Homological Mirror Symmetry and Tropical Geometry

  • Ricardo Castano-Bernard
  • Fabrizio Catanese
  • Maxim Kontsevich
  • Tony Pantev
  • Yan Soibelman
  • Ilia Zharkov

Part of the Lecture Notes of the Unione Matematica Italiana book series (UMILN, volume 15)

Table of contents

  1. Front Matter
    Pages i-xi
  2. Oren Ben-Bassat, Elizabeth Gasparim
    Pages 1-32
  3. David Favero, Fabian Haiden, Ludmil Katzarkov
    Pages 33-42
  4. Stéphane Guillermou, Pierre Schapira
    Pages 43-85
  5. Sergei Gukov, Piotr Sułkowski
    Pages 87-151
  6. M. Kapranov, O. Schiffmann, E. Vasserot
    Pages 153-196
  7. Grigory Mikhalkin, Ilia Zharkov
    Pages 309-349
  8. Back Matter
    Pages 437-437

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.


14J33,53D37,14T05,14N35,14D24 Donaldon-Thomas invariants Fukaya category Homological Mirror Symmetry Tropical Geometry Wall-crossing formulas

Editors and affiliations

  • Ricardo Castano-Bernard
    • 1
  • Fabrizio Catanese
    • 2
  • Maxim Kontsevich
    • 3
  • Tony Pantev
    • 4
  • Yan Soibelman
    • 5
  • Ilia Zharkov
    • 6
  1. 1.Mathematics DepartmentKansas State UniversityManhattanUSA
  2. 2.Mathematisches InstitutUniversität BayreuthBayreuthGermany
  3. 3.Institut des Hautes Etudes ScientifiquesBures-sur-YvetteFrance
  4. 4.Mathematics DepartmentUniversity of PennsylvaniaPhiladelphiaUSA
  5. 5.Department of MathematicsKansas State UniversityManhattanUSA
  6. 6.Department of MathematicsKansas State UniversityManhattanUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-06514-4
  • Copyright Information Springer International Publishing Switzerland 2014
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-06513-7
  • Online ISBN 978-3-319-06514-4
  • Series Print ISSN 1862-9113
  • About this book