# $c_2$ of Calabi-Yau three-folds

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.

• 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. – user25309 Jan 2 at 14:42
• Thanks for the comment! I consider trivial as effective. I imagine such an example has infinite $\pi_1$? – aglearner Jan 2 at 22:41