MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I'm wondering if there are general techniques to calculate the singular cohomology groups of a fiber space (specifically, a non-smooth elliptic fibration) using methods of algebraic topology. More concretely:

Suppose $X, Y$ are smooth, proper irreducible varieties over $\mathbb{C}$, and $X \xrightarrow{\phi} Y$ is a smooth map.

$\textbf{Question 1:}$ Do the singular cohomology groups of $X$ (let's say $\mathbb{Q}$-coefficients) correspond to the cohomology groups of a locally constant sheaf on $Y$?

$\textbf{Question 2:}$ Suppose we don't require that $\phi$ is smooth, but say instead that it just has reduced fibers. Is the same true? Can we even recover $\mathbb{Z}$-coefficients in some cases?

From what I understand, we can compute cohomology of the derived pushforward of the constant sheaf $\mathbb{Z}_X$, and this will give the singular cohomology groups of $X$, but now calculated on $Y$ (however, this is a tautological statement, and I don't really see how it helps with computations). If in our cases, the derived pushforward had the same cohomology as the sheaf $\phi_*\mathbb{Z}_X$ then we we would be done. References will be greatly appreciated!

share|cite|improve this question
Look up "Leray spectral sequence". – Angelo Mar 21 '13 at 4:35
And, by the way, your idea of what a tautological statement is rather different from mine. – Angelo Mar 21 '13 at 7:05
@Angelo, Sorry. That was bad word choice on my part. – comp Mar 21 '13 at 14:27

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.