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Enumerative aspects of the Gross-Siebert program
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van Garrel, Michel, Overholser, D. Peter and Ruddat, Helge (2015) Enumerative aspects of the Gross-Siebert program. In: Laza, R. and Schütt, M. and Yui, N., (eds.) Calabi-Yau Varieties: Arithmetic, Geometry and Physics : Lecture Notes on Concentrated Graduate Courses. Fields Institute Monographs, 34 . Springer. ISBN 9781493928293
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Official URL: https://doi.org/10.1007/978-1-4939-2830-9_11
Abstract
For the last decade, Mark Gross and Bernd Siebert have worked with a number of collaborators to push forward a program whose aim is an understanding of mirror symmetry. In this chapter, we’ll present certain elements of the “Gross-Siebert” program. We begin by sketching its main themes and goals. Next, we review the basic definitions and results of two main tools of the program, logarithmic and tropical geometry. These tools are then used to give tropical interpretations of certain enumerative invariants. We study in detail the tropical pencil of elliptic curves in a toric del Pezzo surface. We move on to a basic illustration of mirror symmetry, Gross’s tropical construction for P2. On the A-model side, we present the proof of Siebert and Nishinou that tropical geometry invariants coincide with classical geometry invariants via toric degenerations. We then summarize Gross’s tropical B-model and the theorem that links the two constructions, emphasizing the common tropical structures underlying both.
Item Type: | Book Item | ||||||
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Subjects: | Q Science > QA Mathematics | ||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||
Library of Congress Subject Headings (LCSH): | Geometry, Algebraic, String models, Number theory | ||||||
Series Name: | Fields Institute Monographs | ||||||
Publisher: | Springer | ||||||
ISBN: | 9781493928293 | ||||||
Book Title: | Calabi-Yau Varieties: Arithmetic, Geometry and Physics : Lecture Notes on Concentrated Graduate Courses | ||||||
Editor: | Laza, R. and Schütt, M. and Yui, N. | ||||||
Official Date: | 1 January 2015 | ||||||
Dates: |
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Volume: | 34 | ||||||
DOI: | 10.1007/978-1-4939-2830-9_11 | ||||||
Status: | Peer Reviewed | ||||||
Publication Status: | Published | ||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||
Date of first compliant deposit: | 17 October 2019 | ||||||
Date of first compliant Open Access: | 23 October 2019 | ||||||
Related URLs: | |||||||
Open Access Version: |
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