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PreprintCC BY 4.0arXiv

On the generalized geometry of almost abelian solvmanifolds

  • Mauro de Andrade Pinto
  • Letícia Camponês do Brasil Maia
  • Leonardo F. Cavenaghi
  • Pedro Antonio Muniz Martins
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Abstract

We study left-invariant generalized complex structures on almost abelian Lie groups GA with Lie algebra gA=ℝ e0ltimesAh, where h is an abelian ideal of codimension one, and on their compact quotients. First, when A is diagonalizable over ℝ, we characterize all admissible types by pairings of its eigenvalues and characterize the structures admitting a closed invariant pure-spinor generator. For general A, we obtain Jordan-theoretic type bounds and a construction using a complex quotient and a symplectic ideal. Finally, in dimension six, we establish nonexistence results and give explicit intermediate-type constructions.

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Cite

BibTeXRIS
BibTeX
@misc{pinto2026generalized,
  title = {{On the generalized geometry of almost abelian solvmanifolds}},
  author = {Mauro de Andrade Pinto and Letícia Camponês do Brasil Maia and Leonardo F. Cavenaghi and Pedro Antonio Muniz Martins},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.00254},
  eprint = {2609.00254},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.00254},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On the generalized geometry of almost abelian solvmanifolds
AU  - Mauro de Andrade Pinto
AU  - Letícia Camponês do Brasil Maia
AU  - Leonardo F. Cavenaghi
AU  - Pedro Antonio Muniz Martins
PY  - 2026
DA  - 2026/09/19
PB  - arXiv
DO  - 10.48550/arxiv.2609.00254
UR  - https://arxiv.org/abs/2609.00254
AB  - We study left-invariant generalized complex structures on almost abelian Lie groups G_A with Lie algebra g_A=ℝ e_0ltimes_Ah, where h is an abelian ideal of codimension one, and on their compact quotients. First, when A is diagonalizable over ℝ, we characterize all admissible types by pairings of its eigenvalues and characterize the structures admitting a closed invariant pure-spinor generator. For general A, we obtain Jordan-theoretic type bounds and a construction using a complex quotient and a symplectic ideal. Finally, in dimension six, we establish nonexistence results and give explicit intermediate-type constructions.
ER  -