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The
higher-dimensional tropical vertex
Hülya Argüz and Mark Gross |
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Geometry & Topology 26 (2022) 2135–2235
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Abstract |
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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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Keywords
mirror symmetry, tropical geometry, Gromov–Witten theory
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Mathematical Subject Classification
Primary: 14J33, 14N35
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References |
Publication
Received: 13 August 2020
Revised: 22 April 2021
Accepted: 24 June 2021
Published: 12 December 2022
Proposed: Jim Bryan
Seconded: Lothar Göttsche, Dan Abramovich
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Authors |
| Hülya Argüz | |
| Université de Versailles
Saint-Quentin-en-Yvelines Versailles France |
Department of Mathematics University of Georgia Athens, GA United States |
| Mark Gross | |
| DPMMS University of Cambridge Cambridge United Kingdom |
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