𝐺-cohomologically rigid local systems are integral

Klevdal, Christian; Patrikis, Stefan

doi:10.1090/tran/8610

$G$-cohomologically rigid local systems are integral
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by Christian Klevdal and Stefan Patrikis PDF

Trans. Amer. Math. Soc. 375 (2022), 4153-4175 Request permission

Abstract:

Let $G$ be a reductive group, and let $X$ be a smooth quasi-projective complex variety. We prove that any $G$-irreducible, $G$-cohomologically rigid local system on $X$ with finite order abelianization and quasi-unipotent local monodromies is integral. This generalizes work of Esnault and Groechenig [Selecta Math. (N. S. ) 24 (2018), pp. 4279–4292; Acta Math. 225 (2020), pp. 103–158] when $G= \mathrm {GL}_n$, and it answers positively a conjecture of Simpson [Inst. Hautes Études Sci. Publ. Math. 75 (1992), pp. 5–95; Inst. Hautes Études Sci. Publ. Math. 80 (1994), pp. 5–79] for $G$-cohomologically rigid local systems. Along the way we show that the connected component of the Zariski-closure of the monodromy group of any such local system is semisimple; this moreover holds when we relax cohomological rigidity to rigidity.

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