HOMOLOGICAL MIRROR SYMMETRY AND TROPICAL GEOMETRY

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)
1.Moduli Stacks of Bundles on Local Surfaces
2.An Orbit Construction of Phantoms, Orlov Spectra, and Knörrer Periodicity
3.Microlocal Theory of Sheaves and Tamarkin’s Non Displaceability Theorem
4.A-Polynomial, B-Model, and Quantization
5.Spherical Hall Algebra of
6.Wall-Crossing Structures in Donaldson–Thomas Invariants, Integrable Systems and Mirror Symmetry
7.Tropical Eigenwave and Intermediate Jacobians
8.Notes on a New Construction of Hyperkahler Metrics
9.Mirror Duality of Landau–Ginzburg Models via Discrete Legendre Transforms
10.Mirror Symmetry in Dimension 1 and Fourier–Mukai Transforms
11.The Very Good Property for Moduli of Parabolic Bundles and the Additive Deligne–Simpson Problem