3 Changed the \\ to \cr for the matrix to display properly.; deleted 104 characters in body

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(The latex doesn't seem to support "pmatrix" or "array", I've put a ";" where the line break should be). 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.

2 added 2 characters in body

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

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

where $a$ and $d$ are smooth real-valued positive functions on $M$, and $b$ is smooth (The latex doesn't seem to support "pmatrix" or "array", I've put a ";" where the line break should be). 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 power 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.

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

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

where $a$ and $d$ are smooth real-valued positive functions on $M$, and $b$ is smooth (The latex doesn't seem to support "pmatrix" or "array", I've put a ";" where the line break should be). 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 power 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.