Let $X$ be a smooth, projective variety and $Y \subset X$ a smooth, effective divisor. Consider now the natural map $i^*:H^2(X) \to H^2(Y)$. Then,

1. When is the image of $c_1(\mathcal{O}_X(Y)) \in H^2(X)$ under the map $i^*$ non-zero?

2. If $Z$ is another smooth projective variety containing $Y$ as an effective divisor (meaning $\dim Z=\dim Y+1$), is $j^*(c_1(\mathcal{O}_Z(Y)))=i^*(c_1(\mathcal{O}_X(Y)))$ where $j^*:H^2(Z) \to H^2(Y)$ is the natural pull-back map?

3. If ($2$) is not true in general, is it true if $Z$ is a deformation of $X$?

Any reference/idea regarding the questions will be most welcome.

Let $X$ be a smooth, projective variety and $Y \subset X$ a smooth, effective divisor. Consider now the natural map $i^\ast:H^2(X) \to H^2(Y)$. Then,

1. When is the image of $c_1(\mathcal{O}_X(Y)) \in H^2(X)$ under the map $i^*$ non-zero?

2. If $Z$ is another smooth projective variety containing $Y$ as an effective divisor (meaning $\dim Z=\dim Y+1$), is $j^*(c_1(\mathcal{O}_Z(Y)))=i^*(c_1(\mathcal{O}_X(Y)))$ where $j^*:H^2(Z) \to H^2(Y)$ is the natural pull-back map?

3. If ($2$) is not true in general, is it true if $Z$ is a deformation of $X$?

Any reference/idea regarding the questions will be most welcome.