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I understand that Chern class for a smooth fibration is the sum of Chern classes of fiber and base; is there a formula for singular fibration, namely when the fiber develops a singularity while moving along the base? I'd expect an additive correction that takes into account singular fibers, but I don't know how to prove it.

The concrete case I'm interested in is fibration of K3 surface on a base $\mathbb P^1$, and see that when K3 develops ADE singularities $\mathbb C^2/\Gamma$ (with $\Gamma$ discrete subgroup of $SU(2)$) we have $c_1=0$.

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For a smooth map $f \colon X \to Y$ there is an exact sequence $$ 0 \to T_{X/Y} \to T_X \to f^*T_Y \to 0. $$ It gives $c(T_X) = c(T_{X/Y})\cdot f^*c(T_Y)$. Note, however, that $c(T_{X/Y})$ is not just the Chern class of the fiber (which fiber?). For instance, one can consider the case when $X = P_Y(E)$ is a projectivization of a vector bundle.

For a non-smooth map the differential $T_X \to f^*T_Y$ is not surjective, so one should take into account the Chern class of its cokernel.

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