Algebraic dimension and complex subvarieties of 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 ∈ 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 ∈ 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 ∈ S 2 on a hypercomplex abelian nilmanifold are also hypercomplex nilmanifolds.

Complex curves in hypercomplex nilmanifolds with H-solvable Lie algebras

“…It was already known that a nilpotent Lie algebra that admits an abelian hypercomplex structure is H-solvable [AV,Proposition 4.5], see also [Rol,Corollary 3.11].…”

“…Example 1.10: To provide an example of an H-solvable Lie algebra with a non-abelian hypercomplex structure, we first define the Kodaira surface, following [Has], see also [AV,Example 1.7]. Consider the Lie algebra g = x, y, z, t , such that the only non-zero commutator is [x, y] = z.…”