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Hodge Theory and Complex Algebraic Geometry I: Volume 1 by Claire Voisin (Paperback, 2007)
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She has done a remarkably good job.'. by Claire Voisin. Claire Voisin is a Professor at the Institut des Hautes Etudes Scientifiques, France. The author then proves the Kaehler identities, which leads to the hard Lefschetz theorem and the Hodge index theorem.