Suppose you have an equivariant closed immersion of toric varieties $Y \subset X$, and suppose further that they are both smooth (meaning that the torus of $X$ restricts to the torus on $Y$). Suppose also that they are defined over $\mathbb C$.
We then get a morphism of singular cohomology groups $H^*(X; \mathbb Z)\to H^*(Y;\mathbb Z)$.
My question is: can we compute the induced map on cohomology based on the combinatorial data of the toric varieties?
If it helps, assume that $Y$ is a hypersurface in $X$, i.e. a divisor. Also, I'd be happy for just references.