Open Access𝐺-cohomologically rigid local systems are integral
Author(s) -
Christian Klevdal,
Stefan Patrikis
Publication year - 2022
Publication title -
transactions of the american mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.798
H-Index - 100
eISSN - 1088-6850
pISSN - 0002-9947
DOI - 10.1090/tran/8610
Subject(s) - algorithm , annotation , computer science , type (biology) , artificial intelligence , mathematics , biology , ecology
Let G G be a reductive group, and let X X be a smooth quasi-projective complex variety. We prove that any G G -irreducible, G G -cohomologically rigid local system on X 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 = G L n 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 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.