Let $X$ be a smooth projective variety, and $D\subset X$ a smooth nef and big divisor. Assume that $X$ and $D$ have the same Picard number.

Under which hypothesis on $D$ may we conclude that there exists an isomorphism $\mathrm{Pic}(X)\rightarrow \mathrm{Pic}(D)$ over $\mathbb{Z}$ ?

Let $X$ be a smooth projective variety, and $D\subset X$ a smooth nef and big divisor. Assume that the restriction map $Pic(X)\rightarrow Pic(D)$ is an isomorphism over $\mathbb{Q}$.

Under which hypothesis may we conclude that there exists an isomorphism $\mathrm{Pic}(X)\rightarrow \mathrm{Pic}(D)$ over $\mathbb{Z}$ ?

I am particularly interested in the case $dim(X)=3$.

Let $X$ be a smooth projective variety, and $D\subset X$ a smooth nef and big divisor. Assume that $X$ and $D$ have the same Picard number.

Under which hypothesis on $D$ may we conclude that there exists an isomorphism $Pic(X)\rightarrow Pic(D)$ over $\mathbb{Z}$ ?

Let $X$ be a smooth projective variety, and $D\subset X$ a smooth nef and big divisor. Assume that $X$ and $D$ have the same Picard number.

Under which hypothesis on $D$ may we conclude that there exists an isomorphism $\mathrm{Pic}(X)\rightarrow \mathrm{Pic}(D)$ over $\mathbb{Z}$ ?