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 integrated 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.
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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 $\tilde{\omega}$ \widetilde{\omega}$ in $\mathbb{CP}^2$, supported in $\mathcal{N}$, such that $\tilde{\omega}$ \widetilde{\omega}$ restrcited to ${p}\times D_2$ (for $p\in\mathbb{CP}^1$) looks like a smooth bump function which integrated 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. |
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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 $\tilde{\omega}$ in $\mathbb{CP}^2$, supported in $\mathcal{N}$, such that $\tilde{\omega}$ restrcited to ${p}\times D_2$ looks like a smooth bump function which integrated 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. |
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