Question

Hi,everyone.Can someone give me some examples of non-Kahler surfaces whose complex structure and metric structure are all clear?

Answer 1

If your surface is fairly explicit you can obtain an explicit hermitian metric on it as well. For example, if we take Francesco's Hopf surface $X$, then a hermitian metric $\omega$ on $X$ can be identified with a $G$-invariant hermitian metric on $\mathbb C^2 \setminus {0}$.

To simplify, let's look at the special case where the group $G$ is generated by $\alpha_1 = \alpha_2 = \alpha$. Let $\omega$ be a hermitian metric on $M = \mathbb C^2 \setminus {0}$. Then we can identify $\omega$ with its coefficient matrix

$$ \Omega = \begin{pmatrix} a & b \cr - \bar b & d \end{pmatrix}, $$

where $a$ and $d$ are smooth real-valued positive functions on $M$, and $b$ is smooth. Looking at the action of the pullback by $(z,w) \mapsto (\alpha z, \alpha w)$ on $\omega$ we find that to be $G$-invariant, the function $a$ has to satisfy the identity

$$ a(\alpha z, \alpha w) = \frac 1{|\alpha|^2} a(z,w).$$

In fact, one finds that $b$ and $d$ have to satisfy the same identity as well.

To construct such functions, let's take $b = 0$ and $d = a$ to simplify. Guesswork, or writing out a Laurent series in $z, \bar z, w, \bar w$ and calculating, leads us to consider $a(z,w) = 1/(|z|^2 + |w|^2)$. Then some quick calculations show that

$$ \omega(z,w) = \frac i2 \frac 1{|z|^2 + |w|^2} ( d z \wedge d \bar z + d w \wedge d \bar w) $$

is a $G$-invariant hermitian metric on $\mathbb C^2 \setminus {0}$, so it gives a hermitian metrix on $X$.

You can probably work out a hermitian metric in the general case by similar methods, but finding the $G$-invariant functions on $M$ is probably slightly more work.

Answer 2

Some standard examples are Hopf surfaces, obtained as quotients of the form

$X=\mathbb{C}^2 - \{0\} /G$,

where $G$ is the subgroup generated by the homothety

$(z_1, z_2) \to (\alpha_1 z_1, \alpha_2 z_2)$,

with $0 < |\alpha_1| \leq |\alpha_2| <1$.

$X$ is a compact complex surface, whose complex structure is induced by the standard one in $\mathbb{C}^2 - \{0\}$. Moreover, one proves that $X$ is diffeomorphic to $S^1 \times S^3$, therefore it does not admit any Kaeler metric.

For further details, see Chapter V of the book "Compact complex surfaces" by Barth-Peters -Van de Ven.

Answer 3

The Kahler condition is a local condition on your metric. Namely, if $J$ is complex multiplication and $g$ is a $J$-invariant metric, define a $2$-form by $\omega(u,v) = g(u, Jv)$. The Kahler condition is that $d \omega=0$.

So, you should be satisfied with local examples. For example, take coordinates $z_1 = x_1+ i y_1$ and $z_2 = x_2 + i y_2$ on $\mathbb{C}^2$. Let $$g = x_2 (dx_1^2 + dy_1^2) + x_1 (dx_2^2 + dy_2^2).$$ This is a metric on the open chart where $x_1$, $x_2 >0$.

This is basically the simplest possible example of a non-Kahler metric. A good exercise is to check that the various Kahler identities do not hold on it.