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This question is on bounding the degree of the Todd class on a complex threefold.

Let $X$ be a smooth compact connected complex surface. Let $c_i=c_i(TX)$ be its $i$-th Chern class. Recall the following two facts. These allow one to bound the degree of the Todd class on a surface in terms of $c_2$.

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, then the Bogomolov-Miyaoka-Yau inequality states that $$c_1^2 \leq 3c_2.$$

Now, I am interested in similar results for 3-dimensional smooth projective connected varieties over $\mathbb{C}$. In this case, the degree of the Todd class of $X$ is the degree of $$\frac{c_1 c_2}{24}.$$

Question. For 3-dimensional smooth projective connected varieties over $\mathbb{C}$, 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 LeBrun. 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 Kotschick, 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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  • $\begingroup$ 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? $\endgroup$
    – Ariyan Javanpeykar
    Commented May 31, 2010 at 16:42
  • $\begingroup$ 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. $\endgroup$
    – Ariyan Javanpeykar
    Commented May 31, 2010 at 16:44
  • $\begingroup$ 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. $\endgroup$
    – Dmitri Panov
    Commented May 31, 2010 at 17:29

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