Invariants of complex structures on nilmanifolds

TL;DR

This paper introduces Riemannian invariants derived from minimal metrics on nilmanifolds with invariant complex structures to distinguish non-isomorphic structures, providing new proofs and classifications in low dimensions.

Contribution

It develops a method using Ricci eigenvalues and polynomial invariants to classify complex structures on nilmanifolds, offering an alternative to existing classification approaches.

Findings

Distinct complex structures in 6D are distinguished using Riemannian invariants.

Alternative proof of non-isomorphism for structures classified in prior work.

Identification of continuous families of structures in 8D.

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 [L1], 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 [ABD]. We also present some continuous families in dimension 8.