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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.

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    $\begingroup$ 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

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