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Suppose we have a real-valued smooth function on a complex torus: $$f: \mathbb{C}^n/(\mathbb{Z}+\sqrt{-1}\mathbb{Z})^n\longrightarrow\mathbb{R},$$ i.e., this $f$ is a real-valued smooth function on $\mathbb{C}^n$ which is invariant under the tranfsormation of the lattice $(\mathbb{Z}+\sqrt{-1}\mathbb{Z})^n$. If the complex Hessian matrix $(\partial_i\partial_{\bar{j}}f)=(\frac{\partial^2f}{\partial z^i\partial\bar{z^j}})$ is a constant valued matrix, could we conclude that this $f$ itself is a constant?

Many thanks in advances!

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  • $\begingroup$ If $\partial \bar \partial f$ is constant, then for any complex line $L\subset \mathbb C^n$, $f|_L$ of $-f|_L$ is subharmonic and bounded on $L\simeq \mathbb C$, hence constant by Liouville theorem. This easily implies that $f$ is constant on $\mathbb C^n$. $\endgroup$
    – Henri
    Commented Sep 11, 2017 at 15:16

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Let me copy my remark to close the topic.

If $\partial \bar \partial f$ is a constant matrix on $\mathbb C^n$, then it follows that for any complex line $L\subset \mathbb C^n$, $\Delta (f|_L)$ is constant, where $\Delta$ is the Laplacian on $L\simeq \mathbb C$ with respect to the restriction of the euclidean metric. In particular, either $f|–L$ or $-f|_L$ is subharmonic. As it is bounded as well, Liouville theorem implies that $f|_L$ is constant. Obviously, one can sweep $\mathbb C^n$ with complex lines intersecting at a given point, so that $f$ has to be constant.

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