Some computations on trivial canonical-bundle solvmanifolds

TL;DR

This paper computes Dolbeault and Bott-Chern cohomology for six-dimensional solvmanifolds with trivial canonical bundle, introduces finite-dimensional subcomplexes for cohomology calculations, and explores formality and Massey products.

Contribution

It provides explicit cohomology computations and structural decompositions for a class of complex solvmanifolds, extending understanding of their geometric properties.

Findings

Finite-dimensional subcomplexes compute cohomology accurately.

Decomposition into indecomposable complexes simplifies analysis.

Characterization of the $ar{ ext{d}}$-lemma and Massey products for these manifolds.

Abstract

We compute the Dolbeault and the Bott-Chern cohomology of six dimensional solvmanifolds endowed with a complex structure of splitting type, introduced by Kasuya, and with trivial canonical bundle. We build, following results by Angella and Kasuya, finite dimensional double subcomplexes $(C_\Gamma^{\bullet,\bullet},\partial,\bar{\partial})\subseteq(\wedge^{\bullet,\bullet}G/\Gamma,\partial,\bar{\partial})$ for which the inclusion is an isomorphism in cohomology. We decompose such double complexes into indecomposable ones. Lastly, we study some notions of formality for this class of manifolds, giving a characterization of the $\partial\bar{\partial}$-Lemma property in general complex dimension, and we compute triple ABC-Massey products on them.