About decomposition theorem BBD with respect to some stratification

Question

I want to follow up a question from here (how to deduce version 1.a. from version 1).

I know a version of decomposition theorem BBD:

Version 1. Let $f:X\to Y$ be a (surjective) proper map of complex algebraic varieties. If $\mathcal{F}$ is a direct sum of shifts of simple perverse sheaves on $X$ (i.e. a semisimple complex on $X$), then so is $f_*\mathcal{F}$.

In Chriss Ginzburg, Representation theory and complex geometry (p. 436), they made the following observation

Version 1.a. When $X$ is smooth, then $\mathcal{F}=\mathbb{C}_X$ is semisimple, suppose there is a stratification $Y=\bigsqcup Y_t$ so that $f_t: f^{-1}(Y_t)\to Y_t$ is a topological fiber bundle (such stratification always exists), then the IC sheaves appearing on $f_*\mathcal{F}$ are $IC(Y_t,\mathcal{L})$ where $\mathcal{L}$ semisimple local system on $Y_t$.

My question is, how to deduce version 1.a. from version 1? (Extra question: Does one really need $\mathcal{F}=\mathbb{C}_X$ in version 1.a. or any semisimple complex is enough)

Some thoughts so far:

• Because of the existence of such stratification, one can always refine it so that $f_*\mathcal{F}$ is constructible wrt to $\bigsqcup Y_t$. Suppose $IC(Z,\mathcal{L})$ (here $\mathcal{L}$ is local system on $Z$) appear in $f_*\mathcal{F}$, then we find its cohomological sheaf $H^i(IC(Z,\mathcal{L}))$ is constructible wrt to $\bigsqcup Y_t$, implying $\DeclareMathOperator\supp{supp}\overline{Z}=\supp IC(Z,\mathcal{L})=\bigsqcup_{t\in I} Y_t$ over some subindex $I$. However, I don't think this is enough to deduce $Z=Y_t$ for some $t$.
• Note that $H^i(IC(Z,\mathcal{L}))|_Z=H^i(IC(Z,\mathcal{L})|_Z)=H^i(\mathcal{L}[\dim Z])$ which is $\mathcal{L}$ if $i=\dim Z$ and $0$ elsewhere. One can suppose $\mathcal{L}$ is irreducible local system, hence has nonzero stalk (otherwise corresponding monodromy representation at the zero stalk is trivial), implying $\supp \mathcal{L}=Z$. Hence, $\supp H^i(IC(Z,\mathcal{L}))|_Z=Z\cap \supp H^i(IC(Z,\mathcal{L}))$ is $Z$ if $i=\dim Z$ and $0$ elsewhere. Hence, $Z\subset \supp H^{\dim Z}(IC(Z,\mathcal{L}))$. Furthermore, as $\overline{Z}=\supp IC(Z,\mathcal{L}):= \overline{\bigcup_{i\in \mathbb{Z}} \supp H^i(IC(Z,\mathcal{L}))}$, we find $\overline{Z}= \overline{\supp H^{\dim Z}(IC(Z,\mathcal{L}))}$.
• Since $\mathcal{F}=\mathbb{C}_X$, restricting to fiber bundle $f_t$, we can use Deligne's theorem $(f_t)_* \mathcal{F}|_{Y_t}=\bigoplus_{q\ge 0} H^q(f_{t,*}\mathcal{F}|_{Y_t})[-q]$ where the cohomological sheaf $H^q(f_{t,*}\mathcal{F}|_{Y_t})$ is a local system ... I'm not sure how much more information does this give on $Z$.

Any help would be much appreciated!

Answer 1

It's helpful to break Version 1.a into three parts, one of which is your Version 1.

Part 2: Suppose there is a stratification $Y=\sqcup Y_t$ so that $f_t: f^{-1}(Y_t)\to Y_t$ is a topological fiber bundle (such stratification always exists), then $f_* \mathbb C_X $ are $IC(Y_t,\mathcal{L})$ is constructible with respect to the stratification $Y_t$.

Part 3: Let $K$ be a complex of sheaves on $Y$ suppose there is a stratification $Y=\sqcup Y_t$ so that $K$ is constructible with respect to $Y$ and $K$ is a sum of shifts of simple perverse sheaves, then the IC sheaves appearing in $K$ are $IC(Y_t,\mathcal{L})$ where $\mathcal{L}$ semisimple local system on $Y_t$.

The deduction of Version 1.a from these three parts is clear.

For part 2, you can assume $Y$ consists of a single stratum, so $X \to Y$ is a topological fiber bundle, and then the claim is that the pushforward is locally constant. This is a local claim in the analytic topology, and a fiber bundle is locally a product bundle, so it reduces to the case of a product bundle, which is standard.

For part 3, you observe correctly that the support of each simple summand must be a union of strata. The next step is to observe that since the support is irreducible, one of the strata may be dense. By constructibility with respect to that stratification, the sheaf is lisse on that strata. Now finally one needs to observe that the IC sheaf of a simple local system may be expressed as the IC sheaf from any dense open subset (now back in the Zariski topology) of the support on which it is lisse. This is because the intermediate extension of a simple local system is the unique way to extend it to a simple perverse sheaf.

For your extra question, one only needs to generalize part 2. A sufficient condition is that there exists a stratification of $X$ and a stratification of $Y$ such that $\mathcal F$ is constructible with respect to the stratification on $X$, the image of each stratum of $X$ is a stratum of $Y$, and each stratum of $X$ is a fiber bundle over the corresponding stratum of $Y$. To prove this, we can reduce by excision to the case when $X$ and $Y$ consist of a single stratum, so $\mathcal F$ is locally constant, then use the same argument as in the previous case.