Let $X$ be a compact Kähler manifold, with $j\_Z: Z\hookrightarrow X$ a submanifold of complex codimension $r$, $\tau: \widetilde{X} \to X$ the blow-up of $X$ along $Z$, with exceptional divisor $j: E \hookrightarrow \widetilde{X}$. The restriction $\tau\_E: E\to Z$ is then a $\mathbb{CP}^{r-1}$-bundle.


Let $h=c\_1(\mathcal{O}\_E(1)) \in H^2(E,\mathbb{Z})$ be the first Chern class of the tautological bundle $\mathcal{O}\_E(1)$ of the $\mathbb{CP}^{r-1}$-bundle $\tau\_E$. One may compute the cohomology groups of $\widetilde{X}$ (see for example [1, Theorem 7.31]) :


$$H^k(X,\mathbb{Z}) \oplus \Big(\oplus\_{i=0}^{r-2} H^{k-2i-2}(Z,\mathbb{Z} ) \Big) \simeq H^k(\widetilde{X}, \mathbb{Z} ), $$
where the isomorphism is given by the sum of
$\tau^\*: H^k(X,\mathbb{Z}) \to H^k(\widetilde{X}, \mathbb{Z} )$ and
$$j\_\* \circ (\cup h^i) \circ \tau\_E^\*: H^{k-2i-2}(Z,\mathbb{Z} ) \to H^k(\widetilde{X}, \mathbb{Z} ). $$


I want to compute the cohomology ring sturcture of $H^\*(\widetilde{X}, \mathbb{Z})$. For example, for any $\alpha \in H^{2k}(X, \mathbb{Z})$, how can I compute $[E]^{n-k} \cup \tau^\*\alpha$? I'd like some references about such computations.


Thanks a lot for your answers!


[1] C. Voisin, Hodge Theory and Complex Algebraic Geometry I. Cambridge University Press, 2003



What's the cohomology ring structure of a blow-up?
This is straightforward. Using $j^\*[E]=-h$ and the projection formula $j\_\*(x\cdot j^\*y)=j\_\*x\cdot y$, we get $[E]^p=j\_\*1\cdot [E]^{p-1}= (-1)^{p-1}j\_\*(h^{p-1})$. Then
$$[E]^{p}\cdot \tau ^\*\alpha = (-1)^{p-1}j\_\*(h^{p-1}\cdot j^\*\tau ^\*\alpha )= (-1)^{p-1}j\_\*(h^{p-1}\cdot p^\*\alpha\_{|Z} )\,,$$ where $p:E\rightarrow Z$ is the projection. You can compute all products in this way.


