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Results tagged with cohomology
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user 78824
A branch of algebraic topology concerning the study of cocycles and coboundaries. It is in some sense a dual theory to homology theory. This tag can be further specialized by using it in conjunction with the tags group-cohomology, etale-cohomology, sheaf-cohomology, galois-cohomology, lie-algebra-cohomology, motivic-cohomology, equivariant-cohomology, ...
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Does hypercohomology of the Koszul complex compute sheaf cohomology?
Recall that we can compute the sheaf cohomology of $\mathcal{O}_X$ using the pushforward
$$
H^i(X,\mathcal{O}_X) = H^i(\mathbb{P}^n,i_*(\mathcal{O}_X))
$$
If $X$ is a complete intersection, then this can …
- 1,716
3
votes
0
answers
229
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How can I find the differential in the Serre spectral sequence for this sphere fibration?
Consider the assocaited sphere bundle $$S(E) \to \mathbb{P}^n$$ for the vector bundle $\mathcal{O}(k)\oplus \mathcal{O}(l) \to \mathbb{P}^n$. Is there a way to determine the differentials
$$
d_4^{p,m} …
- 1,716
6
votes
1
answer
740
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What tools can I use to compute the cohomology of the fibers of a Lefschetz Pencil?
What are some tools I can use to compute cohomology of some smooth projective varieties over $\mathbb{C}$? … For example, how can I study the cohomology of some Lefschetz pencil for the projective scheme
$$
\begin{matrix}
\textbf{Proj}\left( \frac{\mathbb{C}[x,y,z,w]}{(x^8 + y^8 + z^8 + w^8 +x^2y^2z^2w^2)} \right …
- 1,716
4
votes
1
answer
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Are there algorithmic tools for computing poincare residues?
In Schnell's note on Computing Picard-Fuchs Equations he gives a recursive method for computing residues on hypersurfaces. In short, if you have a meromorphic differential form
$$
\frac{dw}{w^k}\wedge …
- 1,716