# Gysin morphism of blow up

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.

• I am not entirely sure, but discussion in Chapters 2 and 19 of the 2nd edition of Fulton's Intersection theory may be helpful Dec 26 '18 at 8:26

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.