Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.23505

mixed Hodge structures on the strictly upper triangular Lie algebras

Taito Shimoji
View original source ↗Read source document ↗Export metadata ↓

Abstract

We show that, for $k\geq 3$, a lattice in the unitriangular Lie group $U_k(\mathbb{R})$ 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.

Explore related subjects

math.AGmath.ATmath.DG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Taito Shimoji. 2026-10-16. mixed Hodge structures on the strictly upper triangular Lie algebras. https://arxiv.org/abs/2610.23505

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Normal complex surface singularities with rational homology disk smoothings

In this paper we show that if the minimal good resolution graph of a normal surface singularity contains at least two nodes (i.e. vertex with valency at least 3) then the singularity does not admit a smoothing with Milnor fiber having rational homology equal to the rational homology of the 4-disk $D^4$ (called a rational homology disk smoothing). Combining with earlier results, this theorem then provides a complete classification of resolution graphs of normal surface singularities with a rational homology disk smoothing, verifying a conjecture of J. Wahl regarding such singularities. Indeed, together with a recent result of J. Fowler we get the complete list of normal surface singularities which admit rational homology disk smoothings.

math.AG↗
arXiv2026

Singularities of orthogonal and symplectic determinantal varieties

Let either $GL(E)\times SO(F)$ or $GL(E)\times Sp(F)$ act naturally on the space of matrices $E\otimes F$. There are only finitely many orbits, and the orbit closures are orthogonal and symplectic generalizations of determinantal varieties, which can be described similarly using rank conditions. In this paper, we study the singularities of these varieties and describe their defining equations. We prove that in the symplectic case, the orbit closures are normal with good filtrations, and in characteristic $0$ have rational singularities. In the orthogonal case we show that most orbit closures will have similar properties, and determine precisely the exceptions to this.

math.AG↗
arXiv2026

On the Hodge Structures of Global Smoothings of Normal Crossing Varieties

We study the Hodge theory of smooth fibers of semistable degenerations of compact complex manifolds with central fibers of Kähler type. If every stratum of the central fiber is a $\partial\overline{\partial}$-manifold, we prove that the local invariant cycle property in all degrees implies that the nearby fibers satisfy the $\partial\overline{\partial}$-lemma. In middle degree, the corresponding property alone yields a canonical Hodge decomposition on the nearby fibers and an explicit formula, in terms of the limiting mixed Hodge structure, for the signature of the cup-product pairing on each Hodge component. This gives a partial converse to Clemens' local invariant cycle theorem for Kähler degenerations. As applications, we study global smoothings of ordinary double points and extend results of Friedman and Li on Clemens manifolds. We also apply our results to the non-Kähler Calabi--Yau manifolds constructed by Hashimoto--Sano and Sano. In particular, these manifolds satisfy the $\partial\overline{\partial}$-lemma, even though they are not in Fujiki class $\mathcal{C}$, and in Sano's higher-dimensional examples we explicitly determine the failure of the Hodge--Riemann bilinear relations on the middle cohomology.

math.AG↗