# Wednesday, April 13

## Spectral sequences continued

:::{.remark}
Recall that for spectral sequences, the diagonal entries $p+q=n$ are the successive quotients in a filtration on $E^n\da \Tot(E_\infty^{\bullet, \bullet})_n$.
Kodaira vanishing: for the original argument, go to characteristic $p$ and look at liftability.
:::

:::{.example title="Deligne-Illusie's proof of Kodaira vanishing"}
We'll have some spectral sequence which we'll want to degenerate at $E_2$.
It STS that $d_r = 0$ for $r\geq 1$, which in fact forces $(E, d)$ to degenerate at $E_1$.
Strategy: find another spectral sequence $(E', d')$ with the same $E_1' \cong E_1$ and a differential $d\neq d'$ which converges to the same thing and more patently stabilizes at $E_1'$.
It then follows that $E$ stabilizes at $E_1$.
Note the $\dim_k E_r^{p, q} \leq \dim_{k} E_{r-1}^{p, q}$ since we're taking kernels mod images.
:::

:::{.lemma title="A 5-term sequence"}
Suppose $E_2^{p, q} \abuts E^n$ for $n=p+q$ is first quadrant.
Then

- $E_2^{0, 0} = E^{\infty}_{0,0}$ and $E_2^{1,0} = E_\infty^{1, 0}$.
- $E_3^{0, 1} = E_\infty^{0, 1}$ and $E_3^{2, 0} = E_\infty^{2, 0}$
- There is a 5-term exact sequence
\[
0\to E_2^{1,0} \to E^1\to E_2^{0, 1} \to E_2^{2, 0} \to E^2
.\]
:::

:::{.example title="?"}
The Leray spectral sequence: for $f\in \Top(X, Y)$ and $\mcf \in \Sh(X; \kvect)$,
\[
E_2^{p, q} = H^p(Y; \RR^q f_* \mcf) \abuts H^{p+q}(X; \mcf)
.\]
This yields
\[
0 \to H^1(X; f_* \mcf) \to H^1(X;\mcf) \to H^0(X; \RR^1 f_* \mcf) \to H^2(X; f_*\mcf) \to H^2(F)
.\]

<!-- Xournal file: /home/zack/SparkleShare/github.com/Notes/Class_Notes/2022/Spring/SheafCohomology/sections/figures/2022-04-13_10-46.xoj -->

Consider the filtration on $E_\infty$:

![](figures/2022-04-13_10-48-19.png)

This yields exact sequences

- $0 \to E_\infty^{1, 0} \to E^1\to E_\infty^{0, 1} \to 0$
- $0\to E_\infty^{2, 0} \to ? \to E_\infty^{1, 1} \to 0$
- $0\to ? \to E^2\to E_\infty^{0, 2}\to 0$.



:::

:::{.remark}
Recall the definition of a double complex: $(C^{\bullet, \bullet}, d_h, d_v)$ where each row is a complex for $d_h$ and each column for $d_v$, and each square skew-commutes.
Note that the sign trick does not change the cohomology.
The totalized complex is is $(\Tot(C), \bd)$ where $C^n \da \bigoplus _{p+q=n} C^{p, q} \mapsvia{\bd} C^{n+1} \da \bigoplus_{p+q=n+1} C^{p, q}$ and the differential is constructed from $C^{p, q} \mapsvia{d_h \oplus d_v} C^{p+1, q} \oplus C^{p, q+1}$.
There is a descending filtration $\complex\Fil \Tot(C)$ where $\Fil_n \Tot(C) = \tau_{\geq n, \bullet} \Tot(C) = \bigoplus _{p\geq n} C^{p, q}$, which is the double complex obtained by truncating all columns to the left of column $n$.
:::