Abstract
Our next goal is to make the Hodge decomposition functorial with respect to holomorphic maps. This is not immediate, since the pullback of a harmonic form along a holomorphic map is almost never harmonic. The trick is to state things in a way that depends only on the complex structure: a cohomology class is of type (p, q) if it can be represented by a form with p dzi’s and by a form with q \(d \bar{Z}j\)’s. Of course, just making a definition is not enough. There is something to be proved. The main ingredients are the previous Hodge decomposition for harmonic forms together with some homological algebra, which we develop here.
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© 2012 Springer Science+Business Media, LLC
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Arapura, D. (2012). Hodge Structures and Homological Methods. In: Algebraic Geometry over the Complex Numbers. Universitext. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-1809-2_12
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DOI: https://doi.org/10.1007/978-1-4614-1809-2_12
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Print ISBN: 978-1-4614-1808-5
Online ISBN: 978-1-4614-1809-2
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