In Voisin's "Hodge Theory and Complex Algebraic geometry I" Corollary 7.6:
$\textbf{Corollary 7.6}$ For every $p\leq n$, $H^{p,0}(X)$ is isomorphic to the space of holomorphic forms of degree $p$ on $X$.
In the proof of Corollary 7.6, Proposition 7.5 was cited:
$\textbf{Proposition 7.5}$ Let $X$ be a compact $\mathbf{Kahler}$ manifold. Let $F^pA^k(X)$ be the set of complex differential forms which are sums of form of type $(r,k-r)$ with $r\geq p$ at every point. Then we have $$F^pH^k(X,\mathbf{C})=\frac{\textrm{Ker}(F^pA^k\to F^pA^{k+1})}{\textrm{Im}(F^pA^{k-1}\to F^pA^{k})}$$
I do not understand the logic, isn't that we have $H^{p,q}(X)\cong H^q(X,\Omega^p_X)$ for all complex manifolds? I don't know where I had the wrong understanding.
It is not obvious that $H^{p,0}$(X) is iso to the $H^0(X,Ω^p)$ (except you define it like that :-)). It is called Dolbeault isomorphism. Which is direct corollary of Hodge theorem. This is always true for compact complex manifold, I mean $H^{p,0}$(X) is iso to the $H^0(X,Ω^p)$. and can be generalized to any holomorphic vector bundle. For noncompact complex manifold, Maybe we need forms with compact support, Which I don't know.