Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces

Abouzaid, Mohammed; Auroux, Denis; Katzarkov, Ludmil

doi:10.1007/s10240-016-0081-9

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

Publications Mathématiques de l'IHÉS, Tome 123 (2016), pp. 199-282.

Résumé

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 $V \times C$ along $H \times 0$ , 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.

MR Zbl

DOI : 10.1007/s10240-016-0081-9

Mots clés : Modulus Space, Toric Variety, Exceptional Divisor, Maslov Index, Instanton Correction

Affiliations des auteurs :

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

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     title = {Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {199--282},
     publisher = {Springer Berlin Heidelberg},
     address = {Berlin/Heidelberg},
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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. http://www.numdam.org/articles/10.1007/s10240-016-0081-9/

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