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Compact solvmanifolds with calibrated and cocalibrated \(\mathrm {G}_2\)-structures

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Abstract

We give a method to obtain new solvable 7-dimensional Lie algebras endowed with closed and coclosed \(\mathrm {G}_2\)-structures starting from 6-dimensional solvable Lie algebras with symplectic half-flat and half-flat \(\mathrm {SU}(3)\)-structures, respectively. Provided the existence of a lattice for the corresponding Lie groups we obtain new examples of compact solvmanifolds endowed with calibrated and cocalibrated \(\mathrm {G}_2\)-structures. As an application of this construction we also obtain a formal compact solvmanifold with first Betti number \(b_1=1\) endowed with a calibrated \(\mathrm {G}_2\)-structure and such that does not admit any invariant torsion-free \(\mathrm {G}_2\)-structure.

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Authors and Affiliations

  1. Departamento de Matemáticas-I.U.M.A., Universidad de Zaragoza, C/Pedro Cerbuna, 12, 50009, Zaragoza, Spain

    Víctor Manero

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  1. Víctor Manero
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Appendix

Appendix

see Table 2.

Table 2 Lie algebras endowed with a closed \(\mathrm {G}_2\)-structure obtained in Proposition 3.3, not coming from the 6-dimensional abelian Lie algebra
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Manero, V. Compact solvmanifolds with calibrated and cocalibrated \(\mathrm {G}_2\)-structures. manuscripta math. 162, 315–339 (2020). https://doi.org/10.1007/s00229-019-01133-w

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