Existence of complex function?

Question

Motivated by a similar question 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.

Answer 1

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.