Quasi-projective nilmanifolds

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 $\Gamma\backslash N\times\mathbb{R}^m$. 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.