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Let $X$ be a smooth, projective variety and $i:Y \hookrightarrow X$ a smooth divisor. Let $Z \subset X$ be a proper, closed subvariety disjoint from $Y$. Let $\pi:\widetilde{X} \to X$ be the blow-up of $X$ along $Z$. Assume that $\widetilde{X}$ is regular. Denote by $j: Y \hookrightarrow \widetilde{X}$ (as $Z$ does not intersect $Y$, the strict transform of $Y$ is isomorphic to itself). Let $\tau \in H^m(Y,\mathbb{C})$. Is $(\pi^* \circ i_*)\tau=j_*\tau$? Here $i_*$ and $j_*$ denotes the associated Gysin morphisms.

Any reference/idea will be most welcome.

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  • $\begingroup$ I am not entirely sure, but discussion in Chapters 2 and 19 of the 2nd edition of Fulton's Intersection theory may be helpful $\endgroup$ – geometer Dec 26 '18 at 8:26
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This is true. To see this, recall that the Gysin map is defined as follows: We have an isomorphism $Th(N_{X/Y})\cong X/(X-Y)$ where $Th$ denotes the Thom space. Then, we have a Thom isomorphism $H^*(Th(V))\cong H^{*-dim(V)}(X)$ for a vector bundle $V\to X$ with complex orientation (in particular every complex vector bundle). The Gysin map is given by $H^m(Y)\cong H^{m+codim(Y,X)}(Th(N_{Y/X}))\cong H^{m+codim(Y,X)}(X/(X-Y))\to H^{m+codim(Y,X)}(X)$.

The good thing about this describtion is that each step is clearly functorial in an appropriate sense. The Thom isomorphism is the same in $X$ and $\tilde{X}$ because the normal bundles are canonically isomorphic. The identification of the thom space of the normal bundle with the quotient $X/(X-Y)$ is the same by excision of the exceptional divisor and $Z$ for $\tilde{X}$ and $X$ repsectively. Finally, the last step is functorial in pairs.

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