Question

Hi, everyone: I am reading a small expository paper on properties of CP², in which the intersection form is defined as an integral of the wedge of two forms $w_1$, $w_2$, and these forms $w_1$, $w_2$ (no problem with compact support, since CP² is compact) seem to have been obtained from the fundamental class [z] of H₂(CP²)-- a copy of CP¹ (embedded in CP²), after which we integrate $w:=w1\wedge w2$ to get the intersection number.

I am curious on whether I am reading the above correctly, i.e., that the volume form in CP² is obtained by using the fund. class [z] in H₂. If not, would someone explain; if this is correct, if we are we using some form of deRham's theorem to turn a purely topological object like [z] into an object like $w$, for which we must have a differentiable structure defined)?

Thanks in Advance.

Answer 1

Consider a suitably small tubular neighbourhood $\mathcal{N}$ of $\mathbb{CP}^1$, thought of as sitting inside $\mathbb{CP}^2$. Then $\mathcal{N}$ locally looks like $\mathbb{CP}^1\times D_2$. The volume form $\omega$ of $\mathbb{CP}^1$ is not necessarily a $2$-form in $\mathbb{CP}^2$. However, one can imagine changing it so that we have new $2$-form $\widetilde{\omega}$ in $\mathbb{CP}^2$, supported in $\mathcal{N}$, such that $\widetilde{\omega}$ restrcited to ${p}\times D_2$ (for $p\in\mathbb{CP}^1$) looks like a smooth bump function which integrates to $1$. This can be taken to be $w_1$ in your case. Now assume you take your copy of $\mathbb{CP}^1$ inside $\mathbb{CP}^2$ and perturb it a bit (i.e., make it transversal to itself) to get another copy. Apply what we said before and get $w_2$ supported in a suitable tubular neighbourhood of this perturbed copy. Now integrating $w_1\wedge w_2$ over $\mathbb{CP}^2$ gives you an integration over balls around points where self-intersections occur. The normalization were so chosen that it counts the intersection number of $\mathbb{CP}^1$ with itself.