A Caltech Library Service

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

Yu, Tony Yue (2016) Enumeration of holomorphic cylinders in log Calabi–Yau surfaces. I. Mathematische Annalen, 366 (3-4). pp. 1649-1675. ISSN 0025-5831. doi:10.1007/s00208-016-1376-3.

[img] PDF - Accepted Version
See Usage Policy.


Use this Persistent URL to link to this item:


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.

Item Type:Article
Related URLs:
URLURL TypeDescription Paper ReadCube access
Yu, Tony Yue0000-0002-6019-8552
Alternate Title:Enumeration of holomorphic cylinders in non-archimedean log Calabi-Yau surfaces
Additional Information:© 2015 Springer. Received 09 May 2015. Revised 19 January 2016. Published 04 February 2016. Issue Date: December 2016. I am very grateful to Maxim Kontsevich for suggesting this direction of research and sharing with me many fruitful ideas. Special thanks to Antoine Chambert-Loir for continuous support. During revision, Sean Keel suggested me a better way to deal with the curve classes. I am equally grateful to Luis Alvarez-Consul, Denis Auroux, Vladimir Berkovich, Benoît Bertrand, Philip Boalch, Olivier Debarre, Lie Fu, Mark Gross, Ilia Itenberg, Mattias Jonsson, François Loeser, Ernesto Lupercio, Grigory Mikhalkin, Johannes Nicaise, Johannes Rau, Yan Soibelman, Jake Solomon, Michael Temkin and Bertrand Toën for their helpful comments, and for providing me opportunities to present this work in various seminars and conferences.
Issue or Number:3-4
Classification Code:Mathematics Subject Classification: Primary 14N35; Secondary 14J32 14J26 14T05 14G22
Record Number:CaltechAUTHORS:20210914-164412813
Persistent URL:
Official Citation:Yu, T.Y. Enumeration of holomorphic cylinders in log Calabi–Yau surfaces. I. Math. Ann. 366, 1649–1675 (2016).
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:110831
Deposited By: George Porter
Deposited On:14 Sep 2021 21:06
Last Modified:14 Sep 2021 21:07

Repository Staff Only: item control page