Differential geometry: Let $S$ be a Kahler surface. That is a 4 real dimensional surface or a complex 2 dimensional one. Since by definition $S$ is complex it comes equipped with a Kahler 2-form $J$. If we choose a coordinate patch with holomorphic coordinates $z, w$ in some neighborhood $U$, then we can write the Kahler form locally as $$ J = f_{z,\bar{z}} \, dz \wedge d\bar{z} + f_{z,\bar{w}}\, dz \wedge d\bar{w} + f_{w,\bar{z}}\, dw \wedge d\bar{z} + f_{w,\bar{w}}\, dw \wedge d\bar{w} $$ Algebraic geometry: Assume from now on that $S$ is also a complex algebraic variety, say $\mathbb{CP}^2$ or some Hirzebruch surface. Then $[J] \in H^2(S,\mathbb{Z})$. Then $[J]$ is also called a polarization, or an ample class and belongs to $\mathcal{O}_S(1)$. The object $\mathcal{O}_S(1)$ is the twisting sheaf over $S$ and sections of it are monomials of degree 1.
My question: Does this simply tell me that the functions $f$ have no poles and they have to be either $z$, $w$, $\bar{z}$ or $\bar w$ (only monomials of degree 1)? And how do we see the "ample" nature of this line bundle from its diffential geometric definition?
If not, what am I missing here? Clearly the line bundle associated to $\mathcal{O}_1(S)$ locally defined by the combinations $da \wedge d\bar{b}$ where $a,b=z,w$.