 
Summary: SPECIAL LAGRANGIAN FIBRATIONS, MIRROR SYMMETRY
AND CALABIYAU DOUBLE COVERS
DENIS AUROUX
Abstract. The first part of this paper is a review of the StromingerYauZaslow
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 LandauGinzburg model mirror to X. In the second part we turn to
more speculative considerations about CalabiYau 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 (CalabiYau) double cover of X branched
along H; unfortunately, the implications for mirror symmetry are far from clear.
To JeanPierre 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 CalabiYau 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
