K\"{a}hler identities for almost complex manifolds

TL;DR

This paper generalizes Kähler identities to compact almost complex manifolds by analyzing the Clifford bundle and Dirac operator, extending classical results beyond Kähler manifolds.

Contribution

It introduces a novel approach using Clifford bundles and Dirac operators to extend Kähler identities to almost complex manifolds.

Findings

Established generalized Kähler identities for almost complex manifolds.

Connected Clifford bundle analysis with classical differential geometry results.

Provided a framework for future studies on geometric structures of almost complex manifolds.

Abstract

We obtain a generalization, for a general compact almost complex manifold, of the well-known K\"{a}hler (or Hodge) identities for K\"{a}hler manifolds involving the commutators of the exterior differential and the Lefschetz operator and its adjoint. The main idea is to study the problem on the Clifford bundle via the Dirac operator, and then translate the results to the exterior bundle.