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Consider the total space $\mathcal{O}(-1)\rightarrow\mathbb{C}\mathbb{P}^1$, it is simply the blow-up $Bl_0\mathbb{C}^2$, now we remove the zero section $E$, we naturally obtain a biholomorphism from $Bl_0\mathbb{C}^2-E$ to $\mathbb{C}^2-\{0\}$. It is now a non-compact space.

Now we consider function $f$ on $Bl_0{\mathbb{C}^2-E}$ with sufficiently high regularity and decay rate such that it's possible to talk about integration by part for $f$. I think we need to first identify the boundary to formulate integration by part. My question is, is that true now we want to view the exceptional divisor $E$ as the boundary of $Bl_0\mathbb{C}^2-E$? Or is there a better formulation?

I'd appreciate any relevant discussions or comments on it, thanks in advance!

Maths007
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  • Often in complex geometry one works with currents (i.e., differential forms with locally integrable coefficients). For example, the Cauchy integral formula can be framed that way instead of by applying Stokes's Theorem to the domain with a little disk removed. At any rate, go to a simpler situation: What do you want to do on $\Bbb C-{0}$? – Ted Shifrin Sep 20 '23 at 20:36
  • Thanks, I believe the $\mathbb{C}^2-{0}$ would be worth looking at, just that in terms of integration, we only need to consider the point singularity for $\mathbb{C}^2-{0}$ case but when removing a section form our total space, there seems to be more complexity. In terms of the setting of currents, maybe we still want to first identify the (regularity of the) boundary if we wanna do IBP? – Maths007 Sep 20 '23 at 21:15
  • One can, for example, interpret the Chern class of the line bundle associated to a divisor as a current that is smooth off the divisor. This is an immediate generalization of the Cauchy integral formula in one dimension. It's hard to read your mind: Can you edit your post with more details about what precisely you're trying to integrate by parts? – Ted Shifrin Sep 20 '23 at 21:35
  • Sorry I've edited the post! – Maths007 Sep 20 '23 at 21:48
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    Ordinarily, you would remove an $\epsilon$ tubular neighborhood of $E$ and work there, then take the limit. This amounts morally to removing an $\epsilon$ ball around $0$ in $\Bbb C^2$. I’m not sure what Kähler form you’re using with these identifications. Ideally, IBP is easier to handle working with differential forms rather than functions. – Ted Shifrin Sep 20 '23 at 22:37
  • Right. Seems like that is the reasonable approach. – Ted Shifrin Sep 21 '23 at 00:24
  • I see! In this case, I'm curios in how we should understand the regularity of its $E$ in $Bl_0\mathbb{C}^2-E$. As a section removed from this space, intuitively, I think the regularity could be very bad, but can we show that more explicitly? (or maybe that intuition is wrong) – Maths007 Sep 21 '23 at 01:03
  • I don’t understand what you’re saying. You know exactly what $E$ is in the product $\Bbb C^2\times \Bbb P^1$. – Ted Shifrin Sep 21 '23 at 01:35
  • Yeah that's right sorry – Maths007 Sep 21 '23 at 05:20

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