How to show a holomorphic function in several variables that vanishes at some points vanishes identically?

Question

Let $D\subset \mathbb{C}^n$ be connected and suppose $f:D\times D\to\mathbb{C}$ is holomorphic in the $2n$ complex variables $(z,w)\in D\times D$. If there is a point $p\in D$ with $\bar{p}\in D$, such that $f(z,\bar{z})=0$ for all $z$ in a neighborhood of $p$, how to show $f(z,w)=0$ for all $(z,w)\in D\times D$?

Answer 1

What you need is to prove that $f(z,w)=0$ in some neighborhood of a specific point $(z_0,\bar{z}_0)$. The best way to do this is to realize that the complex derivative is equal to the real derivative if it exists, and all the complex derivatives can be computed along directions of the set $X = \{ (z,w) : w = \bar{z} \}$.

Think about $(z,w) \in {\mathbb C}^2$ for simplicity. Maybe easiest way is to parametrize $X$ by ${\mathbb R}^2$ as $(x,y) \mapsto (x+iy,x-iy)$. Complexifying this mapping (making $x$ and $y$ complex) will be a biholomorphism to all of ${\mathbb C}^2$ with ${\mathbb C}^2$ taking ${\mathbb R}^2$ to $X$. Now you just have to prove that if a power series $\sum c_{jk} x^jy^k = 0$ for real $x$ and $y$, then it is zero for complex $x$ and $y$, and you do this by computing the derivatives, as the cofficients of the series are gotten by real derivatives in $x$ and $y$.

I will let you fill in the details.

The main point is that $X$ is a generic manifold, that is, $T_p X + J T_p X = T_p {\mathbb C}^{2n}$, where $J$ is the complex structure. Very vaguely, this means that you only need (the real) derivatives from $T_p X$ to compute all the derivatives from $T_p {\mathbb C}^{2n}$. So if a function is holomorphic and vanishes on the $T_p X$ it must also vanish on $J T_p X$, and thus all derivatives must vanish. Any holomorphic function vanishing on a generic manifold must be zero identically.