TITLE: "The complex version of Nash's theorem is not true"
QUESTION [29 upvotes]: The title quote is from p.221 of the 2010 book,
The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions
by Shing-Tung Yau and Steve Nadis. "Nash's theorem" here refers to the Nash embedding theorem
(discussed in an earlier MO question: "Nash embedding theorem for 2D manifolds").
I would appreciate any pointer to literature that explains where and why embedding fails.
This is well-known in the right circles, but I am having difficulty locating sources.  Thanks!

REPLY [10 votes]: In his 1952 Annals paper, Calabi characterised those Kähler manifolds $(X,\omega)$ which admit an isometric embedding into a complex space form. Here, a complex space form is a Kähler manifold of constant holomorphic sectional curvature which is complete with respect to the metric, hence either complex Euclidean space, complex hyperbolic space or complex projective space.
Here, we describe the case of a Kähler manifold $(X,\omega)$ admitting an isometric embedding into complex Euclidean space $\mathbb{C}^N$ for some $N$, as requested. 
Firstly, the Kähler form $\omega$ has to be real-analytic. 
Associated to a real-analytic Kähler manifold $(X,\omega)$, Calabi introduced a diastasis function $D_\omega$ (coming from the Greek word for distance) in order to describe his characterization. This function $D_\omega$ is a partially defined on on $X\times X$ and is given locally as follows: Let a point $u \in X$ be given and choose a local real-analytic Kähler potential $\Phi : U \to \mathbb{R}$ defined in an open neighbourhood $U$ of $u$ in $X$ (i.e. $\sqrt{-1} \partial\bar{\partial} \Phi = \omega$ on $U$). Let $\tilde{\Phi} : U \times \overline{U} \to \mathbb{C}$ denote a polarization of $\Phi$ (i.e. $\tilde{\Phi}$ is a holomorphic function on $U \times \overline{U}$ satisfying $\tilde{\Phi}(x, \overline{x}) = \Phi(x)$ for all $x\in U$, where $\overline{U}$ is the conjugate complex manifold of $U$). Calabi defined
$$D_\omega(x , y) := \tilde{\Phi}(x, \bar{x}) + \tilde{\Phi}(y, \bar{y}) - \tilde{\Phi}(x, \bar{y}) - \tilde{\Phi}(y, \bar{x}) \quad {\text{for }} (x,y)\in U\times U,$$
and showed that the right-hand side is independent of the choice of Kähler potential $\Phi$.
The diastasis function detects isometries, in the sense that a holomorphic map $f : (X, \omega) \to (X^\prime, \omega^\prime) $ between real-analytic Kähler manifolds is an local isometry near a point $x \in X$ if and only if 
$$D_{\omega^\prime}(f(x), f(y)) = D_\omega (x, y) \quad \text{for all } y\in U$$
in some open neighbourhood $U$ of $x$ in $X$.
Since the diastasis of the flat metric $\omega_0 $on $\mathbb{C}^N$ from the origin is given by 
$$D_{\omega_0}(z, 0) = |z_1|^2 + \cdots + |z_N|^2 \quad \text{for } z\in \mathbb{C}^N,$$ 
this leads to the following result. 

Theorem [Calabi]. A real-analytic Kähler manifold $(X,\omega)$ admits an local isometric map near a point $x \in X$ to complex Euclidean space $\mathbb{C}^N$ if and only if there exists local holomorphic functions $f_1,\ldots, f_N$ defined near $x$ such that
  $$D_\omega(x, y) = |f_1(y)|^2 + \cdots + |f_N(y)|^2.$$
  for points $y$ in some open neighbourhood of $x$. 

Regarding embedding (i.e. injective immersion), which is a global question, other than checking the above criterion at some point $x\in X$, there is an added condition, namely that the diastasis function $D_\omega$ extends to a total function on $X\times X$ such that $D_\omega (x, y) \ge 0$ for all $(x, y)\in X\times X$ and equality holds if and only if $x = y$. This added condition ensures injectivity.
Finally, one should note the perhaps surprising observation that if a (connected) real-analytic Kähler manifold $(X,\omega)$  admits an local isometric map near a point $x \in X$ to some complex Euclidean space $\mathbb{C}^N$, then it admits a local isometric map near every point $x \in X$ to $\mathbb{C}^N$, as Calabi showed. This explains why to check the existing of an embedding, it is sufficient to check the sum of squares condition in the highlighted block at some point $x\in X$ plus the global positive-valued extension condition on the diastasis function $D_\omega$.