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In Voisin's beautiful book on Hodge theory she gives a proof that every cohomology class can be represented by a harmonic form by referring to the the following theorem on elliptic operators:

Let $P : E → F \ $ be an elliptic differential operator on a compact manifold. Assume that $E$ and $F$ are of the same rank, and are equipped with metrics. Then $\operatorname{ker} P \subset C^{\infty}(E)\ $ is finite-dimensional, $P( C^{\infty}(E)) \subset C^{\infty}(F) \ $ is closed and of finite codimension, and we have a decomposition as an orthogonal direct sum (for the $L^2$ metric) $$ C^{\infty}(E) = \operatorname{ker} P \oplus P^{\ast}( C^{\infty}(F)), $$ which she then applies to the Laplace operator.

Unfortunately, her reference for this (Demailly - Théorie de Hodge L2 et Théor`emes d'annulation) is in French, which I don't understand.

I am primarily hoping for a reference on elliptic operators which proves this theorem; however, at the same time I also feel like I should know more about elliptic operators in general, and 'how they can be used in geometry' (can they, actually? I'm thinking about Atiyah-Singer here, which I don't understand at all, so I'm not sure.), so if you can give me a reference about that, it would be much appreciated, too.

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    For references for the Atiyah-Singer Index Theorem, see this MO thread: http://mathoverflow.net/questions/1162/atiyah-singer-index-theorem2011-09-11
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    Concerning elliptic differential operators on manifolds and the Atiyah-Singer theorem, you may be interested in my recent answer at http://math.stackexchange.com/a/252225/40582012-12-06

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