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The higher-dimensional tropical vertex

Hülya Argüz and Mark Gross

Geometry & Topology 26 (2022) 2135–2235

We study log Calabi–Yau varieties obtained as a blow-up of a toric variety along hypersurfaces in its toric boundary. Mirrors to such varieties are constructed by Gross and Siebert from a canonical scattering diagram built by using punctured Gromov–Witten invariants of Abramovich, Chen, Gross and Siebert. We show that there is a piecewise-linear isomorphism between the canonical scattering diagram and a scattering diagram defined algorithmically, following a higher-dimensional generalization of the Kontsevich–Soibelman construction. We deduce that the punctured Gromov–Witten invariants of the log Calabi–Yau variety can be captured from this algorithmic construction. This generalizes previous results of Gross, Pandharipande and Siebert on “the tropical vertex” to higher dimensions. As a particular example, we compute these invariants for a nontoric blow-up of the three-dimensional projective space along two lines.

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mirror symmetry, tropical geometry, Gromov–Witten theory
Mathematical Subject Classification
Primary: 14J33, 14N35
Received: 13 August 2020
Revised: 22 April 2021
Accepted: 24 June 2021
Published: 12 December 2022
Proposed: Jim Bryan
Seconded: Lothar Göttsche, Dan Abramovich
Hülya Argüz
Université de Versailles Saint-Quentin-en-Yvelines
Department of Mathematics
University of Georgia
Athens, GA
United States
Mark Gross
University of Cambridge
United Kingdom