$\newcommand{\P}{\mathbb{P}}$Consider the line bundles $\mathcal{O}(k)$ on $\P^n$. I know that there is a map in $[\P^n,\P^\infty]=[\P^n,\P^n]$ classifying this line bundle. Here brackets denote homotopy classes of maps
More precisely, there is a map $\P^n\ \xrightarrow{\ f\ }\ \P^n$ such that $f^*$ (the canonical line bundle over $\P^n$) is $\mathcal{O}(k)$.
Can $f$ be chosen to be holomorphic?
Selected ramblings of mine:
The reason why I ask is because the when there are global sections(i.e. when $k\geq 0$, the standard map to projective space given by the line bundle is to projective space of dimension $\dim\Bbb{C}[x_0,x_1,\ldots,x_n]_k$ which is much bigger than $n$.
Moreover the only way I know how to construct such a map is by using a partition of unity which is not a holomorphic respecting construction.
Suppose $x \in U_{i_o}=D_{x_{i_0}} \subset \P^n$. Let $t_{ij}$ be algebraic transition functions for a line bundle (like that for $\mathcal{O}(k)$). Then $[t_{i_00}:t_{i_01}:\cdots:t_{i_0n}]$ is well defined on the intersections of $\bigcap U_i$ and is a well defined map as soon as one clears denominators. This seems like a good candidate for a classifying map since if one is given a partition of unity $\{\phi_i\}$, then the corresponding classifying map should be $[\phi_0t_{i_00}:\phi_1t_{i_01}:\cdots:\phi_nt_{i_0n}]$.