Looking for an inequality between Chern and Todd classes (something in style of Bogomolov-Miyaoka-Yau) 
*

*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 $c_2\ge 2^2td_2$.


*

*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 $c_n\ge 2^n td_n$)

 A: 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 $$c_1^2(S)=3c_2(S).$$ 
