Special Lagrangian Tori on a Borcea-voisin Threefold

Borcea-Voisin threefolds are Calabi-Yau manifolds. They are constructed by Borcea ([B]) and Voisin ([V]) in the construction of mirror manifolds. In [SYZ], Strominger, Yau and Zaslow propose a geometric construction of mirror manifolds using special Lagrangian tori (called SYZ-construction below). Using degenerate Calabi-Yau metrics Gross and Wilson show that SYZ-construction works for any Borcea-Voisin threefolds ([GW]). In this short note we show that there are special Lagrangian tori on one family of Borcea-Voisin threefolds with respect to non-degenerate Calabi-Yau metrics. These tori cover a large part of the threefolds and are perturbations of the special Lagrangian tori used by Gross and Wilson in [GW]. The method is studying the degeneration of Calabi-Yau metrics using gluing (type I degeneration). There are other examples of special Lagrangian torus in compact Calabi-Yau threefolds. Bryant has a beautiful construction of the special Lagrangian torus in some quintic threefolds (see [Br]). In section 2 we give a family of special Lagrangian submanifolds which cover KCPn .