Normality of monodromy group in generic convolution group
Haohao Liu
TL;DR
The paper investigates how monodromy groups arising from the convolution of relative perverse sheaves on abelian varieties behave in families. By integrating universal local acyclicity, Gabber–Loeser cotori, generic vanishing, and Tannakian convolution, it proves that for uncountably many twists $\chi$, the monodromy $Mon(K,\chi)$ is reductive and normal in the generic convolution group $G_{\omega_{\chi}}(K|_{A_{\eta}})$; a fixed-part theorem then controls invariants to solidify normality. The key innovation is a fixed-part argument that passes from fiberwise monodromy to global normality, paralleling André’s normality phenomena and extending previous lawrence–sawin-type normality results to a wider, relative setting. The results advance the construction of local systems with prescribed monodromy via perverse convolution, with potential applications to arithmetic geometry, inverse Galois problems, and motivic Galois representations.
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 $η$. 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 $L_χ$ on $A$, the monodromy groups of $R^0p_{X*}(K\otimes p_A^*L_χ)$ are normal in the Tannakian group $G(K|_{A_η})$ of the perverse sheaf $K|_{A_η}\in\mathrm{Perv}(A_η)$. This result is inspired from and could be compared to two other normality results: In the same setting, the Tannakian group $G(K|_{A_{\barη}})$ is normal in $G(K|_{A_η})$ (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 André).