$p$-Kähler structures on nilmanifolds and holomorphically parallelizable manifolds

Abstract

We prove the Alessandrini-Bassanelli conjecture on nilmanifolds with nilpotent complex structures and on holomorphically parallelizable solvmanifolds. As a consequence, we classify holomorphically parallelizable solvmanifolds admitting $p$-Kähler structures, for lower values of $p$. We further provide new examples of $(n-2)$-Kähler manifolds, in the setting of compact holomorphically parallelizable manifolds with reductive universal cover.