- 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 $c_n\ge 2^n td_n$)

`$(-1)^n(c_n(X)-1)\ge 2^n(-1)^n\Big( td(T_X) ch(\mathcal{O}_X(-1))\Big)_{top.dim.}$`

After computation of both sides I get some messy school-type inequality. No idea how to prove it in general. But in many numerical cases (of low dimension and codimension) I verified it. I wonder, is there some (indirect?) way to prove the bound without proving this numerical inequality? – Dmitry Kerner Jul 9 '11 at 12:03