arXiv subjects

Taito Shimoji

Publications and source records attributed to Taito Shimoji.

3 recordsLinked to original sources

arXiv2026

Quasi-projective nilmanifolds

Let $M=V\setminus D$ be a smooth quasi-projective variety for some smooth projective variety $V$ and a divisor $D$ with normal crossings. Assume that $M$ is diffeomorphic to a non-compact nilmanifold $\Gamma\backslash N\times\mathbb{R}^m$. We show that $M$ is diffeomorphic to a trivial bundle $T^n\times \mathbb{R}^m$ over a torus $T^n$ if the first cohomology $H^1(V)$ of $V$ vanishes. Moreover, in general, we show that $M$ is diffeomorphic to a trivial bundle $T^{b_1(M)}\times \mathbb{R}^m$ over a $b_1(M)$-dimensional torus $T^{b_1(M)}$, or a trivial bundle $E\times \mathbb{R}^m$ such that $E$ is a torus bundle $E\rightarrow T^{b_1(M)}$ over a torus $T^{b_1(M)}$. Conversely, we consider whether non-compact nilmanifolds are diffeomorphic to a smooth quasi-projective variety. We determine the Lie groups of dimension up to $8$ such that corresponding non-compact nilmanifolds may be diffeomorphic to smooth quasi-projective varieties.

math.AT↗

arXiv2025

On the torsion-free nilpotent fundamental groups of smooth quasi-projective varieties of rank up to seven

Let $X$ be a smooth quasi-projective variety. Assume that the (topological) fundamental group $\pi_1(X, x)$ is torsion-free nilpotent. We show that if the first Betti number $b_1(X) \le 3$, then $\pi_1(X, x)$ is isomorphic to either $\mathbb{Z}^n$ for $n = 1, 2, 3$, a lattice in the Heisenberg group $H_3(\mathbb{R})$ or $\mathbb{R} \times H_3(\mathbb{R})$. Moreover, we prove that $\pi_1(X, x)$ is abelian or $2$-step nilpotent if its rank is less than or equal to seven. More precisely, we determine the real nilpotent Lie groups in which torsion-free nilpotent fundamental groups can be embedded as lattices for ranks up to six and seven, respectively. Our main results are a partial positive answer to a question on nilpotent (quasi-)K\"ahler groups posed by Aguilar and Campana.

math.AG↗

arXiv2025

Gradings on nilpotent Lie algebras associated with the nilpotent fundamental groups of smooth complex algebraic varieties

Let $\Gamma$ be a lattice in a simply-connected nilpotent Lie group $N$ whose Lie algebra $\mathfrak{n}$ is $p$-filiform. We show that $\Gamma$ is either abelian or 2-step nilpotent if $\Gamma$ is isomorphic to the fundamental group of a smooth complex algebraic variety. Moreover as an application of our result, we give a required condition of a lattice in a simply-connected nilpotent Lie group of dimension less than or equal to six to be isomorphic to the fundamental group of a smooth complex algebraic variety.

math.DG↗