Algebraic dimension and sophisticated subvarieties of hypercomplex nilmanifolds
Authors: Anna Abasheva, Misha Verbitsky
Summary: A nilmanifold is a (left) quotient of a nilpotent Lie group by a cocompact lattice. A hypercomplex construction on a manifold is a triple of advanced construction operators satisfying the quaternionic relations. A hypercomplex nilmanifold is a compact quotient of a nilpotent Lie group geared up with a left-invariant hypercomplex construction. Such a manifold admits a complete 2-dimensional sphere S2 of advanced constructions induced by quaternions. We show that for any hypercomplex nilmanifold M and a generic advanced construction L∈S2, the advanced manifold (M,L) has algebraic dimension 0. A stronger result’s confirmed when the hypercomplex nilmanifold is abelian. Contemplate the Lie algebra of left-invariant vector fields of Hodge kind (1,0) on the corresponding nilpotent Lie group with respect to some advanced construction I∈S2. A hypercomplex nilmanifold is named abelian when this Lie algebra is abelian. We show that every one advanced subvarieties of (M,L) for generic L∈S2 on a hypercomplex abelian nilmanifold are additionally hypercomplex nilmanifolds.