Let $X$ be a complex smooth projective variety with trivial topological Euler characteristic $\chi\_{\text{top}}(X)=0$. We assume that $D$ is a smooth irreducible divisor in the linear system $|K\_X|$ of the canonical divisor $K\_X$ of $X$. Is $\chi\_{\text{top}}(D)=0$?



topological Euler characteristic of canonical divisor
This is not true. Consider, for instance a Calabi--Yau threefold $Y$ with $h^{2,1}(Y) = h^{1,1}(Y) + 1$ (an example of such can be found in <https://arxiv.org/abs/1602.06303>, see page 29) and let $X$ be the blowup of $Y$ in a point. Then $\chi\_{\mathrm{top}}(X) = 0$, but the canonical class of $X$ is equal to the exceptional divisor of the blowup, hence the (unique) canonical divisor is isomorphic to $\mathbb{P}^2$.


