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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

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  • $\begingroup$ This is done explicitly for curves in complex $3$-folds in Proposition 14 "Cubic forms and Complex 3-folds" by Okonek and Van de ven. Actually they compute the associated cubic form $F: H^{2}(\tilde{X},\mathbb{Z}) \rightarrow \mathbb{Z}$ given by $F(\alpha) = \int_{\tilde{X}} \alpha^3$. In fact $F$ determines the cohomology ring of the blow up, If the curve is denoted $C=Z$, and $N_{C}$ is its normal bundle then the formula is $F(\pi^*h + x E) = h^3 -3h.C x^2 -deg(N_{C}) x^3$. (This will of course follow from abx's answer after some work). $\endgroup$
    – Nick L
    Commented May 31, 2021 at 11:41

1 Answer 1

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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.

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  • $\begingroup$ I have some additions to abx's answer. For the case I want to compute, $[E]^p \cdot \tau^*\alpha = (-1)^{p-1} j_*(h^{p-1} \cdot p^*\alpha|_Z) $ with $\alpha \in H^{2n-2p}(X,\mathbb{Z}) $, if $p <r$, then $\alpha|_Z=0 $; if $p > r$, then $h^{p-1}=0 $. So the only nonzero case is $p=r$ and we have $h^{r-1}\cdot p^*\alpha|_Z \ne 0$ whenever $\alpha|_Z \ne 0$. $\endgroup$
    – Lineer
    Commented Jun 1, 2021 at 4:59

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