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 $m_{M,P}$.