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Suppose $X$ smooth projective variety of dimension $n$ such that $-K_X$ is ample. If $h^0(-K_X) >0$, then the first Chern class of $X$ can be seen as a cycle of co-dimension $1$ associated to a section of $-K_X$. The second Chern class can be represented by a cycle of co-dimension $2$.

Qestion: Is there any way to identify a cycle associated to the second Chern class of $X$ ?

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  • $\begingroup$ Is it not just the zero locus of a general section $\sigma \in H^0(X, \, \Omega^1_X)$? $\endgroup$ Dec 1, 2021 at 13:38
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    $\begingroup$ @Francesco Polizzi: Since $-K_X$ is ample, $H^0(X,\Omega ^1_X)$ is zero. $\endgroup$
    – abx
    Dec 1, 2021 at 13:43
  • $\begingroup$ I do not know. Could you please give some reference or some hind to show it ? $\endgroup$
    – LAPRAS
    Dec 1, 2021 at 13:45
  • $\begingroup$ If $T_X$ is globally generated, pick $n-1$ general sections $s_1,\ldots ,s_{n-1}$ of $H^0(X,T_X)$; then $c_2$ is represented by $\{x\,|\,s_1(x),\ldots ,s_{n-1}(x)$ linearly dependent$\}$. $\endgroup$
    – abx
    Dec 1, 2021 at 13:47
  • $\begingroup$ @LAPRAS: In case you are asking why $H^0(X,\Omega ^1_X)=0$, this is by Hodge symmetry + Kodaira vanishing. $\endgroup$
    – abx
    Dec 1, 2021 at 13:51

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