Lecture Notes of the Unione Matematica Italiana

Homological Mirror Symmetry and Tropical Geometry

Editors: Castano-Bernard, R., Catanese, F., Kontsevich, M., Pantev, T., Soibelman, Y., Zharkov, I. (Eds.)

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About this book

The relationship between Tropical Geometry and Mirror Symmetry goes back to the work of Kontsevich and Y. Soibelman (2000), who applied methods of non-archimedean geometry (in particular, tropical curves) to Homological Mirror Symmetry. In combination with the subsequent work of Mikhalkin on the “tropical” approach to Gromov-Witten theory and the work of Gross and Siebert, Tropical Geometry has now become a powerful tool. Homological Mirror Symmetry is the area of mathematics concentrated around several categorical equivalences connecting symplectic and holomorphic (or algebraic) geometry. The central ideas first appeared in the work of Maxim Kontsevich (1993). Roughly speaking, the subject can be approached in two ways: either one uses Lagrangian torus fibrations of Calabi-Yau manifolds (the so-called Strominger-Yau-Zaslow picture, further developed by Kontsevich and Soibelman) or one uses Lefschetz fibrations of symplectic manifolds (suggested by Kontsevich and further developed by Seidel). Tropical Geometry studies piecewise-linear objects which appear as “degenerations” of the corresponding algebro-geometric objects.

Table of contents (11 chapters)

  • Moduli Stacks of Bundles on Local Surfaces

    Ben-Bassat, Oren (et al.)

    Pages 1-32

  • An Orbit Construction of Phantoms, Orlov Spectra, and Knörrer Periodicity

    Favero, David (et al.)

    Pages 33-42

  • Microlocal Theory of Sheaves and Tamarkin’s Non Displaceability Theorem

    Guillermou, Stéphane (et al.)

    Pages 43-85

  • A-Polynomial, B-Model, and Quantization

    Gukov, Sergei (et al.)

    Pages 87-151

  • Spherical Hall Algebra of $$\overline{\text{Spec }(\mathbb{Z})}$$

    Kapranov, M. (et al.)

    Pages 153-196

Buy this book

eBook $44.99
price for USA (gross)
  • ISBN 978-3-319-06514-4
  • Digitally watermarked, DRM-free
  • Included format: PDF, EPUB
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $59.99
price for USA
  • ISBN 978-3-319-06513-7
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Homological Mirror Symmetry and Tropical Geometry
Editors
  • Ricardo Castano-Bernard
  • Fabrizio Catanese
  • Maxim Kontsevich
  • Tony Pantev
  • Yan Soibelman
  • Ilia Zharkov
Series Title
Lecture Notes of the Unione Matematica Italiana
Series Volume
15
Copyright
2014
Publisher
Springer International Publishing
Copyright Holder
Springer International Publishing Switzerland
eBook ISBN
978-3-319-06514-4
DOI
10.1007/978-3-319-06514-4
Softcover ISBN
978-3-319-06513-7
Series ISSN
1862-9113
Edition Number
1
Number of Pages
XI, 436
Number of Illustrations and Tables
25 b/w illustrations, 18 illustrations in colour
Topics