# What are some applications of Chow-group-valued characteristic classes?

Let $f:X \to Y$ be a map between smooth projective varieties. The Grothendieck-Riemann-Roch formula is

$$\mbox{ch}(f_{\mbox{!}}{E})\mbox{td}(Y) = f_* (\mbox{ch}(E) \mbox{td}(X) )$$

Some applications of this formula are discussed here. Are there any nice examples that illustrate how much more information you get if you treat this equation as taking place in the Chow group of algebraic cycle classes, rather than ordinary cohomology?