  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 .
