Variation of Tannaka groups of perverse sheaves in family

TL;DR

This paper proves that the simplicity of a perverse sheaf in a family over a smooth variety persists over a non-empty open subset, with applications to the variation of Tannaka groups in abelian schemes.

Contribution

It establishes the stability of the simplicity property of perverse sheaves in families and explores its implications for Tannaka groups in abelian schemes.

Findings

Simplicity of perverse sheaves is open in families over smooth varieties.

The result applies to the variation of Tannaka groups in abelian schemes.

Provides a criterion for the persistence of simplicity in geometric families.

Abstract

Let $k$ be a field of characteristic $0$, let $S$ be a smooth, geometrically connected variety over $k$, with generic point $\eta$, and $f:\mathbb{X}\rightarrow S$ 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 $f:\mathbb{X}\rightarrow S$ is an abelian scheme, we give applications of this result to the variation with $s\in S$ of the Tannaka group of $\mathbb{P}|_{\mathbb{X}_{\bar s}}$.