Geodesic
Mirror symmetry for log Calabi-Yau surfaces I
Gross, Mark1  ; Hacking, Paul2  ; Keel, Sean3
1 DPMMS, Centre for Mathematical Sciences Wilberforce Road CB3 0WB Cambridge UK
2 Department of Mathematics and Statistics, Lederle Graduate Research Tower, University of Massachusetts 01003-9305 Amherst MA USA
3 Department of Mathematics 1 University Station C1200 78712-0257 Austin TX USA
Publications Mathématiques de l'IHÉS, Tome 122 (2015), pp. 65-168

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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).

DOI : 10.1007/s10240-015-0073-1
Keywords: Break Line, Irreducible Component, Theta Function, Toric Variety, Singular Locus

Gross, Mark 1 ; Hacking, Paul 2 ; Keel, Sean 3

1 DPMMS, Centre for Mathematical Sciences Wilberforce Road CB3 0WB Cambridge UK
2 Department of Mathematics and Statistics, Lederle Graduate Research Tower, University of Massachusetts 01003-9305 Amherst MA USA
3 Department of Mathematics 1 University Station C1200 78712-0257 Austin TX USA 
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     title = {Mirror symmetry for log {Calabi-Yau} surfaces {I}},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {65--168},
     publisher = {Springer Berlin Heidelberg},
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     doi = {10.1007/s10240-015-0073-1},
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Gross, Mark; Hacking, Paul; Keel, Sean. Mirror symmetry for log Calabi-Yau surfaces I. Publications Mathématiques de l'IHÉS, Tome 122 (2015), pp. 65-168. doi: 10.1007/s10240-015-0073-1

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