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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.

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    Is cohomology even computed by some combinatorial data e.g. fan? – 2017-02-22
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    If the inclusion is toric (so $Y$ is torus-invariant and the torus action on $X$ restricts to the torus action on $Y$) then everything is great (if so I will expand in an answer). I can't tell if "equivariant" in the question is supposed to describe this situation. – 2017-02-22
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    @Nefertiti Yes, the inclusion is toric (that was what I was trying to say with "equivariant"). – 2017-02-22
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    @Nefertiti I'm sorry for the bump, but did you have an answer for this? – 2017-02-27
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    Sorry for not following up on this yet. After commenting I realised the story is not quite as straightforward as I had thought (although not too bead). I will try to answer in the next few days. – 2017-02-27
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    @Nefertiti I see. Thank you for the effort! – 2017-02-27

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