Holomorphic coordinates on a Kähler manifold

Question

Let $(X,J,\omega)$ be a Kähler manifold. Let $\dim_{\mathbb{R}}(X)=2n$ and we also know that $X$ splits as $X=M\times \mathbb{R}$, where $\dim_{\mathbb{M}}=2n-1$. My question is now: does there exists coordinates $(x_{1},...,x_{n},y_{1},...,y_{n-1},t)$, where $t$ is the global coordinate on $\mathbb{R}$, such that $(z_{1},...,z_{n})$, with $z_{j}=x_{j}+iy_{j},j=1,...,n-1$ and $z_{n}=x_{n}+it$, are holomorphic coordinates?

I considered the global defined vector field $\frac{\partial}{\partial t}$, which never vanishes on $X$ and then look at $J\frac{\partial}{\partial t}$ which also never vanishes on $X$. Then for some point in $X$ choose a complement complex frame i.e. $\{ X_{1},...,X_{n-1},Y_{1},...,Y_{n-1}\}$, with the property that $JX_{i}=Y_{i}$ and $JY_{i}=-X_{i}$. Does this somehow give raise to my desired coordinates somehow? If yes, how does the argument go on? If no, is it even possible to do this?

Greetings Nina

Answer 1

The question makes sense only locally, in which case you are effectively asking the following

Question Q: Let $h: {\mathbb C}^n\to {\mathbb R}$ be a smooth function defined near $0$, which has $0$ as its regular point. Under what conditions there exists a complex-analytic function $f: {\mathbb C}^n\to {\mathbb C}$ (again, defined near $0$), so that $h$ is the real part of $f$. Note that such function $f$ is automatically regular at $0$, i.e., $df(0)\ne 0$. Given this, one can find a locally (near $0$) defined biholomorphic function $F: {\mathbb C}^n\to {\mathbb C}^n$ so that $f$ is the last component of $F$. (This is just the implicit function theorem for holomorphic functions.)

The answer to Question Q is classical and could be found in any book on several complex variables:

Such $f$ exists if and only if $h$ is pluriharmonic.

A randomly chose real analytic function, of course, is not going to be pluriharmonic. Just think about complex 1-dimensional case, where pluriharmonicity is equivalent to harmonicity.