Given a family of curves $\pi: C \to S$, let $\omega_\pi$ be the relative canonical sheaf over $C$. We define the kappa classes $\kappa_{i-1}$ to be $\pi_\ast (c_1(\omega_\pi)^i)$, for $i = 1,2,\dots$.
I have a very basic question that is confusing me:
We have $c_1(\omega_\pi) \in A^1(C)$ or $H^2(C)$. What I don't understand is this: If we restrict $c_1(\omega_\pi)^i$ to a fiber $C_s$, $s \in S$, then this class lives in $A^i(C_s)$ or $H^{2i}(C_s)$. Since each $C_s$ is 1-dimensional, we have $A^i(C_s)=0$, $H^{2i}(C_s)=0$, for all $i > 1$, and so the restriction of $c_1(\omega_\pi)^i$ to each fiber $C_s$ must be zero, for all $i > 1$.
Hence, for $i > 1$, it must follow that $c_1(\omega_\pi)^i$ itself is zero, and thus $\kappa_{i-1}$ is zero... ?!?!?!?!
This last step must be incorrect, since the kappa classes are supposed to be nontrivial in general.
So why is this last step incorrect?