complex-geometryonline-resourcesreference-request
For the past few weeks I have been starting to learn about complex geometry and Hodge theory. I'm doing so with the book of Daniel Huybrechts Complex Geometry, which is an excellent book, however I would like to supplement this book with some online lectures.
Is there anyone who can recommend me a lecture series which I can use to learn about complex geometry and related topics?
I note that you wrote complex differential geometry. There is a difference between complex differential and analytic geometry, the latter having much in common with algebro-geometric scheme theory. However, there is no schism between the two - if you're interested in one, you'd better learn a hell of a lot about the other.
For the former, I feel comfortable recommending:
Zheng - Complex differential geometry : A very diffeo-geometrical introduction to the subject. Does not go into extreme technical details, but does not shy away from difficulties.
Demailly - Complex analytic and differential geometry (available for free on Demailly's website) : This is where you'll find all the technical details. Amazing for a second run on the subject, trying but ultimately rewarding on the first. (Note: Demailly recommends Hörmander's book for the complex analytic technical details needed for his own.)
There are books by Werner Ballmann and Andrei Moroianu on Kähler geometry as well. Both are good. Claire Voisin's book on Hodge theory is closely related to what you want, but more algebraic. I imagine you'll also want to look through Shafarevich's books on algebraic geometry, Mumford's books on the same, and anything written by Joe Harris for motivation and examples. Then you'll want to look at Kobayashi-Nomizu as well.
The problem, as always, is that there are seven million different important things to learn. Thus it is pretty much impossible to write a textbook that trails a coherent narrative and covers all these subjects. Also, beware that a "second course" type book on complex differential geometry does, to my knowledge, not exist. The document that comes closes is perhaps Demailly's notes on applications to algebraic geometry, again available on his website.
You should probably read, or rather violently leaf through, all this all at the same time, and then go bother the local complex geometry types with silly questions. There's no better way to learn a new language than to immerse oneself in it and try to talk to the natives.
[edit:] I see I forgot to mention two things: 1. Griffiths and Harris talk about residues in their book, and 2. "quadratic differentials" is secret code for "deformation theory on Riemann surfaces". There is no introduction for beginners to deformation theory. Anyone who says otherwise is either lying or severly underestimating the technical difficulties involved.
I offer that differential geometry may be a much broader field than algebraic topology, and so it is impossible to have textbooks analogous to Switzer or Whitehead.** So, although it isn't precisely an answer to your question, these are the most widely cited differential geometry textbooks according to MathSciNet. I've roughly grouped them by subject area:
Of the metric geometry books (#1), BBI's book is good for self-study, while Gromov's book is nice to have around and open to random pages.
Kobayashi and Nomizu is a hard book, but it is extremely rewarding, and I don't know of any comparable modern book - I would disagree in the extreme with whoever told you to skip it. It is only aged in superficial ways, such as some notations. Lawson and Michelsohn's book is quite advanced, and K-N vol. 1 (at least) would be a prerequisite. It includes a chapter on the Atiyah-Singer index theorem.
Besse's book covers "special Riemannian metrics", including a review of Riemannian, Kahler, and pseudo-Riemannian geometry. It is more of a reference book, good to look through sometimes.
For classical mechanics, Abraham and Marsden is quite sophisticated, and it is necessary to have a solid geometrical footing (roughly K-N vol 1) before going into it; Arnold's book is more introductory and would probably be very nice for self-study.
The general relativity books in #5 are all introductory and pretty approachable.
I'm not so familiar with the books #6-9. Guckenheimer and Holmes seems very friendly.
Personally, I'd also recommend Chow, Lu, and Ni's "Hamilton's Ricci flow," the content of which is necessary to understand the proofs of the Poincare and geometrization conjectures. The first chapter is an excellent mini-textbook on "classical" Riemannian geometry, reaching just beyond introductory books like Do Carmo's.
** just to underline the point in the first sentence - there are only five general or algebraic topology textbooks (Hatcher, Spanier, Rolfsen, Engelking, and Kelley), four differential topology textbooks (Bredon, Hirsch, Milnor "Morse Theory", and Milnor-Stasheff) and two convex geometry textbooks (Schneider and Ziegler) as widely cited as the above differential geometry textbooks
Best Answer
Books
1.You can refer to Complex Manifolds written by James Morrow and Kunihiko Kodaira.
It is an excellent primer including a lot of calculations and details.
2.You can refer to Principal of Algebraic Geometry written by Griffiths and Harris and Complex Analytic and Differential Geometry written by Jean-Pierre Demailly.
They have more differential-geometric points of view.
3.You can refer to Hodge Theory and Complex Algebraic Geometry 1 written by Claire Voisin.
It may be more difficult and advanced and have more algebraic points of view.
Notes and Lectures
1.Notes about complex manifolds which is a wonderful supplement of the Huybrechts' book.
:https://www.math.stonybrook.edu/~cschnell/pdf/notes/complex-manifolds.pdf
2.New lectures written by Hossein Movasati about the Hodge theory.
:http://w3.impa.br/~hossein/myarticles/hodgetheory.pdf
3.If you interested in Kahler Manifolds,you can see the lectures written by Werner Ballmann.
:http://people.mpim-bonn.mpg.de/hwbllmnn/archiv/kaehler0609.pdf
Videos
1.Hans-Joachim Hein's Complex Geometry that consisting of ten classes is nice. See here ：https://m.youtube.com/results?search_query=complex+geometry
2.Complex Analytic and Algebraic Geometry by Qaisar Latif.
See here： https://m.youtube.com/playlist?list=PLWy-AYZriyMGZjZQ7I8kYhZhtlUzFcWcz