If $X$ is a compact Kahler manifold, then the Hodge decomposition is $$H^n(X,\mathbb{C})=\bigoplus_{i+j=n} H^{i,j}(X).$$ Then the left hand side is homotopy invariant and therefore, so is therefore so is the right. However, as for instance was mentined in these two questions Hodge decomposition and homotopy equivalence Dolbeault Cohomology is invariant under homeomorphisms, the individual $$H^{i,j}(X)$$ are not homotopy invariant, in fact not even diffeomorphism invariant.
Is there a proof that $\bigoplus_{i+j=n} H^{i,j}(X)$ is homotopy invariant without using the Hodge theorem, i.e. the isomorphism to singular cohomology?
