Tropical Geometry and Mirror Symmetry

Tropical geometry provides an explanation for the remarkable power of mirror symmetry to connect complex and symplectic geometry. The main theme of this book is the interplay between tropical geometry and mirror symmetry, culminating in a description of the recent work of Gross and Siebert using log geometry to understand how the tropical world relates the A- and B-models in mirror symmetry. The text starts with a detailed introduction to the notions of tropical curves and manifolds, and then gives a thorough description of both sides of mirror symmetry for projective space, bringing together material which so far can only be found scattered throughout the literature. Next follows an introduction to the log geometry of Fontaine-Illusie and Kato, as needed for Nishinou and Siebert's proof of Mikhalkin's tropical curve counting formulas. This latter proof is given in the fourth chapter. The fifth chapter considers the mirror, B-model side, giving recent results of the author showing how tropical geometry can be used to evaluate the oscillatory integrals appearing. The final chapter surveys reconstruction results of the author and Siebert for "integral tropical manifolds." A complete version of the argument is given in two dimensions. A co-publication of the AMS and CBMS. Table of Contents Cover Tropical geometry and mirror symmetry Contents Preface Introduction Part 1 The three worlds The tropics 1.1. Tropical hypersurfaces 1.2. Some background on fans 1.3. Parameterized tropical curves 1.4. A�ne manifolds with singularities 1.5. The discrete Legendre transform 1.6. Tropical curves on tropical surfaces 1.7. References and further reading The A- and B-models 2.1. The A-model 2.1.1. Stable maps and Gromov-Witten invariants. 2.2. The B-model 2.3. References and further reading Log geometry 3.1. A brief review of toric geometry 3.1.1. Monoids. 3.2. Log schemes 3.3. Log derivations and di erentials 3.4. Log deformation theory 3.5. The twisted de Rham complex revisited 3.6. References and further reading Part 2 Example: P2. Mikhalkin's curve counting formula 4.1. The statement and outline of the proof 4.2. Log world 4.3. Tropical world 4.4. Classical world 4.5. Log world 4.6. The end of the proof 4.7. References and further reading Period integrals 5.1. The perturbed Landau-Ginzburg potential 5.2. Tropical descendent invariants 5.3. The main B-model statement 5.4. Deforming 5.5. Evaluation of the period integrals 5.6. References and further reading Part 3 The Gross-Siebert program The program and two-dimensional results 6.1. The program 6.2. From integral tropical manifolds to degenerations in dimension two 6.3. Achieving compatibility: The tropical vertex group 6.4. Remarks and generalizations 6.5. References and further reading Bibliography Index of Symbols General Index