Homological Mirror Symmetry and Tropical Geometry (Lecture Notes of the Unione Matematica Italiana Book 15)

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. 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 ztropicaly 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 zdegenerationsy of the corresponding algebro-geometric objects Front Matter....Pages i-xi Moduli Stacks of Bundles on Local Surfaces....Pages 1-32 An Orbit Construction of Phantoms, Orlov Spectra, and Knörrer Periodicity....Pages 33-42 Microlocal Theory of Sheaves and Tamarkin’s Non Displaceability Theorem....Pages 43-85 A-Polynomial, B-Model, and Quantization....Pages 87-151 Spherical Hall Algebra of $$\overline{\text{Spec }(\mathbb{Z})}$$ ....Pages 153-196 Wall-Crossing Structures in Donaldson–Thomas Invariants, Integrable Systems and Mirror Symmetry....Pages 197-308 Tropical Eigenwave and Intermediate Jacobians....Pages 309-349 Notes on a New Construction of Hyperkahler Metrics....Pages 351-375 Mirror Duality of Landau–Ginzburg Models via Discrete Legendre Transforms....Pages 377-406 Mirror Symmetry in Dimension 1 and Fourier–Mukai Transforms....Pages 407-428 The Very Good Property for Moduli of Parabolic Bundles and the Additive Deligne–Simpson Problem....Pages 429-436 Back Matter....Pages 437-437