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Mirror symmetry for log Calabi-Yau surfaces I

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Abstract

We give a canonical synthetic construction of the mirror family to pairs (Y,D) where Y is a smooth projective surface and D is an anti-canonical cycle of rational curves. This mirror family is constructed as the spectrum of an explicit algebra structure on a vector space with canonical basis and multiplication rule defined in terms of counts of rational curves on Y meeting D in a single point. The elements of the canonical basis are called theta functions. Their construction depends crucially on the Gromov-Witten theory of the pair (Y,D).

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Authors and Affiliations

  1. DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 0WB, UK

    Mark Gross

  2. Department of Mathematics and Statistics, Lederle Graduate Research Tower, University of Massachusetts, Amherst, MA, 01003-9305, USA

    Paul Hacking

  3. Department of Mathematics, 1 University Station C1200, Austin, TX, 78712-0257, USA

    Sean Keel

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  1. Mark Gross
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  2. Paul Hacking
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  3. Sean Keel
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Correspondence to Mark Gross.

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Gross, M., Hacking, P. & Keel, S. Mirror symmetry for log Calabi-Yau surfaces I. Publ.math.IHES 122, 65–168 (2015). https://doi.org/10.1007/s10240-015-0073-1

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  • DOI: https://doi.org/10.1007/s10240-015-0073-1

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