書誌事項

Tropical geometry and mirror symmetry

Mark Gross

(Regional conference series in mathematics, no. 114)

Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society with support from the National Science Foundation, c2011

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注記

"NSF-CBMS-Regional Conference in the tropical geometry and mirror symmetry held at Kansas State University, Manhattan, Kansas, December 13-17, 2008"--T.p. verso

Bibliography: p. 307-311

Includes indexes

内容説明・目次

内容説明

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.

目次

The three worlds: The tropics The A- and B-models Log geometry Example: $\mathbb{P}^2$: Mikhalkin's curve counting formula Period integrals The Gross-Siebert program: The program and two-dimensional results Bibliography Index of symbols General index

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