# Complex structure on the product of two real torus

Let $$T^{2n}_{\mathbb{R}}$$ be a real torus of dimension $$2n$$, and let $$Z_n$$ be the space consisting of all possible complex structures on $$T^{2n}_{\mathbb{R}}$$. It is known that:

$$Z_n = \mathrm{GL}(2n,\mathbb{Z})\backslash\mathrm{GL}(2n,\mathbb{R})/\mathrm{GL}(n,\mathbb{C})$$

Let $$E$$ be a given elliptic curve, and let $$M_n = E \times T^{2n}_{\mathbb{R}}$$. Take $$f:E \rightarrow Z_n$$ to be a smooth map. Then the expression

$$J := J_E + J_{f(y_1)}$$

evaluated at $$(y_1,y_2)\in M$$ gives an almost complex structure on $$M_n$$. I am wondering when $$J$$ is a complex structure.

I guess that $$J$$ is integrable if and only if $$f$$ is holomorphic in some sense. For example, when $$n=1$$, by the classification of complex surfaces, the only complex structure on $$M_n$$ is the complex structure of a complex torus. Here is a possible explanation: the holomorphic map $$f:E \rightarrow Z_1 = \mathrm{SL}(2,\mathbb{R})/\mathrm{SL}(2,\mathbb{Z}) \in \mathbb{C}$$ can only be constant, and therefore $$J$$ can only be the complex structure of a complex torus. (I don't know whether this is a reasonable explanation. If it is not, please let me know.)

When $$n \geqslant 2$$, $$Z_n$$ may not be a complex space, and I don't know how to understand the integrability of $$J$$ as a property of $$f$$. I am particularly interested in $$n=2$$.