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Enumeration of holomorphic cylinders in log Calabi–Yau surfaces. I

Authors
  • Yu, Tony Yue1
  • 1 Université Paris Diderot-Paris 7, Institut de Mathématiques de Jussieu-Paris Rive Gauche, CNRS-UMR 7586, Case 7012, Bâtiment Sophie Germain, Paris Cedex 13, 75205, France , Paris Cedex 13 (France)
Type
Published Article
Journal
Mathematische Annalen
Publisher
Springer Berlin Heidelberg
Publication Date
Feb 04, 2016
Volume
366
Issue
3-4
Pages
1649–1675
Identifiers
DOI: 10.1007/s00208-016-1376-3
Source
Springer Nature
Keywords
License
Yellow

Abstract

We define the counting of holomorphic cylinders in log Calabi–Yau surfaces. Although we start with a complex log Calabi–Yau surface, the counting is achieved by applying methods from non-archimedean geometry. This gives rise to new geometric invariants. Moreover, we prove that the counting satisfies a property of symmetry. Explicit calculations are given for a del Pezzo surface in detail, which verify the conjectured wall-crossing formula for the focus-focus singularity. Our holomorphic cylinders are expected to give a geometric understanding of the combinatorial notion of broken line by Gross, Hacking, Keel and Siebert. Our tools include Berkovich spaces, tropical geometry, Gromov–Witten theory and the GAGA theorem for non-archimedean analytic stacks.

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