# Cohomology of neighborhood of $\mathbb{C}\mathbb{P}^1$ in $\mathbb{C}\mathbb{P}^n$

Let $$\mathbb{C}\mathbb{P}^1$$ be embedded linearly to $$\mathbb{C}\mathbb{P}^n$$ with $$n>1$$. (Such an embedding is given in coordinates by $$[x:y]\mapsto [x:y:0:\dots: 0]$$.)

Is it true that for any open neighborhood $$U$$ (in the analytic topology) of $$\mathbb{C}\mathbb{P}^1$$ there exists a smaller neighborhood $$V$$ of the latter such that $$H^i(V, \mathcal{O})=0 \mbox{ for any } i>0,$$ where $$\mathcal{O}$$ is the structure sheaf?

• There are arbitrarily small tubular neighborhoods that are disk bundles inside the normal bundle of $\mathbb{CP}^1$ relative to $\mathbb{CP}^n$. You can apply the Leray spectral sequence for projection from the disk bundle to conclude the vanishing that you want. – Jason Starr Oct 16 '18 at 16:33
• Do I understand correctly that $E_2^{p,q}=0$ for $q>0$.? In that case how to show the vanishing of $H^1(\mathbb{C}\mathbb{P}^1)$ with coefficients in the push forward under the projection of the structure sheaf? – makt Oct 16 '18 at 17:32
• I was wrong. I forgot to dualize the normal bundle. Please see Misha Verbitsky's post below. – Jason Starr Oct 16 '18 at 20:25

Take a complement $$M:=CP^3 \ CP^1$$ and consider a projection $$\pi:\; M \mapsto CP^1$$. It is not hard to see that $$M$$ is isomorphic to the total space of the bundle $$O(1)^2$$. The fibers of $$\pi$$ are Stein, hence $$R^i\pi_*F=0$$ for $$i>0$$ and any coherent sheaf $$F$$, and cohomology $$H^i(O_M)$$ are the same as $$H^i(\pi_* O_M)$$.
However, $$\pi_* O_M= Sym^*(O(-1)^2)$$, because the regular functions on the total space of $$O(1)^2$$ are $$Sym^*(O(-1)^2)$$. However, $$H^1(Sym^*(O(-1)^2))$$ is infinite-dimensional.
Same argument works for smaller neighbourhoods $$U\supset CP^1$$, as long as the fibers of $$\pi:\; U \mapsto C P^1$$ remain Stein.
Same is true for a neighbourhood of a rational line $$C$$ in a complex manifold if the normal bundle $$NC$$ is ample, but the proof is more complicated.
• To make sure: when you say that the fibers are Stein, do you refer to a version of the base change theorem? Note that the morphism $\pi$ is not proper. – makt Oct 16 '18 at 22:11