Complex curves in hypercomplex nilmanifolds with H-solvable Lie algebras

Abstract

An operator $I$ on a real Lie algebra $A$ is called a complex structure operator if $I^2=-Id$ and the $\sqrt{-1}$-eigenspace $A^{1,0}$ is a Lie subalgebra in the complexification of $A$. A hypercomplex structure on a Lie algebra $A$ is a triple of complex structures $I,J$ and $K$ on $A$ satisfying the quaternionic relations. We call a hypercomplex nilpotent Lie algebra quaternionic-solvable if there exists a finite filtration by quaternionic-invariant subalgebras with commutative subquotients which converges to zero. We give examples of quaternionic-solvable hypercomplex structures on a nilpotent Lie algebra and conjecture that all hypercomplex structures on nilpotent Lie algebras are quaternionic-solvable. Let $(N,I,J,K)$ be a compact hypercomplex nilmanifold associated to an quaternionic-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 $(N,L)$.