Complex curves in hypercomplex nilmanifolds with H-solvable Lie algebras

Abstract

An operator I on a real Lie algebra g is called a complex structure operator if I 2 = − Id and the √ −1-eigenspace g 1,0 is a Lie subalgebra in the complexification of g. A hypercomplex structure on a Lie algebra g is a triple of complex structures I, J and K on g satisfying the quaternionic relations. We call a hypercomplex nilpotent Lie algebra H-solvable if there exists a sequence of H-invariant subalgebrasWe give examples of H-solvable hypercomplex structures on a nilpotent Lie algebra and conjecture that all hypercomplex structures on nilpotent Lie algebras are H-solvable. Let (N, I, J, K) be a compact hypercomplex nilmanifold associated to an H-solvable hypercomplex Lie algebra. We prove that, for a general complex structure L induced by quaternions, there are no complex curves in a complex manifold (N, L).