Motivated by a similar question [Complex-doubly periodic function in two variables?](https://mathoverflow.net/questions/382480/complex-doubly-periodic-function-in-two-variables), I would like to ask if there exists a non-zero function $(z\_1,z\_2) \mapsto f(z\_1,z\_2)$, where $z\_1,z\_2 \in \mathbb C$ are two complex variables, that satisfies
$$ (\partial\_{z\_2} + \partial\_{z\_1})f =0 \text{ and } (\partial\_{\bar z\_1} - \partial\_{\bar z\_2})f=0.$$


Notice that such a function $f$ is necessarily harmonic.
$$ (\partial\_{\bar z\_2} \partial\_{z\_2} + \partial\_{\bar z\_1} \partial\_{z\_1} )f=0.$$


In addition, I require for $k\_1,k\_2 \in \mathbb R$ the periodicity conditions
\begin{gather\*}
f (z\_1+1,z\_2 ) = f(z\_1,z\_2 ), \quad f (z\_1+i,z\_2 ) = f(z\_1,z\_2 ), \\
\text{and}\quad
f (z\_1,z\_2+1 ) = e^{ik\_1} f(z\_1,z\_2), \quad f (z\_1,z\_2+i ) = e^{ik\_2} f(z\_1,z\_2).
\end{gather\*}


Such a function $f$ must necessarily have poles, unless $f$ is constant, so what I am asking here is if there exists a function $f$ that satisfies the above differential equations up to a set of lower dimension where the functions exhibits poles and in addition all the periodicity conditions.



Existence of complex function?
The answer is 'yes' there do exist such functions that are non-constant with singularities only along surfaces $\Sigma\subset\mathbb{C}^2$, and here is how one can understand them:


First, it helps to change coordinates, though, perhaps, a little more subtly than Fedor Petrov suggested: Let
$$
y\_1 = \tfrac i2({\overline z}\_1+{\overline z}\_2 - z\_1 + z\_2)
\quad\text{and}\quad
y\_2 = \tfrac12({\overline z}\_1+{\overline z}\_2 + z\_1 - z\_2)
$$
Then $(y\_1,y\_2):\mathbb{C}^2\to\mathbb{C}^2$ is a diffeomorphism. Using the complex structure on $\mathbb{C}^2$ for which $y\_1$ and $y\_2$ are holomorphic coordinates, we find that
$$
\frac{\partial f}{\partial{\overline y}\_1} = \frac{\partial f}{\partial{\overline y}\_2} = 0,
$$
so that $f$ is a holomorphic function of $y\_1$ and $y\_2\,$. Moreover, we have the periodicity relations
$$
f(y\_1{+}1,y\_2)= f(y\_1,y\_2{+}1) = f(y\_1,y\_2)
$$
while
$$
f(y\_1{+}i,y\_2)= \mathrm{e}^{ik\_1}f(y\_1,y\_2)
\quad\text{and}\quad 
f(y\_1,y\_2{+}i) = \mathrm{e}^{-ik\_2}f(y\_1,y\_2)
$$


Now, one can construct the general meromorphic $f$ that satisfies these condition as follows:


First, let $p\_1$ and $p\_2$ be (not-identically-vanishing) meromorphic functions on $\mathbb{C}$ that satisfy the period relations
$$
p\_1(y+1) = p\_1(y)\quad\text{and}\quad
p\_1(y+i) = \mathrm{e}^{ik\_1}\,p\_1(y)
$$
and
$$
p\_2(y+1) = p\_2(y)\quad\text{and}\quad
p\_2(y+i) = \mathrm{e}^{-ik\_2}\,p\_2(y).
$$
Techniques for constructing meromorphic functions on $\mathbb{C}$ satisfying such double-periodicity relations are well-known, using $\vartheta$-series or the theory of elliptic curves. The point is that, if $C = \mathbb{C}/\mathbb{Z}[i]$ is the square torus (which is an elliptic curve), then the above relations on meromorphic functions on $\mathbb{C}$ essentially describe the meromorphic sections of two flat complex line bundles $E\_1$ and $E\_2$ over $C$, i.e., elements of $\mathrm{Pic}\_0(C)$.


Given $p\_1$ and $p\_2$, then the general meromorphic $f$ that satisfies the above conditions can be written as a product
$$
f(y\_1,y\_2) = p\_0(y\_1,y\_2)\,p\_1(y\_1)\,p\_2(y\_2),
$$
where $p\_0$ comes from any meromorphic function on $C\times C$, i.e., $p\_0$ is a meromorphic function on $\mathbb{C}^2$ that satisfies
$$
p\_0(y\_1{+}m,y\_2)= p\_0(y\_1,y\_2{+}m) = p\_0(y\_1,y\_2)
$$
for any Gaussian integer $m\in\mathbb{Z}[i]$.


Note that the 'singularities' of $f$ occur along surfaces in $\mathbb{C}^2$ that are complex curves in the $y$-coordinates whose images in $C\times C$ are algebraic curves.


