Algebraic dimension and complex subvarieties of hypercomplex nilmanifolds

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