\( \partial \bar{\partial} \)-lemma and \(p\)-Kähler structures on families of solvmanifolds (Q6624130)

The \(\partial \bar{\partial}\)-lemma is a fundamental result in Kähler geometry, which in particular implies the Hodge decomposition theorem. Compact complex manifolds can be constructed as compact quotients of a solvable Lie group by a lattice, known as solvmanifolds, equipped with a complex structure. However, such manifolds are Kähler only if they are finite quotients of complex tori. Despite this, some solvmanifolds support special metrics, such as balanced metrics or \(p\)-Kähler metrics, which fulfill weaker conditions than Kählerianity. Notably, all known examples of manifolds that satisfy the \(\partial \bar{\partial}\)-lemma also admit a balanced metric. This paper investigates potential connections between \(p\)-Kähler structures and the \(\partial \bar{\partial}\)-lemma, providing families of \(p\)-Kähler manifolds. The main result offers examples of \((n + 1)\)-dimensional manifolds with Kodaira dimension zero that satisfy the \(\partial \bar{\partial}\)-lemma but lack \(p\)-Kähler structures for \(p \neq n\).