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This question is on bounding the degree of the Todd class. Let me explain where this comes from.

Suppose that $X$ is a smooth compact connected complex surface. Let $c_i=c_i(TX)$ be its $i$-th Chern class.

1. If $X$ is not of general type, we have that $c_1^2$ is bounded absolutely from above by 9. See Table 10 of Chapter VI of Compact complex surfaces by Barth, Hulek, Peters and van de Ven.

2. If $X$ is of general type, the Bogomolov-Miyaoka-Yau inequality states that $$c_1^2 \leq 3c_2.$$

Conclusion. Since the degree of the Todd class of $X$ is the degree of $$\frac{c_1^2+c_2}{12},$$ we basically get an upper bound for this if we show that the degree of $c_2$ is bounded from above. Now, the degree of $c_2$ is the topological Euler characteristic. Thus, since my goal was to bound the Todd class from above (in the set-up of my problem), I reduce to bounding the topological Euler characteristic. (That's good!)

Now, I am really interested in a similar result for 3-dimensional smooth compact connected complex varieties.

In this case, the degree of the Todd class of $X$ is the degree of $$\frac{c_1 c_2}{24}.$$ Since the topological Euler characteristic of $X$ is (the degree of) $c_3$, I would like to reduce the problem of bounding this quantity from above to bounding $c_3$. So now for my question.

Question. Do there exist any absolute upper bounds on $c_1c_2$ or any bounds for that matter which are polynomial in $c_3$?

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1 Answer

up vote 11 down vote accepted

The case of complex manifolds of higher dimension is very different from the case of complex surfaces. So the answer to the question about complex $3$ folds is no, there exists a real 6-dimensional simply connected manfiold with integrable complex structures $J_m$ for all $m\in \mathbb Z^+$ such that $c_1c_2=48m$. This is a theorem A from the acticle of Leubrun. Though, the manifolds that he constructs are not algebraic

Topology versus Chern Numbers for Complex 3-Folds

http://arxiv.org/PS_cache/math/pdf/9801/9801133v1.pdf

The question for algebraic manifolds was studied by Kostchik, you may be interested this the following two articles:

TOPOLOGICALLY INVARIANT CHERN NUMBERS OF PROJECTIVE VARIETIES

http://arxiv.org/PS_cache/arxiv/pdf/0903/0903.1587v1.pdf

CHERN NUMBERS AND DIFFEOMORPHISM TYPES OF PROJECTIVE VARIETIES

http://arxiv.org/PS_cache/arxiv/pdf/0709/0709.2857v2.pdf

Finally, it should be added that a systematic attempt to construct various complex 3-fold is given in a very nice article of Okonek and Van de Ven "CUBIC FORMS AND COMPLEX 3-FOLDS" of Okonek, Ch. / Van de Ven, A, L'Enseignement Mathématique Volume / Année: 41 / 1995. The link is given in the comments

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Great answer. Just one tiny question. In the first article you mention, Le Brun writes "For example, if X = CP3, every integer of the form 8j can be realized as c1c2". How does one show this? –  Ari May 31 '10 at 16:42
    
Let me also say that I believe hope isn't completely lost in my case. In fact, my complex 3-fold is fibered over a curve, i.e., there is a projective flat morphism to some smooth projecture curve. If I'm not mistaken, the examples given by Le Brun don't fit in the set up of my example. –  Ari May 31 '10 at 16:44
    
For the first question, check section 4 of the article "CUBIC FORMS AND COMPLEX 3-FOLDS" of Okonek, Ch. / Van de Ven, A, L'Enseignement Mathématique Volume / Année: 41 / 1995. You can download it on the website unige.ch/math/EnsMath/EM_en/welcome.html I agree with the second remark, it is a very strong restriction for complex 3-folds to have a flat morphism to a curve. –  Dmitri May 31 '10 at 17:29
    
What a great article. It's filled with beautiful examples. Thanks alot. –  Ari May 31 '10 at 18:12
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