Get Homological Mirror Symmetry and Tropical Geometry (Lecture PDF

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

the connection among Tropical Geometry and reflect Symmetry is going again to the paintings of Kontsevich and Y. Soibelman (2000), who utilized equipment of non-archimedean geometry (in specific, tropical curves) to Homological replicate Symmetry. together with the next paintings of Mikhalkin at the “tropical” method of Gromov-Witten thought and the paintings of Gross and Siebert, Tropical Geometry has now develop into a strong device. Homological replicate Symmetry is the realm of arithmetic targeted round numerous express equivalences connecting symplectic and holomorphic (or algebraic) geometry. The significant principles first seemed within the paintings of Maxim Kontsevich (1993). approximately talking, the topic may be approached in methods: both one makes use of Lagrangian torus fibrations of Calabi-Yau manifolds (the so-called Strominger-Yau-Zaslow photo, extra built by way of Kontsevich and Soibelman) or one makes use of Lefschetz fibrations of symplectic manifolds (suggested by means of Kontsevich and extra constructed by means of Seidel). Tropical Geometry experiences piecewise-linear items which seem as “degenerations” of the corresponding algebro-geometric objects.

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