Contents Preface ix Introduction xi Part 1. The three worlds 1 Chapter 1. The tropics 3 1.1. Tropical hypersurfaces 3 1.2. Some background on fans 11 1.3. Parameterized tropical curves 13 1.4. Affine manifolds with singularities 19 1.5. The discrete Legendre transform 27 1.6. Tropical curves on tropical surfaces 30 1.7. References and further reading 32 Chapter 2. The A- and B-models 33 2.1. The A-model 33 2.2. The B-model 67 2.3. References and further reading 89 Chapter 3. Log geometry 91 3.1. A brief review of toric geometry 92 3.2. Log schemes 98 3.3. Log derivations and differentials 112 3.4. Log deformation theory 117 3.5. The twisted de Rham complex revisited 126 3.6. References and further reading 129 Part 2. Example: P2. 131 Chapter 4. Mikhalkin’s curve counting formula 133 4.1. The statement and outline of the proof 133 4.2. Log world → tropical world 138 4.3. Tropical world → log world 143 4.4. Classical world → log world 157 4.5. Log world → classical world 165 4.6. The end of the proof 169 4.7. References and further reading 171 Chapter 5. Period integrals 173 5.1. The perturbed Landau-Ginzburg potential 173 vii
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