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RESEARCH PRODUCT

Enumerative Aspects of the Gross-Siebert Program

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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.