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The Leray spectral sequence is a cohomological spectral sequence of the form $$H^p(Y;R^q f_*(F)) \Longrightarrow H^{p+q}(X;F)$$ for abelian sheaves $F$ on a site $X$ and morphisms of sites $f : X \to Y$. Is there an example of a concrete calculation with the Leray spectral sequence for sheaf cohomology? So far I have "only" seen abstract and general arguments which use the Leray spectral sequence; my question is not about these general usages. Often the spectral sequence degenerates directly (at least, in the examples I am aware of), which is not very interesting and doesn't show the real power of spectral sequences. Actually I guess that these cases of the Leray spectral sequence may be replaced by more "direct" arguments.

The cohomological Serre spectral sequence associated to a Serre fibration follows from the Lerre spectral sequence and in algebraic topology there are lots of calculations with the Serre spectral sequence. So I am actually asking for calculations with the Lerre spectral sequence which rather belong to sheaf theory and are not instances of the Serre spectral sequence.

Martin Brandenburg
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  • I don't think this is what you are looking for, but maybe suitable as a side remark: If $f$ is a fiber bundle between spaces and $F$ a constant sheaf, then $R^qf_*(F)$ is the associated system of local coefficents and the spectral sequence should coincide with the Serre spectral sequence under suitable point-set topological restrictions. You will find a ton of concrete nontrivial calculations for that case in textbooks about algebraic topology. – archipelago Jun 07 '15 at 18:34
  • Thank you. I was aware of this. I know explicit calculations with the Serre spectral sequence, but so far I have not seen explicit calculations with the Leray spectral sequence when it does not arise from a fiber bundle. – Martin Brandenburg Jun 07 '15 at 22:35
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    Maybe you can try to see the Cohomology of an elliptic surface by Leray SS? Have you tried that? – lee Nov 29 '15 at 14:18
  • @archipelago do you have any specific algebraic topology references? – 54321user Feb 07 '17 at 03:14

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As per Lee's suggestion, take a smooth projective morphism $f:X \to Y$ with the base a genus 2 curve and fibers elliptic curves. From a theorem of Blanchard-Deligne, the $E_2$-page of the Leray-Serre spectral sequence degenerates, hence giving the isomorphism $$ H^k(X;\underline{\mathbb{Q}}_X) \cong \bigoplus_{k = p + q} H^p(Y;\mathbf{R}^qf_*(\underline{\mathbb{Q}}_X)) $$ And if you'd like, the $E_2$ page of with no monodromy in the cohomology looks like \begin{align*} \begin{matrix} H^0(Y; \underline{\mathbb{Q}}_Y) & H^1(Y; \underline{\mathbb{Q}}_Y) & H^2(Y; \underline{\mathbb{Q}}_Y)\\ H^0(Y; \underline{\mathbb{Q}}_Y\oplus \underline{\mathbb{Q}}_Y) & H^1(Y; \underline{\mathbb{Q}}_Y\oplus \underline{\mathbb{Q}}_Y) & H^2(Y;\underline{\mathbb{Q}}_Y\oplus \underline{\mathbb{Q}}_Y)\\ H^0(Y; \underline{\mathbb{Q}}_Y) & H^1(Y; \underline{\mathbb{Q}}_Y) & H^2(Y; \underline{\mathbb{Q}}_Y) \end{matrix} & = \begin{matrix} \mathbb{Q} & \mathbb{Q}^{\oplus 4} & \mathbb{Q}\\ \mathbb{Q}\oplus \mathbb{Q}& \mathbb{Q}^{\oplus 4} \oplus \mathbb{Q}^{\oplus 4} & \mathbb{Q}\oplus \mathbb{Q}\\ \mathbb{Q} & \mathbb{Q}^{\oplus 4} & \mathbb{Q} \end{matrix} \end{align*}

54321user
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  • For reference, check out https://pcmi.ias.edu/files/Mark%20De%20Cataldo%20Lecture%20Series.pdf – 54321user Dec 24 '16 at 21:13
  • Thank you. So this degenerates again (but for nontrivial reasons, right?). I was interested in Leray spectral sequences which do not degenerate directly. – Martin Brandenburg Apr 12 '17 at 11:21
  • @MartinBrandenburg Apparently a resolution of singularities will do it for you. Check out remark 1.2.3 on page 5 of http://www.ams.org/journals/bull/2009-46-04/S0273-0979-09-01260-9/S0273-0979-09-01260-9.pdf – 54321user Aug 19 '17 at 20:51