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Generalized Kähler almost abelian Lie groups

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Abstract

We study left-invariant generalized Kähler structures on almost abelian Lie groups, i.e., on solvable Lie groups with a codimension-one abelian normal subgroup. In particular, we classify six-dimensional almost abelian Lie groups which admit a left-invariant complex structure and establish which of those have a left-invariant Hermitian structure whose fundamental 2-form is \(\partial {{\overline{\partial }}}\)-closed. We obtain a classification of six-dimensional generalized Kähler almost abelian Lie groups and determine the six-dimensional compact almost abelian solvmanifolds admitting an invariant generalized Kähler structure. Moreover, we prove some results in relation to the existence of holomorphic Poisson structures and to the pluriclosed flow.

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Acknowledgements

The authors would like to thank Ramiro Lafuente and Luigi Vezzoni for useful discussions and Jeffrey Streets for pointing out the reference [37]. The authors are also grateful to an anonymous referee for useful comments. The paper is supported by Project PRIN 2017 “Real and complex manifolds: Topology, Geometry and Holomorphic Dynamics” and by GNSAGA of INdAM.

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  1. Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, 10123, Torino, Italy

    Anna Fino & Fabio Paradiso

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  1. Anna Fino
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Correspondence to Anna Fino.

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Appendix: Six-dimensional almost abelian Lie algebras

Appendix: Six-dimensional almost abelian Lie algebras

Here we provide the classification of six-dimensional non-nilpotent almost abelian Lie algebras. Table 1 features the indecomposable ones, whose classification was obtained in [34] and refined in [36]. In Table 2 one can find six-dimensional non-nilpotent almost abelian Lie algebras which can be decomposed as a direct sum of two or more Lie algebras: these were singled out by studying [32, 33]. For each Lie algebra in Tables 1 and 2 we include the conditions on the parameters (if any) for which the algebra is unimodular.

In Table  3 we give an explicit complex structure for every Lie algebra in Theorem 3.2 (the conditions on the parameters involved in the structure equations are given in Theorem 3.2).

Table 1 Six-dimensional indecomposable non-nilpotent almost abelian Lie algebras
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Table 2 Six-dimensional decomposable non-nilpotent almost abelian Lie algebras
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Table 3 Six-dimensional non-nilpotent almost abelian Lie algebras admitting a complex structure
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Fino, A., Paradiso, F. Generalized Kähler almost abelian Lie groups. Annali di Matematica 200, 1781–1812 (2021). https://doi.org/10.1007/s10231-020-01059-1

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