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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 $n=2$.

Thanks for reading and any comments.

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    $\begingroup$ Catanese proved in early 00s that any complex deformation of a flat torus is a flat torus. This means that if your classifying map is nullhomotopic, it is necessarily constant. $\endgroup$
    – Denis T
    Apr 22, 2023 at 15:34
  • $\begingroup$ Unfortunately, the phrase "complex structure" sometimes means the structure of a complex manifold, and sometimes means an "almost complex structure" — which is defined as an automorphism J of a manifold's tangent bundle such that J^2 = -I. $\endgroup$ Apr 24, 2023 at 3:29

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