Aeppli-Bott-Chern Massey products on non-Kähler solvmanifolds

TL;DR

The paper studies obstructions to Kähler-like geometry on non-Kähler solvmanifolds by computing explicit non-trivial triple ABC-Massey products. It analyzes three families—Bigalke-Rollenske manifolds, generalized Nakamura manifolds, and compact quotients of $\mathbb{C}^{2n}\ltimes_\rho\mathbb{C}^{2m}$—using left-invariant forms and Bott-Chern/Aeppli cohomology to demonstrate non-formality and the absence of astheno-Kähler metrics. In certain Nakamura cases satisfying the $\partial\overline{\partial}$-Lemma, geometrically-BC-formal metrics may exist, but non-lemma cases exhibit non-vanishing ABC-Massey products and metric obstructions. Overall, the work provides explicit, computable obstructions to Kähler-like structure on concrete solvmanifold families and reinforces the role of ABC-Massey products as indicators of non-formality in complex geometry.

Abstract

In this paper, we present explicit computations of non-trivial triple $ABC$-Massey products on non-Kähler solvmanifolds endowed with an invariant complex structure. We prove that the {\em Bigalke-Rollenske manifold}, the {\em generalized Nakamura manifolds} satisfying some suitable assumptions and compact quotients of the solvable Lie group $\mathbb{C}^{2n}\ltimes_ρ \mathbb{C}^{2m}$ have non-vanishing triple $ABC$-Massey products. Furthermore, such manifolds have no astheno-Kähler metric.

Theorems & Definitions (17)

• Theorem 3.1
• proof
• Theorem 4.1
• proof
• Theorem 4.2
• Lemma 4.3
• proof : Proof of Lemma \ref{['BC-coh']}
• Remark 4.4
• Lemma 4.5
• proof : Proof of Lemma \ref{['powers']}