Author: Alexander on 30-12-2016, 04:55
Homological Mirror Symmetry and Tropical Geometry

Homological Mirror Symmetry and Tropical Geometry (Lecture Notes of the Unione Matematica Italiana, Book 15) by Ricardo Castano-Bernard, Fabrizio Catanese, Maxim Kontsevich and Tony Pantev
English | 2014 | ISBN: 3319065130 | 436 pages | PDF | 3 MB


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.

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