When is there a deRham duality relation between the fundamental class and a top form.? 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.
 A: 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. 
