Let $X$ be a complex manifold and we have the natural inclusion of sheaves $\mathbb{C}\to \mathcal{O}_X$, where $\mathbb{C}$ is the constant sheaf and $\mathcal{O}_X$ is the sheaf of holomorphic functions on $X$. As a result we have a natural map of cohomologies $H^p(X,\mathbb{C})\to H^p(X,\mathcal{O}_X)$ for any integer $p$. By Dolbeault theorem we have $H^p(X,\mathcal{O}_X)\cong H^{0,p}_{\bar{\partial}}(X)$ therefore we obtain a natural map $$ H^p(X,\mathbb{C})\to H^{0,p}_{\bar{\partial}}(X). $$
Now we further assume $X$ is a compact Kahler manifold. Then we have the Hodge decomposition $$ H^p(X,\mathbb{C})\cong \bigoplus_{s+t=p}H^{s,t}_{\bar{\partial}}(X). $$ Then we have the natural projection $H^p(X,\mathbb{C})\to H^{0,p}_{\bar{\partial}}(X)$.
My question is: are these two maps $H^p(X,\mathbb{C})\to H^{0,p}_{\bar{\partial}}(X)$ coincide when $X$ is a compact Kahler manifold? Why?