Locally conformal SKT structures

• Autor(a) principal: Ferreira, Ana Cristina
• Data de Publicação: 2022
• Outros Autores: Djebbar, Bachir, Fino, Anna, Larbi Youcef, Nourhane Zineb
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
• Texto Completo: https://hdl.handle.net/1822/82954
• topic: Hermitian metrics; Locally conformal SKT metrics; Nilpotent Lie algebras; Almost abelian Lie algebras; Ciências Naturais::Matemáticas; Science & Technology

A Hermitian metric on a complex manifold is called SKT (strong K ̈ahler with torsion) if the Bismut torsion 3-form H is closed. As the conformal generalization of the SKT condition, we introduce a new type of Hermitian structure, called locally conformal SKT (or shortly LCSKT). More precisely, a Hermitian structure (J, g) is said to be LCSKT if there exists a closed nonzero 1-form α such that dH = α∧H. In this paper, we consider nontrivial LCSKT structures, i.e. we assume that dH ̸= 0 and we study their existence on Lie groups and their compact quotients by lattices. In particular, we classify six-dimensional nilpotent Lie algebras admitting a LCSKT structure and we show that, in contrast to the SKT case, there exists a six-dimensional 3- step nilpotent Lie algebra admitting a nontrivial LCSKT structure. Moreover, we show a characterization of even dimensional almost abelian Lie algebras admitting a nontrivial LCSKT structure, which allows us to construct explicit examples of six-dimensional unimodular almost abelian Lie algebras admitting a nontrivial LCSKT structure. The compatibility between the LCSKT and the balanced condition is also discussed, showing that a Hermitian structure on a six-dimensional nilpotent or a 2n-dimensional almost abelian Lie algebra cannot be simultaneously LCSKT and balanced, unless it is K ̈ahler.