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

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Abstract

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 \(V\times \mathbf {C}\) along \(H\times0\), 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.

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Authors and Affiliations

  1. Department of Mathematics, Columbia University, New York, NY, 10027, USA

    Mohammed Abouzaid

  2. Department of Mathematics, UC Berkeley, Berkeley, CA, 94720-3840, USA

    Denis Auroux

  3. Fakultät für Mathematik, University of Vienna, Vienna, 1090, Austria

    Ludmil Katzarkov

Authors
  1. Mohammed Abouzaid
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  2. Denis Auroux
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  3. Ludmil Katzarkov
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Corresponding author

Correspondence to Mohammed Abouzaid.

Additional information

The first author was partially supported by a Clay Research Fellowship and by NSF grant DMS-1308179.

The second author was partially supported by NSF grants DMS-1264662 and DMS-1406274 and by a Simons Fellowship.

The third author was partially supported by NSF grants DMS-1201475 and DMS-1265230, FWF grant P24572-N25, and ERC grant GEMIS.

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Abouzaid, M., Auroux, D. & Katzarkov, L. Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces. Publ.math.IHES 123, 199–282 (2016). https://doi.org/10.1007/s10240-016-0081-9

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