# Reference request: Chern classes in algebraic geometry

I have encountered Chern classes numerous times, but so far i have been able to work my way around them. However, the time has come to actually learn what they mean.

I am looking for a reference that treats Chern classes in algebraic geometry over $\mathbb{C}$. It is no problem if only varieties are treated and not general schemes. I will be requiring only basic knowledge: definitions and some way to calculate them.

Thanks!

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I guess like everything else this in Griffiths--Harris, treated via the differential-geometric approach. It wouldn't be my choice for learning this stuff, but different people have different tastes. –  Asal Beag Dubh Apr 15 '13 at 20:52
@AsalBeagDubh, thanks but i would immensely prefer an algebro-geometric approach, if it exists.. Does it? Is this the approach of Chern-Weil that you mentioned in the other comment? –  Joachim Apr 15 '13 at 21:38
No, Chern-Weil is (IIRC) what's in Griffiths--Harris --- connections, curvature, and all the rest. I think Fulton is the best reference in English for the purely algebro-geometric approach. –  Asal Beag Dubh Apr 15 '13 at 21:53

The best short introduction (in my opinion) to get you going with Chern classes in algebraic geometry is Zach Tietler's "An informal introduction to computing with Chern classes", which can be found here:

http://works.bepress.com/cgi/viewcontent.cgi?article=1001&context=zach_teitler

This is a purely algebraic treatment with lots of basic examples.

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In my opinion the best reference is by far Grothendieck's La théorie des classes de Chern.

This seminal article was published in 1958, is purely algebraic and is valid in characteristic $p$.
Needless to say it doesn't necessitate any differential geometry: no curvature of connections here!

This article was written before Grothendieck introduced scheme theory and is incredibly elementary, probably the simplest text he has ever written!
It relies on the purely geometric idea that given a vector bundle $E$ on the variety $X$, you should consider the associated projective bundle $\mathbb P(E)$ over $X$, lift $E$ to $\mathbb P(E)$, quotient out the tautological line bundle and iterate.
This idea is "childish", an adjective Grothendieck loves to apply to his work, and incredibly powerful.

Even differential geometers/algebraic topologists use it: Bott and Tu introduce characteristic classes by means of Grothendieck's construction in their celebrated Differential Forms in Algebraic Topology.
And incidentally their treatment is also an excellent introduction to Chern classes: the ideas are from Grothendieck but there are more applications/examples in their book.

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Somehow I never knew this existed. Wonderful! –  Asal Beag Dubh Apr 15 '13 at 21:55
Often original works are the best ones. –  Martin Brandenburg Apr 15 '13 at 23:50

Milnor's book Characteristic Classes is always a good choice. There were some lecture notes by Chern that I remember liking quite a bit but I haven't been able to find the title. Both of these titles use the Chern-Weil approach to Chern classes. I think this is one of the more accessible approaches.

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+1 for Milnor, but as far as I recall there's nothing about Chern-Weil theory in there; the whole theory is done topologically. –  Asal Beag Dubh Apr 15 '13 at 20:02
I think we're just using different titles for the same thing. Using elementary symmetric polynomials of eigenvalues for a matrix of 2forms to create characteristic classes is very topological but is also called the Chern-Weil approach. Are you thinking of Chern-Simmons maybe? –  Schmitty Apr 15 '13 at 22:16
en.wikipedia.org/wiki/Chern-Weil_theory "computes topological invariants of vector bundles and principal bundles in terms of connections and curvature". Unless I misremember, in Milnor there are no connections in sight. –  Asal Beag Dubh Apr 15 '13 at 22:19
Hey- this is kind of a silly debate. Let's call this whatever we like. It's always been called Chern-Weil around me but that's hardly a sacred cow. –  Schmitty Apr 16 '13 at 15:54

There is an appendix in Hartshorne that gives the basic properties, and is quite brief but sufficient to learn how to do some basic computations. I believe a standard, detailed reference for algebraic geometers is Fulton's Intersection Theory. And while I haven't read them yet, browsing through Gathmann's notes there seems to be a nice exposition in the final chapter.

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A non-standard reference is Hirebuech's book:

Hirzebruch, Friedrich (1995) [1956], Topological methods in algebraic geometry, Classics in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-3-540-58663-0, MR 1335917

I heard Tom Farrell suggest this is a good book from a pure algebraic point of view.

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