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Non-archimedean quantum K-invariants

Porta, Mauro and Yu, Tony Yue (2020) Non-archimedean quantum K-invariants. . (Unpublished)

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We construct quantum K-invariants in non-archimedean analytic geometry. Our approach differs from the classical one in algebraic geometry via perfect obstruction theory. Instead, we build on our previous works on the foundation of derived non-archimedean geometry and Gromov compactness. We obtain a list of natural geometric relations of the stacks of stable maps, directly at the derived level, with respect to elementary operations on graphs, namely, products, cutting edges, forgetting tails and contracting edges. They imply immediately the corresponding properties of K-theoretic invariants. The derived approach produces highly intuitive statements and functorial proofs, without the need to manipulate perfect obstruction theories. The flexibility of our derived approach to quantum K-invariants allows us to impose not only simple incidence conditions for marked points, but also incidence conditions with multiplicities, which we discuss in the final section of the paper. This leads to a new set of enumerative invariants, which is not yet considered in the literature, to the best of our knowledge. For the proofs, we had to further develop the foundations of derived non-archimedean geometry in this paper: we study derived lci morphisms, relative analytification, and deformation to the normal bundle. Our motivations come from non-archimedean enumerative geometry and mirror symmetry.

Item Type:Report or Paper (Discussion Paper)
Related URLs:
URLURL TypeDescription Paper
Porta, Mauro0000-0002-1239-3409
Yu, Tony Yue0000-0002-6019-8552
Additional Information:We are very grateful to Federico Binda, Antoine Chambert-Loir, Benjamin Hennion, Maxim Kontsevich, Gérard Laumon, Y.P. Lee, Jacob Lurie, Etienne Mann, Tony Pantev, Francesco Sala, Carlos Simpson, Georg Tamme, Bertrand Toën and Gabriele Vezzosi for valuable discussions. We would like to thank Marco Robalo in particular for many detailed discussions and for his enthusiasm for our work. The authors would also like to thank each other for the joint effort. This research was partially conducted during the period when T.Y. Yu served as a Clay Research Fellow. We have also received supports from the National Science Foundation under Grant No. 1440140, while we were in residence at the Mathematical Sciences Research Institute in Berkeley, California, and from the Agence Nationale de la Recherce under Grant ANR-17-CE40-0014, while we were at the Université Paris-Sud in Orsay, France.
Funding AgencyGrant Number
Clay Mathematics InstituteUNSPECIFIED
Agence Nationale de la Recherche (ANR)ANR-17-CE40-0014
Subject Keywords:Quantum K-invariant, Gromov-Witten, non-archimedean geometry, rigid analytic geometry
Classification Code:2020 Mathematics Subject Classification. Primary 14N35; Secondary 14A30, 14G22, 14D23
Record Number:CaltechAUTHORS:20210914-164513718
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:110839
Deposited By: George Porter
Deposited On:14 Sep 2021 19:38
Last Modified:14 Sep 2021 19:38

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