Hodge decomposition theorem on compact $d$-K\"ahler manifolds

TL;DR

This paper proves the Hodge decomposition theorem for compact d-Kähler manifolds, linking de-Rham and Dolbeault cohomology groups, thereby extending classical Hodge theory to a broader class of complex manifolds.

Contribution

It establishes the Hodge decomposition theorem specifically for compact d-Kähler manifolds, a significant generalization in complex differential geometry.

Findings

Hodge decomposition holds on compact d-Kähler manifolds

Established relationships between de-Rham and Dolbeault cohomology

Extended classical Hodge theory to d-Kähler setting

Abstract

In this article, we will explore the fundamental concepts, including various basic concepts on $d$-complex manifolds, along with several differential operators and examine the relationships between them. A $d$-K\"ahler manifold is a $d$-complex manifold equipped with a metric that satisfies a specific condition. We prove the Hodge decomposition theorem on compact $d$-K\"ahler manifolds, which establishes a crucial relationship between certain de-Rham cohomology groups and Dolbeault cohomology groups on a compact $d$-K\"ahler manifold .