Remarks on the homological mirror symmetry for tori

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Abstract: Let us consider an

n

-dimensional complex torus

T_{J = T}^{2 n} : = m a t h b b C^{n} / 2 p i (m a t h b b Z^{n} o p l u s T m a t h b b Z^{n})

. 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

T_{J = T}^{2}

is defined by using

- f r a c 1 T

or

T

. However, if we assume

n g e q 2

and that

T

is a singular matrix, we can not define a mirror partner of

T_{J = T}^{2 n}

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.

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