Topological mirror symmetry

摘要

The Strominger-Yau-Zaslow conjecture proposes that mirror symmetry can be explained by the existence in a mirror pair of Calabi-Yau manifolds, of dual special Lagrangian T~n-fibrations. (See [19,8,6,7] for further clarification of this conjecture). Recently, Zharkov in [21] proved that non-singular Calabi-Yau hypersurfaces in toric varieties have topological T~n-fibrations, and Ruan in [17] has shown further that the quintic threefold has a Lagrangian torus fibration. He in addition announced in that paper an extension of these results to arbitrary Calabi-Yau threefolds which are hypersurfaces in toric varieties. This gives initial evidence for the SYZ conjecture.