Complex curves in hypercomplex nilmanifolds with [formula omitted]-solvable Lie algebras

Abstract

An operator I on a real Lie algebra g is called a complex structure operator if I2=−Id and the −1-eigenspace g1,0 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 subalgebrasg1H⊃g2H⊃⋯⊃gk−1H⊃gkH=0, such that [giH,giH]⊂gi+1H. We 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).