# Basic question about tautological classes (kappa classes)

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?

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It looks like you are trying to pull back a cohomology class on the total space $C$ along the inclusion of a one dimensional subvariety. As you say, this will be zero if the degree is too big.
I think the essential point is that even if a cohomology class has trivial pullback along all one dimensional subvarieties, it is not necessarily zero. As an easier example, note that $\mathbb{P}^2$ has nontrivial cohomology classes in degree 4, whose pullback along any curve vanishes.