I am trying to find an explicit way to view global holomorphic sections of $\Omega^{1} \otimes \mathcal{O} (2)$ over $\mathbb{CP}^{2}$. I guess what I mean by "explicit" would be a formulation over an affine open $U_i \subset \mathbb{CP}^{2}$. According to what I found in Okoneck, Schneider and Spindler, there is a 3dimensional space of such sections, but I want this for a computation in differential geometry.
2 Answers
If $x,y,z$ are coordinates on $P^2$ then the 3 sections of $\Omega(2)$ are given by $xdy  ydx$, $ydzzdy$, and $zdxxdz$.

$\begingroup$ That is elegant, and the sort of answer I wanted. Can you give me a reference as to why this is true? $\endgroup$ Nov 29, 2010 at 5:43

$\begingroup$ It follows from Yuhao's answer plus the Koszul complex exactness. $\endgroup$ Nov 29, 2010 at 6:08

$\begingroup$ I think it is necessary to clarify the notations here: as $x,y,z$ are only sections of $\mathcal{O}(1)$, "d" of them doesn't make sense in the usual way, in fact, as sections of $\Omega(2)$ the above sections are $x^2d(\dfrac{y}{x})$, etc. which can be written formally as $xdyydx$. $\endgroup$ Nov 29, 2010 at 6:22

$\begingroup$ If $P^2 = (V  \{0\})/C^*$ then you can think about $x$, $y$, and $z$ as about the coordinates on $V$. Then $dx$, $dy$ and $dz$ are forms on $V$. $\endgroup$– SashaNov 29, 2010 at 6:27

$\begingroup$ Between Sasha's answers and Torsten and Yuhao's explanations, I believe this makes sense to me now. To Yuhao, as a section of $\mathcal{O}(1)$ is a linear polynomial in the homogeneous coordinates, I think that the formulation of Sasha is correct, and dx has meaning in terms of the pullback via a local section of $\mathbb{C}^3\\{(0,0,0)}\to \mathbb{P}^2$ over a coordinate neighborhood. Thanks for the answers. $\endgroup$ Nov 30, 2010 at 4:36
Use the Euler sequence:
$0 \to \Omega^1_{\mathbb{P}^n_A/A} \to \mathcal{O}_{\mathbb{P}^n_A}(1)^{\oplus n+1} \to \mathcal{O}_{\mathbb{P}^n_A} \to 0.$
Everything can be seen explicitly from here. Tensoring the sequence with $\mathcal{O}(2)$ gives an exact sequence, which is still exact if you take global sections because every monomial of degree 2 is a multiple of a monomial of degree 1.:) So the dim you want = 96 =3.

$\begingroup$ The dimension I can see, but what I am looking for is explicit formulas, like Sasha's  which is probably exactly what I need. $\endgroup$ Nov 29, 2010 at 5:45

1$\begingroup$ Well, the 3 sections Sasha write down is precisely a basis for the kernel of the map obtained by taking global sections. The wikipedia page of Euler sequence contains a description of the maps of the Euler sequence, which is precisely what you missed, I guess. $\endgroup$ Nov 29, 2010 at 6:01

1$\begingroup$ You get the kernel of the map of global sections $\mathcal O_{\mathbb P^n_A}(1)^{\oplus n+1}\to \mathcal O_{\mathbb P^n_A}(2)$ by using the exactness of the Koszul complex of the $x_0,\dots,x_n$. $\endgroup$ Nov 29, 2010 at 6:07