Variation of Tannaka groups of perverse sheaves in family

Abstract

Let $k$ be a field of characteristic $0$, let $S$ be a smooth, geometrically connected variety over $k$, with generic point $\eta$, and $f:\mathbb{X}\to S$ a morphism separated and of finite type. Fix a prime $\ell$. Let $\mathbb{P}$ be an $f$-universally locally acyclic relative perverse ${\overline{\mathbb{Q}}}_{\ell }$-sheaf on $\mathbb{X}/S$. We prove that if for some (equivalently, every) geometric point $\bar{\eta }$ over $\eta$ the restriction $\mathbb{P}{|}_{{\mathbb{X}}_{\bar{\eta }}}$ is simple as a perverse ${\overline{\mathbb{Q}}}_{\ell }$-sheaf on ${\mathbb{X}}_{\bar{\eta }}$, then there is a non-empty open subscheme $U\subset S$ such that, for every geometric point $\bar{s}$ on $U$, the restriction $\mathbb{P}{|}_{{\mathbb{X}}_{\bar{s}}}$ is simple as a perverse ${\overline{\mathbb{Q}}}_{\ell }$-sheaf on ${\mathbb{X}}_{\bar{s}}$. When $f:\mathbb{X}\to S$ is an abelian scheme, we give applications of this result to the variation with $s\in S$ of the Tannaka group of $\mathbb{P}{|}_{{\mathbb{X}}_{\bar{s}}}$.

Lemma 4.2 in v1 is wrong. We modify the proof of Theorem 4.1