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mixed Hodge structures on the strictly upper triangular Lie algebras
Abstract
We show that, for k≥ 3, a lattice in the unitriangular Lie group Uk(ℝ) is the fundamental group of a smooth quasi-projective complex algebraic variety if and only if k=3. The proof uses the conjugation symmetry and weight conditions of mixed Hodge structure on the Malcev Lie algebra proven by Morgan. This result identifies a family of groups that must be excluded in the search for smooth quasi-projective varieties with torsion-free nilpotent fundamental groups of high nilpotency class.
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BibTeX
@misc{shimoji2026mixed,
title = {{mixed Hodge structures on the strictly upper triangular Lie algebras}},
author = {Taito Shimoji},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23505},
eprint = {2610.23505},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23505},
}
RIS
TY - UNPB
TI - mixed Hodge structures on the strictly upper triangular Lie algebras
AU - Taito Shimoji
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.23505
UR - https://arxiv.org/abs/2610.23505
AB - We show that, for k≥ 3, a lattice in the unitriangular Lie group U_k(ℝ) is the fundamental group of a smooth quasi-projective complex algebraic variety if and only if k=3. The proof uses the conjugation symmetry and weight conditions of mixed Hodge structure on the Malcev Lie algebra proven by Morgan. This result identifies a family of groups that must be excluded in the search for smooth quasi-projective varieties with torsion-free nilpotent fundamental groups of high nilpotency class.
ER -