Motivation behind spectral sequences

Question

It is well known that spectral sequence is very important in algebraic geometry and complex geometry, but its definition seems very unnatural. For example, in Voisin's book Hodge theory and complex algebraic geometry, I, p.202, the author defines

$E_r^{p,q}:=\frac{{Z_r}^{p,q}}{{B_r}^{p,q}}$,

where $Z_r^{p,q}=\{x\in F^pA^{p+q}|dx\in F^{p+r}A^{p+q}\}$, and $B_r^{p,q}=Z_{r-1}^{p+1,q-1}+dZ_{r-1}^{p-r+1,q+r-2}$.

It can be checked that $B_r^{p,q}\subset Z_r^{p,q}$. Note that $d$ sends $Z_r^{p,q}$ to $Z_r^{p+r,q-r+1}$ and $B_r^{p,q}$ to $B_r^{p+r,q-r+1}$, so we have a differential $d_r:E_r^{p,q}\to E_r^{p+r,q-r+1}$ satisfying $d_r^2=0$.

Although I can verify all the facts stated above, but I don't know the motivation behind this definition, especially why we should define $B_r^{p,q}$ as the sum of $Z_{r-1}^{p+1,q-1}$ and $dZ_{r-1}^{p-r+1,q+r-2}$ and why the indices appear so bizarre. I believe there should be a natural motivation under which all these terms become natural and easy to remember, does anyone know it?

Remark: In Griffiths & Harris, the authors say that the spectral sequence is a generalization of a long exact cohomology sequence and promise they will come back to this topic after introducing some more definitions, but seems never come back.

Answer 1

I suspect a previous comment of mine led to this question, so let me say a few words here. The basic problem is this: Suppose $(A^\bullet, F)$ is a filtered complex, then one wants to relate the (co)homology of the associated graded $E_1=H^*(Gr_F A^\bullet)$ to the associated graded of the (co)homology $E_\infty=Gr_F H^*(A)$. It would wonderful if they were equal, and sometimes they are*, but they usually aren't. Instead, the spectral sequence gives a procedure that starts with $E_1$ and eventually gets to $E_\infty$. A bit more precisely, $E_1$ carries a differential. Taking cohomology gives $E_2$, which again carries a differential. Taking cohomology gives $E_3$ and so on. With appropriate conditions, this process will stabilize to $E_\infty$. This is obviously a very rough sketch, and is just meant to supply motivation and not computational skill.

• The good news is that in the case you seem to be interested in (in another question), namely the Frölicher spectral sequence for a compact $\partial\overline{\partial}$-manifold, $E_1$ does equal $E_\infty$.