$p$-K\"ahler and balanced structures on nilmanifolds with nilpotent complex structures

TL;DR

This paper investigates the existence of $p$-K"ahler and balanced structures on nilmanifolds with nilpotent complex structures, revealing new insights into their geometric properties and spectral sequence behavior.

Contribution

It determines the conditions under which $p$-K"ahler structures exist on such nilmanifolds and shows the independence of balanced metrics from the Fr"olicher spectral sequence degeneracy step.

Findings

Identifies the optimal $p$ for $p$-K"ahler structures on nilmanifolds.

Shows no direct relation between balanced metrics and spectral sequence degeneracy.

Demonstrates that the degeneracy step can be arbitrarily large on balanced manifolds.

Abstract

Let $(X,J)$ be a nilmanifold with a left-invariant nilpotent complex structure. We study the existence of $p$-K\"ahler structures (which include K\"ahler and balanced metrics) on $X$. More precisely, we determine an optimal $p$ such that there are no $p$-K\"ahler structures on $X$. Finally, we show that, contrarily to the K\"ahler case, on compact complex manifolds there is no relation between the existence of balanced metrics and the degeneracy step of the Fr\"olicher spectral sequence. More precisely, on balanced manifolds the degeneracy step can be arbitrarily large.