Let $X$ be a compact Riemann surface and $L$ be a holomorphic line bundle on $X$. One can define its Chern class $c_1(L) \in H^{1,1}(X,\mathbb{C})$, use either pure cohomology or by means of Hermitian metrics.
This $c_1(L)$ is way more special than a random $(1,1)$-form.
- It is "real", in the sense that it is locally $f\mathrm{d}z \mathrm{d}\overline{z}$ with $f$ real-valued.
- It is "integral", in the sense that it lies in $H^{1,1}(X,\mathbb{C}) \bigcap H^2(X,\mathbb{Z})$ (Lefschetz (1,1) theorem), and $\displaystyle \int_X c_1(L)=\deg(L)$ is an integer (Gauss-Bonnet).
But $c_1(L)$ is only a class of smooth $(1,1)$-form and smooth forms can be very strange. I want to know how bad can our special $c_1(L)$ be.
For example, can $c_1(L)$ have infinitely many zeroes; can we nicely cover $X$ with $U_i$ on which $c_1(L)$ is positive or negative...
I'd be appreciated if anyone could give me some reference on this topic.
To be precise, $c_1(L)$ is a class in $H^2(M, \mathbb Z)$. I think you are asking how bad can a representative in $c_1 (L)$ can be.
Let $h_0$ be a fixed Hermitian form on $L$, let $\Theta_0 = \frac{i}{2\pi} \bar\partial \partial \log h_0 $ be the curvature two form. Now let $\omega$ be any $(1, 1)$ form in the same class $c_1(L)$. You are asking if there is a Hermitian form $h$ on $L$ such that $$\tag{1} \Theta = \frac{i}{2\pi} \bar\partial \partial \log h = \omega?$$ Write $h = e^u h_0$ for some $u : M \to \mathbb R$. Then (1) can be re-written as
$$\tag{2} \bar \partial \partial u =2\pi i( \Theta_0-\omega) .$$
Thus you are asking, give any $\omega \in c_1(L)$,is there a $u$ which satisfy the above PDE (2)? The answer is yes and if I remembered correctly, a proof can be found in Donaldson's Riemann surfaces.
A generalization of the above theorem is the $\partial \bar\partial$-lemma.