Normality of monodromy group in generic convolution group

TL;DR

This paper proves that for a family of perverse sheaves on an abelian scheme, the monodromy groups associated with generic character twists are normal in the Tannakian group, extending known normality results in algebraic geometry.

Contribution

It establishes the normality of monodromy groups for generic character sheaves in a Tannakian setting, generalizing previous normality results for algebraic and Hodge-theoretic structures.

Findings

Monodromy groups are normal in the Tannakian group for uncountably many character sheaves.

The result connects to and extends Lawrence-Sawin's and André's normality theorems.

Provides a new perspective on the structure of perverse sheaves on abelian varieties.

Abstract

On an abelian variety $A$, sheaf convolution gives a Tannakian formalism for perverse sheaves. Let $X$ be an irreducible algebraic variety with generic point $\eta$. Let $K$ be a family of perverse sheaves (more precisely, a relative perverse sheaf) on the constant abelian scheme $p_X:A\times X\to X$. We show that for uncountably many character sheaves $L_{\chi}$ on $A$, the monodromy groups of $R^0p_{X*}(K\otimes p_A^*L_{\chi})$ are normal in the Tannakian group $G(K|_{A_{\eta}})$ of the perverse sheaf $K|_{A_{\eta}}\in\mathrm{Perv}(A_{\eta})$. This result is inspired from and could be compared to two other normality results: In the same setting, the Tannakian group $G(K|_{A_{\bar{\eta}}})$ is normal in $G(K|_{A_{\eta}})$ (due to Lawrence-Sawin). For a polarizable variation of Hodge structures, outside a meager locus, the connected monodromy group is normal in the derived Mumford-Tate group (due to Andr\'e).