Let $Y$ be a smooth compact Calabi-Yau three-fold (over $\mathbb C$, with $\pi_1(Y)=0$). Is it true that $c_2(Y)$ is Poincare dual to an effective curve? If not, can one construct a counter-example?

Note, that the answer to the same question for Fano $3$-folds is positive, as is stated here: Do all Fano threefolds have effective $c_2$? However, I don't understand if that answer applies to Calabi Yaus too.

  • 1
    $\begingroup$ Some remark: it is possible to have $c_2(Y)=0$ even for $Y$ Calabi-Yau in a strict sense ($K_Y$ trivial and $H^1(Y,\mathcal{O}_Y)=0$): take a $Y$ of the form quotient of an abelian 3-fold by a finite group. $\endgroup$ – user25309 Jan 2 at 14:42
  • $\begingroup$ Thanks for the comment! I consider trivial as effective. I imagine such an example has infinite $\pi_1$? $\endgroup$ – aglearner Jan 2 at 22:41

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.