From: mmurray@maths.adelaide.edu.au (Michael K Murray)
Subject: Re: Bott's Theorem
Date: 2 May 2001 10:20:01 -0500
Newsgroups: sci.math.research
Summary: Bott-Borel-Weyl theorem: sheaf cohomology on homogeneous spaces

In article <9cm9l5$hbm$1@fanth.sci.kun.nl>, grooten@sci.kun.nl (Grooten
MI) wrote:

> Hello,
> 
> I'm looking for a theorem by Bott. it says when the cohomology groups of the
> sheaf of holomorphic functions with some twist on a projective space are 0.
> I need to know what these groups are in case they are not 0. Does anyone
> know this theorem or does someone perhaps know a reference to this theorem?
> 
> Thank you,
> Martijn Grooten

It sounds like what has been generalised to become the Bott-Borel-Weyl
theorem.  This tells you when the sheaf or Dolbeault cohomology 
of a homogeneuous bundle on G/P vanishes and if it doesn't
vanish it tells you what which irreducible representation of G it is.

But thats all overkill if want you want is line bundles on projective space.
For that try 

   AUTHOR Okonek, Christian, 1952-
      TITLE Vector bundles on complex projective spaces / Christian Okonek,
            Michael Schneider, Heinz Spindler  
  PUBLISHED Boston : Birkhauser, c1980 
 PHYS DESCR vii, 389 p. ; 23 cm.
            Progress in mathematics ; v. 3


Michael