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Theories of manifolds w/ extra structure and singularities

Many different objects in mathematics can be described as manifolds with extra structure. Among the most famous examples of these are smooth manifolds, Riemannian manifolds, complex manifolds, and ...
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Hodge Theory (Voisin)

I have a strong understanding of Representation Theory but am interested in learning from Voisin, Hodge Theory and Complex Algebraic Geometry. What are the prerequisites to learning from this textbook?...
user39691's user avatar