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 $$\frac{c\_1 c\_2}{24}.$$ 



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> **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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Can one bound the Todd class of a 3-dimensional variety polynomially in c_3
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


