I'm trying to prove Chow's theorem (closed analytic subvarieties of projective space are algebraic) in a special case by the following differential geometric method (imitating the argument for the Kodaira embedding theorem in Griffiths-Harris).
I want to show the following: let $X$ be a compact complex manifold, $L$ a positive line bundle on it, $V\subset X$ a compact complex submanifold and let $x\in X$ be a point not in $V$. Then for $n\gg0$, there exists a section $\sigma\in H^0(X,L^{\otimes n})$ such that $\sigma|_V\equiv 0$ and $\sigma(x)\ne 0$.
The strategy I have in mind is as follows. If $V$ is of codimension $1$, then the ideal sheaf of $V$ defines a line bundle $\mathcal O_X(-V)$, and thus, $L^{\otimes n}\otimes\mathcal O_{X}(-V)$ is positive for $n\gg 0$ (and thus, is globally generated for $n\gg 0$), which proves the result. If $V$ is of codimension $\ge 2$, then let $\pi:Y\to X$ be the blow-up of $X$ along $V$, and let $E$ be the exceptional divisor. We know that $E \cong \mathbb P(N_{X}V)$, and that $\mathcal O_Y(E)|_E = N_{E}Y$ is the relative $\mathcal O(-1)$ bundle on $E\to V$. Now, $\mathcal O_Y(-E)$ is positive on each fibre of $E\to V$ (and these are the directions contracted by the blowdown map $\pi$). So, I'm led to guess that $\tilde L = \pi^*L^{\otimes n}\otimes\mathcal O_Y(-E)$ is positive for $n\gg 0$ (I have not yet done the computation to check this). If this is the case, then we are again done.
So my question is: is $\tilde L$ positive for $n\gg 0$?
