A Caltech Library Service

The Frobenius structure theorem for affine log Calabi-Yau varieties containing a torus

Keel, Sean and Yu, Tony Yue (2019) The Frobenius structure theorem for affine log Calabi-Yau varieties containing a torus. . (Unpublished)

[img] PDF - Submitted Version
See Usage Policy.


Use this Persistent URL to link to this item:


We show that the naive counts of rational curves in any affine log Calabi-Yau variety U, containing an open algebraic torus, determine in a surprisingly simple way, a family of log Calabi-Yau varieties, as the spectrum of a commutative associative algebra equipped with a compatible multilinear form. This is directly inspired by a very similar conjecture of Gross-Hacking-Keel in mirror symmetry, known as the Frobenius structure conjecture. Although the statement involves only elementary algebraic geometry, our proof employs Berkovich non-archimedean analytic methods. We construct the structure constants of the algebra via counting non-archimedean analytic disks in the analytification of U. We establish various properties of the counting, notably deformation invariance, symmetry, gluing formula and convexity. In the special case when U is a Fock-Goncharov skew-symmetric X-cluster variety, we prove that our algebra generalizes, and in particular gives a direct geometric construction of, the mirror algebra of Gross-Hacking-Keel-Kontsevich.

Item Type:Report or Paper (Discussion Paper)
Related URLs:
URLURL TypeDescription Paper
Yu, Tony Yue0000-0002-6019-8552
Additional Information:We benefited tremendously from profound, detailed technical discussions with Mark Gross, Paul Hacking, Johannes Nicaise and Maxim Kontsevich. The beautiful Frobenius structure conjecture is due to Hacking, as is the idea of using degeneration to the toric case to prove non-degeneracy of the trace pairing. We enjoyed fruitful conversations with M. Baker, V. Berkovich, M. Brown, A. Chambert-Loir, F. Charles, A. Corti, A. Durcos, W. Gubler, E. Mazzon, M. Porta, J. Rabinoff, M. Robalo, B. Siebert, Y. Soibelman, M. Temkin and J. Xie. Keel would like to especially thank B. Conrad and S. Payne for detailed email tutorials on rigid geometry. Keel was supported by NSF grant DMS-1561632. T.Y. Yu was supported by the Clay Mathematics Institute as Clay Research Fellow. Much of the research was carried out during the authors’ trips to IHES and IAS.
Funding AgencyGrant Number
Clay Mathematics InstituteUNSPECIFIED
Subject Keywords:Frobenius structure, mirror symmetry, log Calabi-Yau, skeletal curve, non-archimedean geometry, rigid analytic geometry, cluster algebra, scattering diagram, wall-crossing, broken lines
Classification Code:2010 Mathematics Subject Classification. Primary 14J33; Secondary 14G22, 14N35, 14J32, 14T05, 13F60
Record Number:CaltechAUTHORS:20210914-164521010
Persistent URL:
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:110842
Deposited By: George Porter
Deposited On:14 Sep 2021 19:32
Last Modified:14 Sep 2021 19:32

Repository Staff Only: item control page