On the generalized cohomologies of almost abelian solvmanifolds
Abstract
Generalized complex structures give rise to generalized Dolbeault, Bott--Chern and Aeppli cohomologies. We study these cohomologies for invariant generalized complex structures on almost abelian Lie algebras gA=ℝ e0ltimesAh, where h is an abelian ideal of codimension one, and on almost abelian solvmanifolds Γbackslash GA. We first show that the invariant generalized Dolbeault cohomology can be identified with the Chevalley--Eilenberg cohomology of the complex Lie algebra ℒ, the +i-eigenspace of the structure, with coefficients in a character determined by the canonical line. We prove that ℒ is itself an almost abelian Lie algebra; thus the cohomology reduces to the kernel and cokernel of an explicit derivation. Assuming that A is diagonalizable over ℝ, we obtain a combinatorial formula for the cohomology dimensions and compare with compact quotients. Other comparison results are discussed, noting that the cohomology of invariant forms always injects into that of Γbackslash GA.
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BibTeX
@misc{pinto2026generalized,
title = {{On the generalized cohomologies 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 = oct,
doi = {10.48550/arxiv.2610.20153},
eprint = {2610.20153},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20153},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the generalized cohomologies 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 - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.20153
UR - https://arxiv.org/abs/2610.20153
AB - Generalized complex structures give rise to generalized Dolbeault, Bott--Chern and Aeppli cohomologies. We study these cohomologies for invariant generalized complex structures on almost abelian Lie algebras g_A=ℝ e_0ltimes_Ah, where h is an abelian ideal of codimension one, and on almost abelian solvmanifolds Γbackslash G_A. We first show that the invariant generalized Dolbeault cohomology can be identified with the Chevalley--Eilenberg cohomology of the complex Lie algebra ℒ, the +i-eigenspace of the structure, with coefficients in a character determined by the canonical line. We prove that ℒ is itself an almost abelian Lie algebra; thus the cohomology reduces to the kernel and cokernel of an explicit derivation. Assuming that A is diagonalizable over ℝ, we obtain a combinatorial formula for the cohomology dimensions and compare with compact quotients. Other comparison results are discussed, noting that the cohomology of invariant forms always injects into that of Γbackslash G_A.
ER -