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I'm looking for a text book which has detailed explanations on the subject. If you know it, please let me know. Since I don't have an easy access to a university library, I prefer a book which can be bought(and not so expensive) in an online bookstore like Amazon. Of course, free downloadable books are welcome! Thanks.

EDIT I'm particularly interested in the theory of harmonic integrals on non-compact complete Kähler manifolds.

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I believe this is called "Hodge theory" today. You'll find a wealth of survey articles and books by googling this term. For recent books, "Complex geometry" by D. Huybrechts is a good introduction, "Hodge theory" by C. Voisin is excellent, and for all the details see J.-P. Demailly's "Complex analytic and differential geometry" (available for free on his website). You may want to start by considering the compact case; there exists a Hodge theory on non-compact manifolds, but there are additional technical difficulties to overcome. – Gunnar Þór Magnússon May 23 '12 at 11:11
Thank you, sir. I'd like to know the theory of harmonic integrals(or Hodge theory) of non-compact complete Kähler manifolds. I'm sorry I didn't make my question clear enough. – Makoto Kato May 23 '12 at 12:36
Ok. Then try Chapter 8 of Demailly's book (p. 367 seems particularily relevant to your interests) and google $L^2$ cohomology and Hodge theory. Know also that there exists a "mixed Hodge theory", that tries to explain the geometry of a "compactifyable" manifold, or what one gets by removing a hypersurface (or less) from a compact manifold. Both, or a mix of the two, may be what you seek. – Gunnar Þór Magnússon May 24 '12 at 2:14
I found what I was looking for in the Demailly's book. Thank you so much. – Makoto Kato May 24 '12 at 8:46

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