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 $X$ ?

  • $\begingroup$ 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
  • 2
    $\begingroup$ @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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.