 
Summary: MIRROR SYMMETRY AND TDUALITY IN THE COMPLEMENT
OF AN ANTICANONICAL DIVISOR
DENIS AUROUX
Abstract. We study the geometry of complexified moduli spaces of special La
grangian submanifolds in the complement of an anticanonical divisor in a compact
Kšahler manifold. In particular, we explore the connections between Tduality and
mirror symmetry in concrete examples, and show how quantum corrections arise in
this context.
1. Introduction
The StromingerYauZaslow conjecture [26] asserts that the mirror of a CalabiYau
manifold can be constructed by dualizing a fibration by special Lagrangian tori. This
conjecture has been studied extensively, and the works of Fukaya, Kontsevich and
Soibelman, Gross and Siebert, and many others paint a very rich and subtle picture
of mirror symmetry as a Tduality modified by "quantum corrections" [14, 18, 19].
On the other hand, mirror symmetry has been extended to the non CalabiYau set
ting, and in particular to Fano manifolds, by considering LandauGinzburg models,
i.e. noncompact manifolds equipped with a complexvalued function called superpo
tential [16]. Our goal is to understand the connection between mirror symmetry and
Tduality in this setting.
For a toric Fano manifold, the moment map provides a fibration by Lagrangian
