Algebraic dimension and complex subvarieties of hypercomplex nilmanifolds

TL;DR

This paper investigates the algebraic and complex subvariety structures of hypercomplex nilmanifolds, showing that generic complex structures lead to algebraic dimension zero and that subvarieties are also hypercomplex nilmanifolds.

Contribution

It proves that generic complex structures on hypercomplex nilmanifolds have algebraic dimension zero, and that subvarieties are hypercomplex nilmanifolds in the abelian case.

Findings

Generic complex structures have algebraic dimension zero.

Subvarieties of hypercomplex abelian nilmanifolds are also hypercomplex nilmanifolds.

Results apply to both general and abelian hypercomplex nilmanifolds.

Abstract

A nilmanifold is a (left) quotient of a nilpotent Lie group by a cocompact lattice. A hypercomplex structure on a manifold is a triple of complex structure operators satisfying the quaternionic relations. A hypercomplex nilmanifold is a compact quotient of a nilpotent Lie group equipped with a left-invariant hypercomplex structure. Such a manifold admits a whole 2-dimensional sphere $S^2$ of complex structures induced by quaternions. We prove that for any hypercomplex nilmanifold $M$ and a generic complex structure $L\in S^2$, the complex manifold $(M,L)$ has algebraic dimension 0. A stronger result is proven when the hypercomplex nilmanifold is abelian. Consider the Lie algebra of left-invariant vector fields of Hodge type (1,0) on the corresponding nilpotent Lie group with respect to some complex structure $I\in S^2$. A hypercomplex nilmanifold is called abelian when this Lie algebra is abelian. We prove that all complex subvarieties of $(M,L)$ for generic $L\in S^2$ on a hypercomplex abelian nilmanifold are also hypercomplex nilmanifolds.