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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 degree)?$d_i$'s)?

Let $td_2$ be the top-dimensional Todd class, i.e. $td_2=\frac{c^2_1+c^2}{12}$. The bound 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 $c_n\ge 2^n td_n$)
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# Looking for an inequality between Chern and Todd classes (something in style of Bogomolov-Miyaoka-Yau)

1. Consider a smooth projective surface $S\subset\Bbb P^N_{\Bbb C}$ which is a complete intersection. 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 degree)?

Let $td_2$ be the top-dimensional Todd class, i.e. $td_2=\frac{c^2_1+c^2}{12}$. The bound 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 $c_n\ge 2^n td_n$)