Quasi-projective nilmanifolds
Taito Shimoji
TL;DR
Problem: characterize when a smooth quasi-projective variety $M=V\setminus D$ is diffeomorphic to a non-compact nilmanifold $\Gamma\backslash N\times \mathbb{R}^m$, and determine which such nilmanifolds can arise as smooth quasi-projective varieties. Approach: combine Nomizu's Lie-algebra computation of de Rham cohomology for nilmanifolds with Deligne–Morgan mixed Hodge structures via minimal models to derive topological and algebraic constraints, then classify low-dimensional cases. Key contributions: (i) diffeomorphism types: if $H^1(V)=0$ then $M\simeq T^n\times \mathbb{R}^m$, and in general $M\simeq T^{b_1(M)}\times \mathbb{R}^m$ or $E\times \mathbb{R}^m$ with $E\to T^{b_1(M)}$ a torus bundle; (ii) the fundamental group is torsion-free nilpotent with nilpotency class at most $2$; (iii) a dimension-bounded classification for ${\rm dim\ }N\leq 8$ of possible complexifications $\mathfrak n_{\mathbb{C}}$ and explicit bigradings compatible with quasi-projective structure; (iv) obstructions ruling out certain $3$-step nilpotent Lie algebras as hosts for quasi-projective diffeomorphic nilmanifolds. Significance: links topology, Lie theory, and algebraic geometry to constrain the topology of smooth quasi-projective varieties and to guide higher-dimensional analysis of possible nilmanifold models.
Abstract
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 $Γ\backslash N\times\mathbb{R}^m$. 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.