[Math] Path to learning Hodge Theory via Voisin’s text.

Question

I'm currently in the process of learning a bit of Homological Algebra and Smooth Manifold theory via Tu's book. Being curious I searched around and ran into Voisin's Book on "Hodge Theory and Complex Algebraic Geometry". Looking through the text, it contained a lot of concepts from complex geometry and algebraic geometry.

My question is what would be the best way to get to Voisin's text. Would I need to understand some basic complex geometry via Huybrecht and possibly Algebraic Geometry from Cox and Hartshorne? Or does understanding manifold theory and Homological Algebra sufficient enough?

Any advice and comments would be greatly appreciated.

Answer 1

This semester we followed Voisin's book for a reading group on Hodge theory. Here is what you need in order to understand the book :

The really important topic is real smooth manifolds. For this, I would recommand Introduction to smooth manifolds by J. Lee which is a great book and will introduces everything you need. Especially, you need to have a very good understanding of the tangent bundle of a manifold, more generally of a vector bundle and of De Rham cohomology. Then, Voisin will explain everything you need to know about complex manifolds, assuming you understand the real case. There are even an introductory chapter about complex analysis !

Next you need to know some algebraic geometry for motivation. This is a huge topic, but already a good knowledge on compact Riemann surfaces is enough, for example as in the book by Forster Lectures on Riemann surfaces, chapter 2, where some Hodge theory already implicitely appears. Other nice examples to know are K3 surfaces or abelian varieties.

Other topics to be familiar with :

• Algebraic topology. You can skip it if you understand well enough De Rham cohomology but nevertheless it's one of the motivation of the topic (understand the topology of algebraic varieties). A great article to read is "The topology of complex projective varieties, after S. Lefschetz" by Lamotke.
• Differential geometry, maybe a bit of Riemannian geometry : the definition of a Riemannian metric with a few properties, the Levi-Civita connection and parallel transport should be enough.
• Sheaf theory. This is covered in Forster, and also Voisin recall everything including the definition of a sheaf.
• Spectral sequences : again Voisin covers everything from scratch. You don't need to know anything.

With this you should be able to read most of the book.

Also I should say that you don't need any functional analysis (unless you really want to understand the analytical details, but then Voisin's book is not the best place for this).

Good luck !