Project description
A non-archimedean approach to mirror symmetry
Mirror symmetry is one of the most mysterious dualities in mathematics. Roughly, it predicts that given any Calabi-Yau variety, there exists a mirror Calabi-Yau variety such that a rich list of geometric relations holds between the two. Despite continual progress in the subject, a fundamental question remains unclear: the extent to which mirrors exist, and how to construct the mirror variety. The EU-funded NAMirror project proposes a new approach to answer this question, based on latest developments from non- archimedean geometry, in particular the theory of Berkovich spaces, as well as derived non-archimedean geometry. The goal is to conceive and pursue a full-fledged theory of non-archimedean mirror symmetry, which will lead to new results unattainable from existing methods.
Objective
Mirror symmetry is one of the most mysterious dualities in mathematics. Roughly, it predicts that given any Calabi-Yau variety, there exists a mirror Calabi-Yau variety such that a rich list of geometric relations hold between the two, involving Hodge numbers, Gromov-Witten invariants, variation of Hodge structures, Floer homology (Fukaya category), coherent sheaves, stability conditions and so on. Despite continual progress in the subject, a fundamental question remains unclear: to what extent do mirrors exist, and how to construct the mirror variety?
Here we propose a new approach to answer this question, based on latest developments from non-archimedean geometry, in particular the theory of Berkovich spaces, as well as derived non-archimedean geometry. Our goal is to conceive and pursue a full-fledged theory of non-archimedean mirror symmetry, which will lead to new results unattainable from existing methods.
We propose to work out a general mirror construction, starting directly from a non-archimedean Strominger-Yau-Zaslow torus fibration, conjectured by Kontsevich-Soibelman, by counting non-archimedean analytic disks with boundaries on SYZ torus fibers. First we need to establish the existence of such counts in full generality, based on non-archimedean Gromov-Witten theory and tail conditions. Then we have to prove various properties of the mirror algebra, including associativity, radius of convergence and singularity estimates. Finally we propose to use wall-crossing formulas to glue local mirror algebras together to obtain the global mirror variety. A long-term goal is to show that the mirror construction is an involution, the best exhibition of mirror duality.
We also aim for applications outside mirror symmetry, in particular towards the moduli of KSBA stable pairs in birational geometry. Our project is intimately related to the ongoing Gross-Siebert program based on logarithmic geometry. We also expect fruitful future interactions with their program.
Fields of science (EuroSciVoc)
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CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: The European Science Vocabulary.
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Keywords
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Project’s keywords as indicated by the project coordinator. Not to be confused with the EuroSciVoc taxonomy (Fields of science)
Programme(s)
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Multi-annual funding programmes that define the EU’s priorities for research and innovation.
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H2020-EU.1.1. - EXCELLENT SCIENCE - European Research Council (ERC)
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Calls for proposals are divided into topics. A topic defines a specific subject or area for which applicants can submit proposals. The description of a topic comprises its specific scope and the expected impact of the funded project.
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ERC-STG - Starting Grant
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Call for proposal
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(opens in new window) ERC-2020-STG
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75794 PARIS
France
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