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A nilmanifold is a (left) quotient of a nilpotent Lie group by a cocompactlattice. A hypercomplex structure on a manifold is a triple of complexstructure operators satisfying the quaternionic relations. A hypercomplexnilmanifold is a compact quotient of a nilpotent Lie group equipped with aleft-invariant hypercomplex structure. Such a manifold admits a whole2-dimensional sphere $S^2$ of complex structures induced by quaternions. Weprove 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 strongerresult is proven when the hypercomplex nilmanifold is abelian. Consider the Liealgebra of left-invariant vector fields of Hodge type (1,0) on thecorresponding nilpotent Lie group with respect to some complex structure $I\inS^2$. A hypercomplex nilmanifold is called abelian when this Lie algebra isabelian. We prove that all complex subvarieties of $(M,L)$ for generic $L\inS^2$ on a hypercomplex abelian nilmanifold are also hypercomplex nilmanifolds.

Algebraic dimension and complex subvarieties of hypercomplex nilmanifolds