  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.
