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I was wondering: is there a good place to find exercises in Hodge theory? Mostly computations and proving small (preferably nifty) theorems, is what I have in mind. Something roughly like the Problems in Group Theory book by Dixon, or many other such problem books.

The biggest difficulty I'm having is that I don't have an adequate number of problems to do, as Griffiths & Harris's Principles has none, and Voisin's otherwise excellent books have only a couple of exercises each chapter, and I've not been able to find a book or paper on the arxiv that does.

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@CS: I changed "problems" in the title to "exercises" and also once (but not twice) in the text. When I read the title, I honestly thought you were asking for research questions in Hodge theory (have you tried the Hodge Conjecture? It's a doozy). By the time I read your question completely, it was clear what you meant, but I hope you don't mind this edit in the spirit of increased clarity. (If it does not go without saying, please feel free to roll back if you like.) – Pete L. Clark Feb 15 '10 at 18:17
No, that's fine. I was basing the title off of Dixon's book, which is my favorite group theory text because it's pretty much nothing but problems to work through. Your edit does make things a bit clearer. – Charles Siegel Feb 15 '10 at 20:02
up vote 5 down vote accepted

One suggestion: "Period mappings and Period Domains", by Carlson, Muller-Stach, and Peters, in the Cambridge studies in advanced mathematics series. It's a very nice read, and each chapter comes with examples and problems.

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Just got a chance to flip through someone's copy, and it looks promising. – Charles Siegel Feb 15 '10 at 20:03

(This started as a comment but then got a bit too long.)

In brief: I'd be interested to know the answer myself. But it would be a (pleasant) surprise to me if there is something exactly of that sort covering a sizable portion of Hodge theory.

Other sources, apart from GH and Voisin, which may be useful are Demailly's Complex analytic and algebraic geometry (exercises on Hodge theory promised in the table of contents but actually absent in the version I have) and Kulikov and Kurchanov's Complex algebraic varieties (Springer EMS, Algebraic geometry 3).

I think one can extract a couple of problems from every chapter of these; if you are looking for computations, you might want to take a look at e.g. VI.10 of Demailly (complex curves, Abel-Jacobi map, Weierstrass points).

Kulikov and Kurchanov have examples of algebraic non-projective and Moishezon non-algebraic varieties (1.3; strictly speaking, not a part of Hodge theory, but good to know), Kaehler non-Hodge tori (end of 1.7); one could extract a couple of problems from chapter 3 on Torelli theorems. Depending on who the intended audience is, one can present a part of the proof of the non-rationality of a cubic threefold as a series of problems.

Finally, Shafarevich's Basic algebraic geometry has about a dozen exercises on Hodge theory in VIII.4. Again, if you allow for not directly related but good-to-know things (e.g. examples of surfaces isomorphic in the analytic but not algebraic category), then the whole of chapter VIII there can be useful.

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There are at least 20 exercises on Hodge theory of Kahler manifolds in Huybrechts book "Complex geometry", section 3, maybe this is what you are looking for. In general this book has a lot of exercises and seem to be well written.

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