Let $X$ be a smooth compact complex manifold of dimension $N$ and let $Y$ be an ample irreducible divisor in $X$. Set $U:=X\setminus Y$ and let $j\colon U\to X$ denote the open embedding morphism. Recall that a coholomogy class $b\in H^{k}(X,\mathbb{Q})$, $k\geq N$, is called primitive if $b\smile c_1(Z)^{N-k+1}=0$. Here $c_1(Z)$ is the first Chern class of the line bundle associated with the divisor $Z$. Let $H^k_{\mathrm{prim}}(X,\mathbb{Q})$ denote the subgroup of primitive elements of the cohomology group $H^k(X,\mathbb{Q})$.

In the case of $Y$ is smooth, one can easily see that the homomorphism $j^*\colon H^k(X,\mathbb{Q})\to H^k(U,\mathbb{Q})$, $k\leq N$, is an injection when it is restricted to $H^k_{\mathrm{prim}}(X,\mathbb{Q})$. This claim can be obtained by using the Gysin exact sequence, weak and hard Lefschetz theorems. I wonder it is still true that the homomorphism $j^*:H^k_{\mathrm{prim}}(X,\mathbb{Q})\to H^k(U,\mathbb{Q})$, $k\leq N$, is an injection in the case of a singular divisor.