# Extending IC sheaves across smooth divisors with normal crossings

I am trying to understand paragraph 1.6 of Lusztig's paper "Character Sheaves I". The basic setup is that $$X$$ is a smooth irreducible variety over a field $$k=\overline{k}$$, $$D_i, i=1,...,r$$ are smooth divisors with normal crossings and $$\mathcal{L}$$ is $$\overline{\mathbb{Q}}_l$$-local system of rank one such that the corresponding monodromy action factors through a finite quotient of order invertible in $$k$$. Now the IC sheaf $$IC(X,\mathcal{L})$$ is represented by a single constructible $$\overline{\mathbb{Q}}_l$$-sheaf $$\overline{\mathcal{L}}$$.

Then he claims that if the local monodromy of $$\mathcal{L}$$ around one $$D_i$$ is trivial, then $$\overline{\mathcal{L}}$$ restricted to $$U\bigcup D_i$$ is a local system (this is a special case of his claim). What is the local monodromy of $$\mathcal{L}$$ around a $$D_i$$ and how to show his claim?

• "smooth divisors with normal crossings" would be more accurate Jun 4 at 7:59

The local monodromy around $$D_i$$ can be obtained by taking a $$\eta$$ a geometric generic point of $$D_i$$, $$R$$ the etale local ring of $$X$$ at $$\eta$$ with uniformizer $$\pi$$, then pulling $$\mathcal L$$ back to $$\operatorname{Spec} R[ \pi^{-1} ]$$. We then obtain a representation of the étale fundamental group of $$\operatorname{Spec} R[ \pi^{-1} ]$$. The image of this representation is called the local monodromy.

Rather than $$U \cup D_i$$, it would be more precise to say $$X \setminus \cup_{j\neq i} D_j$$. To check that $$\overline{\mathcal L}$$ restricted to this space is a local system, we first check that for $$j \colon U \to X \setminus \cup_{j\neq i} D_j$$ the open immersion, $$j_* \mathcal L$$ is a local system, and then that $$j_* \mathcal L$$ is the restriction of $$\overline{\mathcal L}$$.

The pullback of $$j_*\mathcal L$$ to $$\operatorname{Spec} R$$ is the pushforward from $$\operatorname{Spec} R[ \pi^{-1} ]$$ to $$\operatorname{Spec} R$$ of the restriction of $$\mathcal L$$ to $$\operatorname{Spec} R[ \pi^{-1} ]$$. So it is the pushforward of a constant sheaf from the generic point to the whole spectrum and thus is a constant sheaf. Thus $$j_* \mathcal L$$ is locally constant at the generic point of $$D_i$$. Thus it is locally constant on some neighborhood of the generic point. The complement of the largest open set on which $$j_* \mathcal L$$ is locally constant is a closed set contained in $$D_i$$ but not containing the generic point of $$D_i$$ and thus has codimension $$\geq 2$$.

If that complement is nonempty, consider a generic point $$\eta'$$ of that complement and a Henselian local ring $$R'$$ at $$\eta$$. The lisse sheaf gives a representation of the fundamental group of the punctured spectrum of $$R'$$, which by purity (since the codimension is $$\geq 2$$) is trivial. So again the sheaf is the constant sheaf and the pushforward is just the constant sheaf, contradicting the assumption that the sheaf is not locally constant at $$\eta$$, so the complement is indeed empty and $$j_* \mathcal L$$ is lisse.

Since the intermediate extension is the unique perverse extension with no irreducible components supported outside $$U$$, and $$j_* \mathcal L$$, being lisse, is perverse and has no irreducible components supported outside $$U$$, $$j_* \mathcal L$$ must give the intermediate extension.