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Mirror symmetry for log Calabi-Yau surfaces I

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arxiv 1106.4977 v3 pith:HEYTQFRH submitted 2011-06-24 math.AG

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keywords curvescuspfamilymirrorrationalalgebraanti-canonicalcalabi-yau
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We give a canonical synthetic construction of the mirror family to a pair (Y,D) of a smooth projective surface with an anti-canonical cycle of rational curves, as the spectrum of an explicit algebra defined in terms of counts of rational curves on Y meeting D in a single point. In the case D is contractible, the family gives a smoothing of the dual cusp, and thus a proof of Looijenga's 1981 cusp conjecture.

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

    math.AG 2019-08 accept novelty 8.0 of 10

    Naive counts of rational curves in an affine log Calabi-Yau variety containing a torus uniquely determine a Frobenius algebra, now proved via non-archimedean analytic disk counting.

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