# Universal property for complex blowup in smooth category

If $M$ is a smooth complex manifold and $N$ is a smooth complex surface, we may ask when a holomorphic map $f:M\rightarrow N$ lifts to a map $f:M\rightarrow [N:p]$, where $[N:p]$ denotes the blowup of $N$ in a point $p\in N$.

The answer, which provides a universal property for $[N:p]$, is that the map lifts whenever the pullback of the ideal sheaf of $p$, i.e. $f^{-1}I_p\otimes_{f^{-1}O_N}O_M$, is locally free of rank 1 on $M$. In other words, the functions $x,y$ cutting out $p$ pull back to $M$ to define a divisor.

Question: Let $M$ be a smooth real manifold and $N$ a smooth complex surface. When does a smooth map $f:M\rightarrow N$ lift to the complex blowup $[N:p]$?

I believe it is true that the map lifts whenever $f^{-1}I_p\otimes_{f^{-1}O_N} C^\infty_M$ is locally free of rank 1 over the sheaf of $C^\infty$ complex-valued functions, in other words it generates a complex line bundle. I would especially like to know if this result appears in the literature.

-