Comptes Rendus
no. 2
Differential geometry
A remark on the Bismut–Ricci form on 2-step nilmanifolds
[Une remarque sur la forme de Bismut–Ricci des espaces homogènes sous l'action d'un groupe nilpotent de classe ≤2]

Présenté par : the Editorial Board

Mattia Pujia1 ; Luigi Vezzoni1
1 Dipartimento di Matematica G. Peano, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy
Comptes Rendus. Mathématique, Volume 356 (2018) no. 2, pp. 222-226.

Nous observons que, sur un groupe de Lie nilpotent de classe ≤2, équipé d'une structure de Kähler forte avec torsion (SKT), invariante à gauche, la partie (1,1) de la forme de Bismut–Ricci est définie semi-négative. Comme application, nous donnons une démonstration simplifée de la non-existence d'une métrique statique SKT sur un espace homogène sous l'action d'un groupe nilpotent de classe ≤2. Nous montrons également l'existence d'une solution à long terme du flot plurifermé dans ces mêmes espaces.

In this note, we observe that, on a 2-step nilpotent Lie group equipped with a left-invariant SKT structure, the (1,1)-part of the Bismut–Ricci form is seminegative definite. As an application, we give a simplified proof of the non-existence of invariant SKT static metrics on 2-step nilmanifolds and of the existence of a long-time solution to the pluriclosed flow in 2-step nilmanifolds.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2018.01.002
Mattia Pujia 1 ; Luigi Vezzoni 1

1 Dipartimento di Matematica G. Peano, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy 
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Mattia Pujia; Luigi Vezzoni. A remark on the Bismut–Ricci form on 2-step nilmanifolds. Comptes Rendus. Mathématique, Volume 356 (2018) no. 2, pp. 222-226. doi : 10.1016/j.crma.2018.01.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.01.002/

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☆ This work was supported by G.N.S.A.G.A. of I.N.d.A.M.

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