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.Comment: 83 pages; v2: added appendix discussing the analytic structure on
moduli of objects in the Fukaya category; v3: further clarifications in
response to referee report; v4: further clarifications throughout, especially
sections 4 and 7 and appendix A; added appendix B on the geometry of reduced
space