Let $X$ be a normal projective variety over the complex number field and $D \subset X$ be an ample $\mathbb{Q}$-Cartier divisor. Assume that $\dim X \geq 3$. Consider the restriction map between the Picard groups ${\rm Pic} X \rightarrow {\rm Pic} D$. Is this restriction map injective?
In fact, I'm particularly interested in the case where $X$ is a Fano 3-fold with only terminal singularities and $D \in |-K_X|$ is a general anticanonical element with only Du Val singularities. Actually, I want to know the case where $X$ is non-Gorenstein with such $D$.
If there are related articles about restriction map to ample Weil divisor, please let me know about it.
(add) maybe I found the answer in the case $X \setminus D$ has only l.c.i. singularities, so in particular it's OK in the $\mathbb{Q}$-Fano 3-fold case. It was written in the Fulton's article on the topology of algebraic varieties.
