I am reviewing de Rham cohomology, so I picked an example complex projective space, $\mathbb{C}P^2$. Even before I compute the cohomology groups I need to make sure I understand the chain complexes.
$$ 1 \stackrel{d}{\to} \Omega^0(M) \stackrel{d}{\to} \Omega^1(M) \stackrel{d}{\to}\Omega^2(M) \stackrel{d}{\to} 1 $$
What could be the $0$-forms,
those are the constants.those are functions $f: \mathbb{C}P^2 \to \mathbb{R}$.The only reasonable $1$-forms I can think of are $dx, dy, dz$ . Perhaps I have to say $f(x) \, dx$ etc.
The area forms should be: $ f\, \big( dx \wedge dy\big), g\, \big( dy \wedge dz\big), h \big( dz \wedge dx \big) $ .
There should be no 3-forms.
My chain complex looks like this (I may have confused the exterior algebra and the spaces of differential forms) These notations are nearly right, a $2$-form is a map $\omega : T_p^*(M)\to \mathbb{R}$, so here I'm just writing the cotangent spaces:
$$ 1 \to \langle dx, dy, dz \rangle \to \langle dx \wedge dy,\; dy \wedge dz,\; dz \wedge dx \rangle \to 1 $$
Then start questions like, are the 1-forms in my $\Omega^1$ space independent? Should I say that $\Omega^{-1} = 1$. And more problems:
$$ \mathbb{C} P^2 = \{ [x:y:z] : x,y,z \in \mathbb{C}\}/\mathbb{C}^\times $$
So my the forms I have written are not elements of $\Omega^1(\mathbb{C}P^2)$. Instead I have written the 1-forms for $\mathbb{C}^3$. Here is my revised version:
$$ 1 \to \big\langle \frac{dx}{x}, \frac{dy}{y}, \frac{dz}{z} \big\rangle \to \big\langle \frac{dx}{x} \wedge \frac{dy}{y},\; \frac{dy}{y} \wedge \frac{dz}{z},\; \frac{dz}{z} \wedge \frac{dx}{x} \big\rangle \to 1 $$
One question I have is what these numbers are measuring. Certainly if we have:
$$ \frac{dx}{x} = d \big( \log x \big) = d \big( \log r \big) + i \, d\theta $$
Then it suggests that these one-forms are computing angles, but what could the wedge of two angles means $d\theta \wedge d\phi $ ? So there are a few questions swirling around my mind.
This is a fairly algebraic question as no limits are being taken... My main question is not how are the homology groups computed, at this moment, I am only interested in building the de Rham chain complex for $\mathbb{C}P^2$.