Hodge Structures and Homological Methods

Part of the Universitext book series (UTX)


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.


Manifold Stein Tate Betti 


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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of MathematicsPurdue UniversityWest LafayetteUSA

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