A nonzero holomorphic function on ${\mathbb{C}}^2$ whose zero set is an unbounded hyperplane

The function $f(z_0,z_1)=z_0$ is holomorphic on ${\mathbb{C}}^2$ and is not identically zero, since $f(1,0)=1$.

Its zero set is exactly $\{(z_0,z_1)\in {\mathbb{C}}^2:z_0=0\}=\{0\}\times \mathbb{C}$.

This zero set is unbounded, because $(0,t)$ belongs to it for every complex $t$, and no point of it is isolated in it: if $(0,w)$ is a zero and $\epsilon >0$, then $(0,w+\epsilon \mathrm{/}2)$ is a different zero within Euclidean distance $\epsilon \mathrm{/}2$.

The zero set has empty interior, so this witness does not contradict [L2]; it shows instead that in several variables a nonzero holomorphic function can vanish on a whole positive-dimensional complex hyperplane. Therefore the refuted claim fails.