Algebraic varieties with certain topological properties

Question

Does there exist a smooth, projective, complex algebraic variety $X$, with two cohomology classes $\alpha,\beta \in H^{*}(X,\mathbb{Z})$ neither $\alpha$ nor $\beta$ is torsion but the product $\alpha \cup \beta$ is non-trivial and torsion?

Answer 1

Let $Y$ be a smooth complex projective surface whose cohomology group $H^2(Y;\mathbb{Z})$ has a nonzero torsion element $\gamma$ and whose torsion-free quotient has rank at least $2$, e.g., this holds for an Enriques surface. Let $D$ and $E$ be nontorsion elements whose cup product is zero. By the Hodge Index Theorem, such elements exist.

Let $X$ be $Y\times Y$. Let $\alpha$ be $\text{pr}_1^*\gamma +\text{pr}_2^*D$. Let $\beta$ be $\text{pr}_2^*E$.