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2015 | OriginalPaper | Buchkapitel

Enumerative Aspects of the Gross-Siebert Program

verfasst von: Michel van Garrel, D. Peter Overholser, Helge Ruddat

Erschienen in: Calabi-Yau Varieties: Arithmetic, Geometry and Physics

Verlag: Springer New York

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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 \(\mathbb{P}^{2}\). 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.
Fußnoten
1
If we strove for maximal generality, we would assume π to be flat and locally finitely presented.
 
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Metadaten
Titel
Enumerative Aspects of the Gross-Siebert Program
verfasst von
Michel van Garrel
D. Peter Overholser
Helge Ruddat
Copyright-Jahr
2015
Verlag
Springer New York
DOI
https://doi.org/10.1007/978-1-4939-2830-9_11

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