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

2012/04/30 by Abouzaid, Mohammed, Auroux, Denis, Katzarkov, Ludmil · 4 citations
#Algebraic Geometry (math.AG) #FOS: Mathematics #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.1205.0053

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×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.

Cited by

• Plumbings of lens spaces and crepant resolutions of compound A_n singularities
• Singularities of Landau-Ginzburg models for complete intersections and derived categories
• Lagrangian Floer theory, from geometry to algebra and back again
• Semi-orthogonality in Fukaya-Seidel mirrors to blowups of abelian varieties

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