I'm going through a proof of a vanishing theorem by Sommese ($H^{p,q}(X,L) = 0$ for $p+q > n+k$ if $L$ is $k$-ample) and have hit the following brick wall:
I've got a complex projective manifold $X$, a meromorphic map $f : X \to \mathbb P^n$ and an ample line bundle $L$ over $X$. I want to show that for any big multiple $mL$ there is a global section $\sigma$ of $mL$ which is not identically zero on any $f^{-1}(y)$ for $y \in f(X \setminus (\text{singularities of f}))$. Basically we're proving this vanishing theorem by induction on $k$ and such a section gives the induction step.
Now, for any fixed $y$ we let $Z$ be the closure of an irreducible component of $f^{-1}(y)$ and find an $N_0$ such that
$$0 \to H^0(X, \mathcal I_Z \otimes mL) \to H^0(X, mL) \to H^0(Z, mL|_Z) \to 0$$
is exact for all $m \geq N_0$, which gives us what we want. The problem is that this $N_0$ depends on $y$.
Sommese says in his book that we get around this by invoking the Hilbert scheme on $\mathbb P^n$ and a result by Grothendieck. He also gives a reference to an article by Hironaka where this result is proved. Both methods are very general and make heavy use of schemes, which I don't know well enough to be able to follow.
I've mostly convinced myself that I can reduce this to where $f$ is holomorphic, and I think some sort of a deformation argument might be possible if $f$ can be made behave nicely enough, but I don't see how to proceed.
I'd really like some comments or suggestions on if it's possible to do this with analytic methods; a bit violently if you will. My thesis advisor suggested this theorem as a topic to present to other students with a rather diffeo-geometric background, and diving into schemes just to prove this one lemma seems like too big a detour for that audience.
