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SPECIAL LAGRANGIAN FIBRATIONS, MIRROR SYMMETRY AND CALABI-YAU DOUBLE COVERS
 

Summary: SPECIAL LAGRANGIAN FIBRATIONS, MIRROR SYMMETRY
AND CALABI-YAU DOUBLE COVERS
DENIS AUROUX
Abstract. The first part of this paper is a review of the Strominger-Yau-Zaslow
conjecture in various settings. In particular, we summarize how, given a pair (X, D)
consisting of a Kšahler manifold and an anticanonical divisor, families of special La-
grangian tori in X \ D and weighted counts of holomorphic discs in X can be used
to build a Landau-Ginzburg model mirror to X. In the second part we turn to
more speculative considerations about Calabi-Yau manifolds with holomorphic in-
volutions and their quotients. Namely, given a hypersurface H representing twice
the anticanonical class in a Kšahler manifold X, we attempt to relate special La-
grangian fibrations on X \ H and on the (Calabi-Yau) double cover of X branched
along H; unfortunately, the implications for mirror symmetry are far from clear.
To Jean-Pierre Bourguignon on his 60th birthday, with my most sincere
gratitude for the time he spent guiding me through the process of becoming
a mathematician.
1. Introduction
The phenomenon of mirror symmetry was first evidenced for Calabi-Yau manifolds,
i.e. Kšahler manifolds with holomorphically trivial canonical bundle. Subsequently it
became apparent that mirror symmetry also holds in a more general setting, if one

  

Source: Auroux, Denis - Department of Mathematics, Massachusetts Institute of Technology (MIT)

 

Collections: Mathematics