Reference request for "Hodge Theorem"

Question

I have been told about a theorem (it was called Hodge Theorem), which states the following isomorphism:

$H^q(X, E) \simeq Ker(\Delta^q).$

Where $X$ is a Kähler Manifold, $E$ an Hermitian vector bundle on it and $\Delta^q$ is the Laplacian acting on the space of $(0,q)$-forms $A^{0,q}(X, E)$.

Unfortunately I couldn´t find it in the web. Anyone knows a reliable reference for such a theorem? (In specific I´m looking for a complete list of hypothesis needed and for a proof.)

Thank you!

Answer 1

The Hodge theorem is proved in detail in chapter 0 of Principles of Algebraic Geometry by Griffiths and Harris. It's also worth mentioning that in chapter 1 they prove the Kodaira vanishing theorem and Kodaira-Serre duality more or less as corollaries to the Hodge theorem.