I’m currently studying Complex Geometry by Daniel Huybrechts and I can’t understand why it is crucial to prove that Hodge Decomposition does not depend on the chosen metric. I’m talking about Corollary 3.2.12 (p. 129).
We have already proved that (p.128) $\mathcal{H}_{\bar{\partial}}^{p,q}(X,g) \simeq H^{p,q}(X)$ for any Kahler metric. And this in my opinion shows that the decomposition does not depend on the metric. We also know that (p. 126 prop 3.2.6) $\mathcal{H}_{\bar{\partial}}^{k}(X,g) = \bigoplus\limits_{p+q=k}\mathcal{H}_{\bar{\partial}}^{p,q}(X,g)$.
Why should the decomposition depend on the metric?