# 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!

• 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. – user64687 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. – user64687 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.

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.

• Somehow I never knew this existed. Wonderful! – user64687 Apr 15 '13 at 21:55
• Often original works are the best ones. – Martin Brandenburg Apr 15 '13 at 23:50
• do you happen to know if there is an english translation? – hjhjhj57 Feb 11 '16 at 1:38
• @hjhjhj No, I an not aware of any translation. – Georges Elencwajg Feb 11 '16 at 7:55
• What a shame. Thanks anyway, Georges. – hjhjhj57 Feb 11 '16 at 8:02

A non-standard reference is Hirzebruch'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 purely algebraic point of view.