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Hello,

I am trying to learn about Chow ring on smooth projective manifolds (over an algebraic closed field). Do any one know of good references for this? Thank you.

Edit: Thank you for the suggestions. I am working in complex geometry, and am familiar with Griffiths-Harris book. I also did detail reading of some chapters in Hartshorne book. I looked at Fulton's book before, and will read again in more detail as your suggestions. I like to have a reference that contains from beginning to up-to-date results about Chow ring, and from your comments, I imply that Fulton's book is the one?

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    $\begingroup$ Fulton's "Intersection theory" should be more than adequate. $\endgroup$
    – Donu Arapura
    Commented Oct 25, 2012 at 15:58
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    $\begingroup$ Fulton's Intersection Theory will clearly have any result bonho needs, but it's unlikely to be a good introduction to the subject unless s/he is very well-versed in abstract algebraic geometry. $\endgroup$
    – Michael Joyce
    Commented Oct 25, 2012 at 16:17
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    $\begingroup$ You might also find this blog post helpful: rigtriv.wordpress.com/2009/03/22/… $\endgroup$
    – Michael Joyce
    Commented Oct 25, 2012 at 16:21
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    $\begingroup$ Agreed. (As an aside, when asking a question like this, it is useful to provide some information about your background.) $\endgroup$
    – Donu Arapura
    Commented Oct 25, 2012 at 17:32
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    $\begingroup$ It is very rare to have a single good reference that contains "beginning to up-to-date results" about anything; most references that start from the beginning of the subject are not kept up to date, even if they went all the way to the cutting edge when they were written. $${}$$ Fulton's book, for instance, discusses no developments since 1983, according to the Preface to the Second Edition. Fulton does refer for more up-to-date discussion to an Introduction to Intersection Theory in Algebraic Geometry he revised in 1996, but even that would today be sixteen years out of date. $\endgroup$
    – Charles Staats
    Commented Oct 26, 2012 at 22:33

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Bonho,

Let me turn my comment into an answer. My initial suggestion was Fulton's Intersection Theory. This certainly has the most complete treatment of Chow groups, but it is certainly not easy reading, as Michael Joyce pointed out. He gives some suggestions in his comments which should be more suitable as an introduction. In addition since you mention your background is on the complex side, you might also look at Voisin's Hodge theory and complex algebraic geometry I, II. Chap 9 of the second volume has an introduction to Chow groups for complex algebraic varieties.

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