Talk:Coherent sheaf cohomology

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

Todo

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 $\mathbb {A} ^{1}$ into one of these vector spaces (since we can use the derived category interpretation). This shows that ${\text{Vect}}_{r}(-)$ is not $\mathbb {A} ^{1}$ invariant since if our map contains $0$ 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)

Here are some additional resources for computing cohomology:

• https://math.stackexchange.com/questions/2228606/does-serre-duality-for-hodge-numbers-follow-from-serre-duality-for-coherent-shea
• http://www.math.purdue.edu/~dvb/preprints/book-chap17.pdf — Preceding unsigned comment added by 128.138.65.80 (talk) 22:52, 4 December 2017 (UTC)

For example, take a complete intersection surface $Y$ of bidegree $(d_{1},d_{2})$ in $\mathbb {P} ^{4}$. Then, using the Hodge theory notes there are short exact sequences on the degree $d_{1}$ hypersurface $X$ from the cotangent sequence

${\begin{aligned}0\to {\mathcal {O}}_{X}(-d_{1})\to \Omega |_{X}\to \Omega _{X}\to 0\\0\to \Omega _{X}(-d_{1})\to \Omega ^{2}|_{X}\to \Omega _{X}^{2}\to 0\\0\to \Omega _{X}^{2}(-d_{1})\to \Omega ^{3}|_{X}\to \Omega _{X}^{3}\to 0\end{aligned}}$

Then, using the cotangent sequence again we find

${\begin{aligned}0\to {\mathcal {O}}_{Y}(-d_{2})\to \Omega _{X}|_{Y}\to \Omega _{Y}\to 0\\0\to \Omega _{Y}(-d_{2})\to \Omega _{X}^{2}|_{Y}\to \Omega _{Y}^{2}\to 0\end{aligned}}$

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

${\begin{aligned}0\to \Omega (-d_{1})\to \Omega \to \Omega |_{X}\to 0\\0\to \Omega ^{1}\to {\mathcal {O}}(-1)^{\oplus (n+1)}\to {\mathcal {O}}\to 0\end{aligned}}$

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