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? I haven't studied differential geometry in a while but am decently comfortable with the basics of the subject.
Any textbook recommendations for a sort of roadmap for this textbook?

## 1 Answer

Voisin's book is quite (but not always) self-contained and well-written. For the supplementary references, you may use the first chapters of Griffiths-Harris "principles of algebraic geometry" which can help you to understand and motivate the complex backgrounds of complex Hodge theory. The book by Carlson, Müller-Stach, Peters, "Period mappings and Period domains" is more readable and self-contained than Voisin's book. The book by Bertin, Demailley, Illusie and Peters, "Introduction to Hodge theory" is less famous but a very good reference specially if you are interested in interactions between complex Hodge theory and Hodge theory in characterstic $p$. Note that nowadays, there are a lot of online lecture notes that are simplified and can help you to understand the content of Voisin's book much more easily. For example this or this which is more detailed. By googling you can find even more references. Also, your background of representation theory can help you a lot in Hodge theory, as it constantly appears in studying Hodge theory and in Voisin's book (for example the very important notion of local systems is nothing but studying representations of the fundamental group).