TITLE: $H^2(X,\mathbb{Z})$,$H_2(X,\mathbb{Z})$of smooth complex projective variety$X$
QUESTION [8 upvotes]: is there some example that: $X$ is a smooth complex projective variety, second singular cohomology $H^2(X,\mathbb{Z})$ has nontrivial torsion subgroup?

REPLY [3 votes]: Feng, try this:
Consider a surface $\Sigma$ in $\mathbb{P}^3$. There is an action of $\mu_p$ (roots of unity) on $\mathbb{P}^3$ where the coordinate $z_i$ is sent to $\zeta^i z_i$ under the action of $\zeta \in \mu_p$. Assume $\Sigma$ is fixed by this action. Then $H^2(\Sigma/\mu_p,\mathbb{Z})_{tors} = \mathbb{Z}/p\mathbb{Z}$. (Recall that by Lefschetz theorem $\Sigma$ is simply connected.) Why? Consider a nontrivial character of $\mu_p$. This gives you descent data for descending the trivial line bundle on $\Sigma$ to $\mathcal{L}$ on $\Sigma/\mu_p$. This will have $p$-torsion Chern class $c_1(\mathcal{L})$ in $H^2$.