The article under review presents a classification (up to finite cover) of flat compact complete Hermite-Lorentz manifolds up to complex dimension \(4\).
Section 2 gives a presentation, valid in any dimension, of unipotent simply transitive groups of Hermite-Lorentz affine motion and a proposition about their conjugacy classes is proved.
In Section 3 one reduces, as for the Lorentzian case, the study of crystallographic groups to the study of lattices in simply transitive unipotent Lie groups. According to the main theorem from Section 3, the unipotent hypothesis from Section 2 only leaves out the easy abelian by cyclic case. This section ends with the classification of the latter case.
In Section 4 the classification, up to an isomorphism, of those \(H\) from Section 2 (unipotent groups acting simply transitively on \(a(\mathbf{C}^{n+1}),\) the affine space associated to the complex vector space \(\mathbf{C}^{n+1}\) endowed with a Hermitian form of signature \((n,1)\)) is started and their classification for the dimensions 2 and 3 is completed. In dimension 4 the classification of some particular cases, called degenerate cases, is given.
Section 5 completes the classification of the unipotent simply transitive groups \(H\) in dimension 4.
Section 6 deals with the study the Hermite-Lorentz crystallographic groups in the nilpotent case.
Finally in Section 7 the author gives some topological considerations about some compact flat Hermite-Lorentz manifolds that are virtually nilpotent, namely that they are finitely covered by torus bundles over tori.