc., Providence, RI (2007) 497–615