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Secondary fan, theta functions and moduli of Calabi-Yau pairs

Hacking, Paul and Keel, Sean and Yu, Tony Yue (2020) Secondary fan, theta functions and moduli of Calabi-Yau pairs. . (Unpublished)

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We conjecture that any connected component Q of the moduli space of triples (X,E=E₁+⋯+E_n,Θ) where X is a smooth projective variety, E is a normal crossing anti-canonical divisor with a 0-stratum, every E_i is smooth, and Θ is an ample divisor not containing any 0-stratum of E, is unirational. More precisely: note that Q has a natural embedding into the Kollár-Shepherd-Barron-Alexeev moduli space of stable pairs, we conjecture that its closure admits a finite cover by a complete toric variety. We construct the associated complete toric fan, generalizing the Gelfand-Kapranov-Zelevinski secondary fan for reflexive polytopes. Inspired by mirror symmetry, we speculate a synthetic construction of the universal family over this toric variety, as the Proj of a sheaf of graded algebras with a canonical basis, whose structure constants are given by counts of non-archimedean analytic disks. In the Fano case and under the assumption that the mirror contains a Zariski open torus, we construct the conjectural universal family, generalizing the families of Kapranov-Sturmfels-Zelevinski and Alexeev in the toric case. In the case of del Pezzo surfaces with an anti-canonical cycle of (−1)-curves, we prove the full conjecture.

Item Type:Report or Paper (Discussion Paper)
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URLURL TypeDescription Paper
Yu, Tony Yue0000-0002-6019-8552
Additional Information:We enjoyed fruitful conversations with P. Achinger, V. Alexeev, M. Baker, V. Berkovich, F. Charles, A. Corti, A. Durcos, W. Gubler, J. Kollár, M. Porta, J. Rabinoff, D. Ranganathan, M. Robalo and Y. Soibelman. We were heavily inspired and influenced by our long-term collaborations with M. Gross, M. Kontsevich and B. Siebert. Hacking was supported by NSF grants DMS-1601065 and DMS-1901970. Keel was supported by NSF grant DMS-1561632. T.Y. Yu was supported by the Clay Mathematics Institute as Clay Research Fellow. Some of the research was conducted while Keel and Yu visited the Institute for Advanced Study in Princeton, and some while the three authors visited the Institut des Hautes Études Scientifiques in Bures-sur-Yvette.
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Clay Mathematics InstituteUNSPECIFIED
Classification Code:2020 Mathematics Subject Classification. Primary 14J33; Secondary 14J10, 14J32, 14E30, 14G22
Record Number:CaltechAUTHORS:20210914-164517148
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:110840
Deposited By: George Porter
Deposited On:14 Sep 2021 21:04
Last Modified:14 Sep 2021 21:04

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