The setting is the following: $X \subset \mathbb{CP}^r$ is an algebraic subvariety of dimension $n$, codimension $e=r-n$, and degree $d$. Call $j$ the inclusion. Then, Poincaré duality induces a map \begin{equation} j_*:H^i(X,\mathbb{C}) \rightarrow H^{i+2e}(\mathbb{CP}^r,\mathbb{C}). \end{equation} Denote by $\xi \in H^2(\mathbb{CP}^r,\mathbb{C})$ the hyperplane class and by $\psi$ its pullback to $X$. Then I am given two identities:

(1) $j_*j^*(x)=x \cup (d \xi^e)$;

(2) $j^*j_*(y)=y\cup(d \psi^e)$.

While I see that the first one comes from projection formula and the class of $X$ being $d \xi ^e \in H^*(\mathbb{CP}^r,\mathbb{C})$, I can not figure out the second one. I have been pushing symbols for a while, but without success. The only thing I got is that both sides of (2), if capped with $[X]$ and pushed forward, give the same thing. How can I show the second relation?


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