Complex structures on $2$-step nilpotent Lie algebras

Início: 09/06/2023 17:00

Término: 09/06/2023 18:00

Palestrante: Maria Laura Barberis (UNC)

E-mail do Palestrante: barberis@famaf.unc.edu.ar

Resumo: There is a notion of nilpotent complex structures on nilpotent Lie algebras introduced by Cordero-Fernández-Gray-Ugarte (2000). Not every complex structure on a nilpotent Lie algebra $mathfrak{n}$ is nilpotent, but when $mathfrak{n}$ is $2$-step nilpotent any complex structure on $mathfrak{n}$ is nilpotent of step either $2$ or $3$ (a fact proved by J. Zhang in 2022). The class of nilpotent complex structures of step $2$ strictly contains the space of abelian and bi-invariant complex structures on a $2$-step nilpotent Lie algebra. In this work in progress, we obtain a characterization of the $2$-step nilpotent Lie algebras whose corresponding Lie groups admit a left invariant complex structure. We consider separately the cases when the complex structure is nilpotent of step $2$ or $3$. Some applications of our results to Hermitian geometry are discussed, for instance, it turns out that the $2$-step nilpotent Lie algebras constructed by Tamaru from Hermitian symmetric spaces admit pluriclosed (or SKT) metrics. We also show that abelian complex structures are frequent on naturally reductive $2$-step nilmanifolds, while it is known (Del Barco-Moroianu) that these do not admit orthogonal bi-invariant complex structures.