# The Lefschetz hyperplane section theorem for class groups

Let $X$ be a normal projective variety over $\mathbb{C}$ and $D \subset X$ be a reduced ample Cartier divisor which is normal. Then we have a restriction homomorphism $r \colon {\rm Cl} X \rightarrow {\rm Cl} D$ and $r_{\mathbb{Q}} \colon {\rm Cl} X \otimes \mathbb{Q} \rightarrow {\rm Cl} D \otimes \mathbb{Q}$.

Question Assume that $\dim X \ge 3$. Is $r_{\mathbb{Q}}$ injective?

Actually, I can assume that $\dim X = 3$, $X$ and $D$ have only rational hypersurface singularities and $H^1(X, \mathcal{O}_X) = 0 = H^2(X, \mathcal{O}_X)$.

(Added again) My tentative argument contained a gap.

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