Given a fibered surface $X \rightarrow C$, with generic fiber $Y$ and a vector bundle $E$ on $X$.


Then the first Chern class $c\_1(E)$ is a divisor on $X$, so one can restrict this divisor to the generic fiber $c\_1(E)\_{|Y}$.


On the other hand one can restrict $E$ to the generic fiber and look at its first Chern class as a divisor on $Y$, i.e. $c\_1(E\_{|Y})$.


Do we always have the equality $c\_1(E)\_{|Y}=c\_1(E\_{|Y})$ as divisors on $Y$, if $X$, $C$ and $Y$ are "nice" enough?



Restriction of divisors to the generic fiber
Yes, even when $X$, $C$ and $Y$ are as nasty as they can be. Chern classes are functorial.


