# Reference request for “Hodge Theorem”

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!

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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.

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Thank you! It is exactly what I was looking for! –  Giovanni De Gaetano Apr 25 '12 at 11:47
Another reference is Voisin's Hodge Theory and Complex Algebraic Geometry. I think it's a bit friendlier than Griffiths and Harris. –  Dylan Moreland Apr 25 '12 at 13:21

There is a proof of Hodge theorem in John Roe's book, Elliptic Operators, topology, and asymptotic expansion of heat kernel. The proof is only two page long and very readable. However he only proved it for the classical Laplace operator, and the statement holds for any generalized Laplace operator. Another place you can find a reference is Richard Melrose's notes on microlocal analysis, which you can find in his homepage. But the proof is difficult to read without some background.

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