1. Consider a smooth projective surface $S\subset\Bbb P^N\_{\Bbb C}$ which is a complete intersection of hypersurfaces of degrees $(d\_1,..,d\_{k\ge2})$ with $d\_i\ge2$ for all i. Is it true that for such surfaces $c^2\_1\le 2c\_2$? (i.e. much better than BMY)
At least asymptotically (i.e. for high enough $d\_i$'s)?


Let $td\_2$ be the top-dimensional Todd class, i.e. $td\_2=\frac{c^2\_1+c^2}{12}$. The inequality as above can be written as $c\_2\ge 2^2td\_2$.


1. More generally, let $X\subset\Bbb P^N\_{\Bbb C}$ be a smooth complete intersection of dimension $n$. Let $c\_n$ and $td\_n$ be its top-dimensional Chern and Todd classes. What are the known inequalities on $c\_n$ and $td\_n$? (I would like to have smth like $c\_n\ge 2^n td\_n$)



Looking for an inequality between Chern and Todd classes (something in style of Bogomolov-Miyaoka-Yau)
By the formulae in [Barth-Peters-Van de Ven, Chapter V] one has, for a surface which is complete intersection of type $(d\_1, \ldots, d\_{n-2})$ in $\mathbb{P}^n$:
$$c\_1^2(X)= \big(\sum d\_i-(n+1)\big)^2 \prod d\_i,$$
$$c\_2(X)=\bigg[\binom{n+1}{2}-(n+1)\sum d\_i+\sum d\_i^2 +\sum\_{i < j} d\_id\_j \bigg]\prod d\_i.$$


Then one obtains that the inequality
$$c\_1^2(X) \leq 2 c\_2(X)$$
is equivalent to
$$n+1 \leq \sum d\_i^2,$$
and this is of course almost always true, since the right-hand term is $\geq 4(n-2)$.
So the answer to $1.$ seems to be *yes*.


**ADDED.** Actually, this is also written in the book by Barth-Peters- Van de Ven. In Chapter V, at the beginning of the Section "The Geography of Chern Numbers", they say:



> 
> "
>  The simplest examples, like complete intersections and double coverings of $\mathbb{P}^2$, pratically always yield a point of $D\_1$ [where $D\_1$ is the region in the $(c\_1, c\_2)$-plane given by $c\_1^2 \leq 2c\_2$]. Indeed, for a long time only few examples were known of surfaces with Chern pairs $(c\_1^2, c\_2)$ in $D\_2$ [i.e., such that $2c\_2 < c\_1^2 \leq 3c\_2$]." 
> 
> 
> 


For your question in the last comment, instead, the answer is clearly *no* if $S$ is ACM. In fact, every smooth surface $S$ with $H^1(S, \mathcal{O}\_S)=0$ is ACM for some embedding in the projective space. Now take for instance a fake projective plane. It satisfies $p\_g(S)=q(S)=0$, so it is ACM, but $$c\_1^2(S)=3c\_2(S).$$ 


