Geodesic
Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces
Abouzaid, Mohammed1 ; Auroux, Denis2  ; Katzarkov, Ludmil3 
1 Department of Mathematics, Columbia University 10027 New York NY USA
2 Department of Mathematics, UC Berkeley 94720-3840 Berkeley CA USA
3 Fakultät für Mathematik, University of Vienna 1090 Vienna Austria
Publications Mathématiques de l'IHÉS, Tome 123 (2016), pp. 199-282 Cet article a éte moissonné depuis la source Numdam

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We consider mirror symmetry for (essentially arbitrary) hypersurfaces in (possibly noncompact) toric varieties from the perspective of the Strominger-Yau-Zaslow (SYZ) conjecture. Given a hypersurface H in a toric variety V we construct a Landau-Ginzburg model which is SYZ mirror to the blowup of V×C along H×0, under a positivity assumption. This construction also yields SYZ mirrors to affine conic bundles, as well as a Landau-Ginzburg model which can be naturally viewed as a mirror to H. The main applications concern affine hypersurfaces of general type, for which our results provide a geometric basis for various mirror symmetry statements that appear in the recent literature. We also obtain analogous results for complete intersections.

DOI : 10.1007/s10240-016-0081-9
Keywords: Modulus Space, Toric Variety, Exceptional Divisor, Maslov Index, Instanton Correction

Abouzaid, Mohammed 1 ; Auroux, Denis 2 ; Katzarkov, Ludmil 3

1 Department of Mathematics, Columbia University 10027 New York NY USA
2 Department of Mathematics, UC Berkeley 94720-3840 Berkeley CA USA
3 Fakultät für Mathematik, University of Vienna 1090 Vienna Austria 
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     author = {Abouzaid, Mohammed and Auroux, Denis and Katzarkov, Ludmil},
     title = {Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {199--282},
     year = {2016},
     publisher = {Springer Berlin Heidelberg},
     address = {Berlin/Heidelberg},
     volume = {123},
     doi = {10.1007/s10240-016-0081-9},
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     mrnumber = {3502098},
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Abouzaid, Mohammed; Auroux, Denis; Katzarkov, Ludmil. Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces. Publications Mathématiques de l'IHÉS, Tome 123 (2016), pp. 199-282. doi: 10.1007/s10240-016-0081-9

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