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

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 .