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

Yu, Tony Yue (2021) Enumeration of holomorphic cylinders in log Calabi–Yau surfaces, II : Positivity, integrality and the gluing formula. Geometry & Topology, 25 (1). pp. 1-46. ISSN 1364-0380. doi:10.2140/gt.2021.25.1.

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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.

Item Type:Article
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URLURL TypeDescription Paper
Yu, Tony Yue0000-0002-6019-8552
Additional Information:© 2021 MSP. Received: 5 September 2016. Revised: 17 November 2019. Accepted: 21 February 2020. Published: 2 March 2021. I am very grateful to Maxim Kontsevich for sharing with me many ideas. I am equally grateful to Vladimir Berkovich, Antoine Chambert-Loir, Mark Gross, Bernd Siebert and Michael Temkin for valuable discussions. The smoothness argument in Section 5 I learned from Sean Keel. I would like to thank him in particular. This research was partially conducted during the period the author served as a Clay Research Fellow.
Funding AgencyGrant Number
Clay Mathematics InstituteUNSPECIFIED
Subject Keywords:cylinder, enumerative geometry, nonarchimedean geometry, Berkovich space, Gromov–Witten, Calabi–Yau
Issue or Number:1
Classification Code:Mathematical Subject Classification 2010: Primary: 14N35; Secondary: 14G22, 14J32, 14T05, 53D37
Record Number:CaltechAUTHORS:20210914-164412666
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:110829
Deposited By: George Porter
Deposited On:14 Sep 2021 21:25
Last Modified:14 Sep 2021 21:46

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