Solvable fundamental groups of algebraic varieties and Kähler manifolds

1997/06/24 by Donu Arapura, Arapura, Donu, Madhav V. Nori +1 · 4 citations
Mathematics · #Algebraic Geometry and Number Theory #Geometry and complex manifolds #Advanced Algebra and Geometry

paper · pdf · doi:10.48550/arxiv.alg-geom/9706011

Abstract

This is a long overdue write up of the following: If the fundamental group of a normal complex algebraic variety (respectively Zariski open subset of a compact Kähler manifold) is a solvable group of matrices over Q (respectively polycyclic group), then it is virtually nilpotent. This should be interpreted as saying that a large class of solvable groups will not occur as fundamental groups of such spaces. The essential strategy is to first "complete" these groups to algebraic groups (generalizing constructions of Malcev and Mostow), and then check that the identity component is unipotent. This is done by Galois theoretic methods in the algebraic case, and is reduced to homological properties of one dimensional characters in the analytic case.

Cited by

• Hyperbolicity and fundamental groups of complex quasi-projective varieties (III): applications
• On the torsion-free nilpotent fundamental groups of smooth quasi-projective varieties of rank up to seven
• Kähler Solvmanifolds
• Automorphism groups of solvable groups of finite abelian ranks

Related

• Solvable Quotients of Kahler Groups
• Representations of Fundamental Groups of Compact Kahler Manifolds in Solvable Matrix Groups
• K-theory of Solvable Groups
• The Identity Problem in virtually solvable matrix groups over algebraic numbers
• Automorphism groups of solvable groups of finite abelian ranks