Let $\tilde X$ be the blow up of $X\subset \mathbb{CP^{n+1}}$ along some closed subscheme $C\subset X$, $f : \tilde X\to X$ be the blow down map, and $f_*:H_k(\tilde X)\to H_k(X)$ be the induced map on homology groups. Is it true that $f_*$ is injective?

Edit

Should add the condition that $X$ is of dimension $n$ and $C$ is of dimension $n-2$, and only consider the homology group of degree $n-2$.

No!

Example: blowing up a point in $\mathbb{CP}^2$ gives $\widetilde{\mathbb{CP}^2}=\mathbb{CP}^2\#\overline{\mathbb{CP}^2}$ so its $H_2$ is $\mathbb{Z}^2$, generated by the exceptional divisor $E$ and the $\mathbb{CP}^1$ at infinity, but $H^2(\mathbb{CP}^2)=\mathbb{Z}$ is generated by the $\mathbb{CP}^1$ at infinity.so $f_*\colon H_2(\widetilde{\mathbb{CP}^2})\to H_2(\mathbb{CP}^2)$ cannot be injective. In fact $f$ shrinks the exceptional divisor $E$ to a point so $f_*[E]=0$ in $H_2$.

Maybe you are thinking about the map induced by the strict transform?

  • But anyway I see my original guess is no true. Please check the condition I add in my questions. – User X Sep 13 at 7:17
  • Yes you are right in the original case. – User X Sep 13 at 7:27
  • Have you forgotten some factor of 2 converting between the topological dimension (homology degree) and the dimension over $\mathbb{C}$? – user10354138 Sep 13 at 7:37
  • No. More precisely, $dim_\mathbb C X=n$, $dim_\mathbb C C=n-2$ and degree of homology is $n-1$. (otherwise your previous counter example will still work) – User X Sep 13 at 7:39
  • If you blow up $\mathbb{P}^3$ along a line, the exceptional fiber over a point is a $\mathbb{P}^1$ which is a topological 2-sphere. That should be nontrivial in homology but got killed by blowdown? – user10354138 Sep 13 at 18:53

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