Talk:Coherent sheaf cohomology

Latest comment: 6 years ago by 128.138.65.80 in topic Todo

"Coherent cohomology", "holomorphic Euler characteristic", and "Serre's vanishing theorem" redirect here. BTotaro (talk) 17:19, 20 May 2016 (UTC)Reply

Todo

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This page should include several computations of sheaf cohomology. This should include

  • cohomology of projective space
  • cohomology of the module from the structure sheaf of a projective variety
  • hodge theory computations

This page should also describe applications. One is the fact that the sheaf cohomology modules can be used to construct examples of families of extensions parameterized by the affine line using the fact that we can just a morphism of   into one of these vector spaces (since we can use the derived category interpretation). This shows that   is not   invariant since if our map contains   and a non-zero point, we have a trivial and non-trivial extension of vector bundles, hence they are not isomorphic. — Preceding unsigned comment added by 128.138.65.203 (talk) 01:46, 10 October 2017 (UTC)Reply

Here are some additional resources for computing cohomology:

For example, take a complete intersection surface   of bidegree   in  . Then, using the Hodge theory notes there are short exact sequences on the degree   hypersurface   from the cotangent sequence

 

Then, using the cotangent sequence again we find

 

These give long exact sequences, which can then be used to compute the hodge numbers of this complete intersection hypersurface. The other two sequences to consider are

 

Notice that tensoring the euler sequence will show how to compute the cohomology of the first sequence.