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Enumeration of holomorphic cylinders in log Calabi–Yau surfaces, II: Positivity, integrality and the gluing formula

Tony Yue Yu

Geometry & Topology 25 (2021) 1–46

We prove three fundamental properties of counting holomorphic cylinders in log Calabi–Yau surfaces: positivity, integrality and the gluing formula. Positivity and integrality assert that the numbers of cylinders, defined via virtual techniques, are in fact nonnegative integers. The gluing formula roughly says that cylinders can be glued together to form longer cylinders, and the number of longer cylinders equals the product of the numbers of shorter cylinders. Our approach uses Berkovich geometry, tropical geometry, deformation theory and the ideas in the proof of associativity relations of Gromov–Witten invariants by Maxim Kontsevich. These three properties provide evidence for a conjectural relation between counting cylinders and the broken lines of Gross, Hacking and Keel.

cylinder, enumerative geometry, nonarchimedean geometry, Berkovich space, Gromov–Witten, Calabi–Yau
Mathematical Subject Classification 2010
Primary: 14N35
Secondary: 14G22, 14J32, 14T05, 53D37
Received: 5 September 2016
Revised: 17 November 2019
Accepted: 21 February 2020
Published: 2 March 2021
Proposed: Dan Abramovich
Seconded: Mark Gross, Paul Seidel
Tony Yue Yu
Laboratoire de Mathématiques d’Orsay
Université Paris-Sud
Université Paris-Saclay