I had this simple question when formulating the Todd class question.


  Does there exist an example of proper morphism $f:X\to Y$ together with nontrivial homology class $t\in H^*(X)$ such that for all coherent sheaves on $X$ the equality $f_*(\mathop{\text{ch}}(u)\cdot t) = 0$ holds?


You look at the case when $X=D$ is a Cartier divisor on $Y$ (so that the relative tangent bundle -- as an element of the K-group -- is the normal bundle $\mathcal N_{D/X}=\mathcal O_D(D)$ (conveniently a line bundle, so is its own Chern root), and $\mathcal F=\mathcal O_D$. And the Todd class pops out right away.

Indeed from the exact sequence $0\to \mathcal O_Y(-D) \to \mathcal O_Y\to \mathcal O_D\to 0$, you get that 
$$ch(f_! \mathcal O_D)=ch(O_Y(D))=
ch(\mathcal O_Y) - ch(\mathcal O_Y(-D)) = 1- e^{-D}.$$
And you need to compare this with the pushforward of $[D]$ in the Chow group, which is $D$. The ratio 
$$ \frac{D}{1-e^{-D}} = Td( \mathcal O(D) )$$ 
is what you are after. Now you have just discovered the Todd class. I suspect that this is how Grothendieck discovered his formula, too -- after seeing that this case fits with Hirzebruch's formula, that the same Todd class appears in both cases.
