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

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.

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Keywords

cylinder, enumerative geometry, nonarchimedean geometry, Berkovich space, Gromov–Witten, Calabi–Yau

Mathematical Subject Classification 2010

Primary: 14N35

Secondary: 14G22, 14J32, 14T05, 53D37

References

Publication

Received: 5 September 2016

Revised: 17 November 2019

Accepted: 21 February 2020

Published: 2 March 2021

Proposed: Dan Abramovich

Seconded: Mark Gross, Paul Seidel

Authors

Tony Yue Yu
Laboratoire de Mathématiques d’Orsay Université Paris-Sud CNRS Université Paris-Saclay Orsay France

Volume 25, issue 1 (2021)

Tony Yue Yu

Abstract