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



Examples of non-Kahler surfaces with explicit non-Kahler metric
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.


