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Let $X$ be a smooth projective variety over an algebraically closed field $k$ and $\mathcal E$ a family on torsion free coherent sheaves on $X$ parametrized by a smooth curve (over $k$) i.e. a coherent sheaf on $C\times X$ inducing torsion free coherent sheaf $\mathcal E_c$ on $X$ for all $c$ in $C$. Is there a relation in $\mathrm{CH}^i(X)$ between $c_i(\mathcal E)_*c$ (where $c$ is looked at as a zero cycle on $C$) and $c_i(\mathcal E_c)$ ?

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  • $\begingroup$ What do you mean by $c_i(\mathcal{E})*c$? If it is $c_i(\mathcal{E})_{|c\times X}$, this is just the standard functoriality of Chern classes. $\endgroup$
    – abx
    Jan 22, 2016 at 13:09
  • $\begingroup$ Thank you for your answer. You mean using the pullback by the inclusion $c\times X\hookrightarrow C\times X$ ? $\endgroup$
    – pi_1
    Jan 22, 2016 at 14:21
  • $\begingroup$ Yes, that's what I meant. $\endgroup$
    – abx
    Jan 22, 2016 at 14:35
  • $\begingroup$ Ok, thank you again $\endgroup$
    – pi_1
    Jan 22, 2016 at 15:20

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