Why is the blow up of a submanifold of $\mathbb{P}^n$ again projective

Question

I saw somewhere that the blow up (at any point) of a submanifold of $\mathbb{P}^n$ is still projective. I have the feeling that this is a consequence of the Kodaira embedding theorem, any thoughts?

Answer 1

Yes, this follows from Kodaira embedding. If $L$ is a positive holomorphic line bundle on $X$ and $\hat{X} = \operatorname{Bl}_{\{x\}} X$ is a one-point blow-up with projection $\sigma: \hat{X} \to X$ and exceptional divisor $E = \sigma^{-1}(x)$ then $\mathcal{O}_{\hat{X}} ( -E) \otimes \sigma^*L^k$ is positive for $k$ large enough, so $\hat{X}$ is again projective by Kodaira embedding.

To see that last bit notice that the curvature of the pullback connection on $\sigma^*L$ is positive on all vector except those in $TE$. But in a small coordinate neighborhood $U$ centered on $x$, $\hat{U} = \sigma^{-1}(U)$ is given by the incidence variety $\{(y, l) \in U \times \mathbb{P}^{n-1} \mid y \in l\}$ which has a projection $p$ to $\mathbb{P}^{n-1}$.
One can see that $p^* \mathcal{O}_{\mathbb{P}^{n-1}}(1)$ has a section whose zero set is exactly $E$, so $p^* \mathcal{O}_{\mathbb{P}^{n-1}}(-1) \cong \mathcal{O}_{\hat{U}}(E)$ and $p^* \mathcal{O}_{\mathbb{P}^{n-1}}(1) \cong \mathcal{O}_{\hat{U}}(-E)$ . Pulling back the canonical positive metric on $\mathcal{O}_{\mathbb{P}^{n-1}}(1)$ we get a metric on $\mathcal{O}_{\hat{U}}(-E)$ that is positive on vectors in $TE$.
Then you use a bump function of unity that is $1$ on a neighborhood of $\{0\} \times \mathbb{P}^{n-1}$ to extend this to a metric on $\mathcal{O}_{\hat{X}}(-E)$ and pick a large $k$ such that the overall metric on $\mathcal{O}_{\hat{X}} ( -E) \otimes \sigma^*L^k$ is positive