# Invariants of complex structures on nilmanifolds

## Edwin Alejandro Rodríguez Valencia

Address: FaMAF and CIEM, Universidad Nacional de Córdoba, Medina Allende s/n, 5000 Córdoba, Argentina

E-mail: earodriguez@famaf.unc.edu.ar

Abstract: Let $(N, J)$ be a simply connected $2n$-dimensional nilpotent Lie group endowed with an invariant complex structure. We define a left invariant Riemannian metric on $N$ compatible with $J$ to be minimal, if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics with the same scalar curvature. In [7], J. Lauret proved that minimal metrics (if any) are unique up to isometry and scaling. This uniqueness allows us to distinguish two complex structures with Riemannian data, giving rise to a great deal of invariants.
We show how to use a Riemannian invariant: the eigenvalues of the Ricci operator, polynomial invariants and discrete invariants to give an alternative proof of the pairwise non-isomorphism between the structures which have appeared in the classification of abelian complex structures on $6$-dimensional nilpotent Lie algebras given in [1]. We also present some continuous families in dimension $8$.

AMSclassification: primary 53C15; secondary 32Q60, 53C30, 22E25, 37J15.

Keywords: complex, nilmanifolds, nilpotent Lie groups, minimal metrics, Pfaffian forms.

DOI: 10.5817/AM2015-1-27