Some Remarks on Nijenhuis Bracket, Formality, and K\"ahler Manifolds

TL;DR

This paper explores the relationship between Nijenhuis brackets, formality, and K"ahler manifolds by analyzing derivations, tensor fields, and conditions under which the $d ext{}ar{d}$-lemma holds, leading to a new characterization of K"ahler structures.

Contribution

It establishes a novel link between the vanishing Nijenhuis torsion of certain tensor fields and the K"ahler condition, providing new criteria for formality and complex structure compatibility.

Findings

Vanishing Nijenhuis torsion implies the orthogonal component forms a complex structure.

Conditions on tensor fields ensure the $dar{d}$-lemma holds, implying formality.

Characterization of K"ahler structures via tensor field properties.

Abstract

One (actually, almost the only effective) way to prove formality of a differentiable manifold is to be able to produce a suitable derivation $\delta$ such that $d\delta$-lemma holds. We first show that such derivation $\delta$ generates a (1,1)-tensor field (we denote it by $R$). Then, we show that the supercommutation of $d$ and $\delta$ (which is a natural, essentially necessary condition to get a $d\delta$-lemma) is equivalent to vanishing of the Nijenhujis torsion of $R$. Then, we are looking for sufficient conditions that ensure the $d\delta$-lemma holds: we consider the cases when $R$ is self adjoint with respect to a Riemannian metric or compatible with an almost symplectic structure. Finally, we show that if $R$ is scew-symmetric with respect to a Riemannian metric, has constant determinant, and if its Nijenhujis torsion vanishes, then the orthogonal component of $R$ in its polar decomposition is a complex structure compatible with the metric, which gives us a new characterization of K\"ahler structures