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