Question

I'm reading Siu's peper"Fujita conjecture and the extension theorem of Ohsawa-Takegoshi".He refered to a "well-known techniques of using Rieamann-Roch to produce singular metrics".

The situation is: $L$ is a positive line bundle over an $n$-dimensional compact complex manifold M,$\epsilon$ is a sufficiently small positive rational number,$P$ is a given point on $M$.

He claims that we can find a multivalued holomorphic section of $(n+\epsilon)L$ such that the section vanishes to order at least $n$ at $P$;

or equivalent to say for some positive integer $m$ with $m \epsilon$ is a integer, we can find a holomorphic section of $m(n+\epsilon)L$ which vanishes to order at least $mn$ at $P$.

Why this statement is true?

Here vanishing order $k$ means when the section is represented by a local holomorphic function,the function germ lies eaxctly in the k-th power of the maximal ideal $m_{M,P}$.

Answer 1

I would suggest you two references; the first one is Demailly's survey on Hodge theory ([B3] here: http://www-fourier.ujf-grenoble.fr/~demailly/books.html ); exercise 15.11 explains exactly how to construct sections with desired vanishing order at some point.

This is a consequence of Nadel's theorem applied to singular metrics on ample line bundles; the idea is to impose the metric to have a logarithmic pole at your point, so that Skoda's lemma (the easy part) ensures you that any section of L twisted by the multiplier ideal of the metric has to vanish enough along P. As a consequence, you can deduce Kodaira's embedding theorem.

The second reference is the one already mentionned by Donu Arapura. More precisely, there is a lemma in Lazarsfeld' PAG I (edit: Proposition 1.1.31 p.23). It shows how elementary linear algebra (+ RR) can give a lower bound on the vanishing order of some section of a (say nef and big) line bundle with big enough top intersection.

Answer 2

The proof Kodaira's embedding theorem in Griffiths-Harris is a good place to see how positivity + Riemann-Roch can be used to produce sections with many properties.