# Can one bound the todd class of a 3-dimensional variety polynomially in c_3

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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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? –  Ariyan Javanpeykar 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. –  Ariyan Javanpeykar 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. –  Ariyan Javanpeykar May 31 '10 at 18:12