
TL;DR
This paper characterizes when certain non-compact nilmanifolds can be realized as smooth quasi-projective varieties, showing they are often trivial or torus bundles over tori, with specific results for low-dimensional cases.
Contribution
It provides a classification of quasi-projective nilmanifolds as trivial or torus bundle structures and identifies low-dimensional Lie groups whose nilmanifolds can be diffeomorphic to such varieties.
Findings
Nilmanifolds are diffeomorphic to trivial torus bundles over tori when $H^1(V)$ vanishes.
General nilmanifolds are either trivial torus bundles or torus bundles over tori.
Low-dimensional nilmanifolds (up to dimension 8) are classified regarding their potential to be diffeomorphic to smooth quasi-projective varieties.
Abstract
Let be a smooth quasi-projective variety for some smooth projective variety and a divisor with normal crossings. Assume that is diffeomorphic to a non-compact nilmanifold . We show that is diffeomorphic to a trivial bundle over a torus if the first cohomology of vanishes. Moreover, in general, we show that is diffeomorphic to a trivial bundle over a -dimensional torus , or a trivial bundle such that is a torus bundle over a torus . Conversely, we consider whether non-compact nilmanifolds are diffeomorphic to a smooth quasi-projective variety. We determine the Lie groups of dimension up to such that corresponding non-compact nilmanifolds may be…
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Taxonomy
TopicsGeometry and complex manifolds · Algebraic Geometry and Number Theory · Homotopy and Cohomology in Algebraic Topology
