Let $K$ be a finitely generated field, and $X/K$ a smooth proper variety. Let $G$ denote the absolute Galois group $\mathrm{Gal}(\bar{K}/K)$. Let $\ell$ be a prime different from $\mathrm{char}(K)$. A longstanding famous conjecture (due to, I think, Tate, but also Grothendieck–Serre) is:
Conjecture. The representation of $G$ on $H_{\mathrm{ét}}^{i}(X_{\bar{K}}, \mathbb{Q}_{\ell})$ is semisimple.
The conjecture is known if $X$ is an abelian variety (hence if $X$ is a curve), and a few other cases.
One way of approaching this might be to mimic Deligne's proof of Weil I, and one readily reduces to the case where:
- $f \colon X \to \mathbb{P}^{1}$ is a Lefschetz pencil (finitely many singular fibres, and each singular fibre has a unique singularity, that is a quadratic singularity);
- with fibres of dimension $n$;
- by induction the conjecture is known for varieties of dimension $\le n$.
The groups $H_{\mathrm{ét}}^{i}(X_{\bar{K}}, \mathbb{Q}_{\ell})$, for $i \ne n+1$ are semisimple via an induction argument.
One can then study $H_{\mathrm{ét}}^{n+1}(X_{\bar{K}}, \mathbb{Q}_{\ell})$ via the Leray spectral sequence of $f$. Here one should use perverse sheaves, in order to obtain $E_{2}$-degeneration of the spectral sequence.
Lei Fu does some computations on this spectral sequence in “On the semisimplicity conjecture and Galois representations”. Using those computations, one remains with a factor $H_{\mathrm{ét}}^{1}(\mathbb{P}^{1}, R^{n}f_{*}\mathbb{Q}_{\ell})$ for which one has to prove semisimplicity. Lei Fu reduces this to his own conjecture about Galois representations related to function fields.
My question is whether there is any literature that has pursued this path along Deligne's Weil I. With this I mean the following: Consider the subsheaf $\mathcal{E} \subset R^{n}f_{*}\mathbb{Q}_{\ell}$ of vanishing cycles. Then we have a surjection $H_{\mathrm{ét}}^{1}(\mathbb{P}^{1}, \mathcal{E}) \to H_{\mathrm{ét}}^{1}(\mathbb{P}^{1}, R^{n}f_{*}\mathbb{Q}_{\ell})$.
Has anyone studied the semisimplicity of $H_{\mathrm{ét}}^{1}(\mathbb{P}^{1}, \mathcal{E})$? Is there literature about this? Are there any further/partial results?
