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

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 realvalued 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. 


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 BarthPeters Van de Ven. 


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 nonKahler metric. A good exercise is to check that the various Kahler identities do not hold on it. 

