Remarks on the homological mirror symmetry for tori

Abstract

Let us consider an n-dimensional complex torus TJ=T2n≔ℂn∕2π(Zn⊕TZn). Here, T is a complex matrix of order n whose imaginary part is positive definite. In particular, when we consider the case n=1, the complexified symplectic form of a mirror partner of TJ=T2 is defined by using −1T or T. However, if we assume n≥2 and that T is a singular matrix, we cannot define a mirror partner of TJ=T2n as a natural generalization of the case n=1 to the higher dimensional case. In this paper, we propose a way to avoid this problem, and discuss the homological mirror symmetry.