I am trying to work some examples, and i chose $\mathbb{C}P^1$. Representing points by homogeneous coordinates $[z]=[z^1:z^2]$, the tangent space can be desribed as: $$ T_{[z]}\mathbb{C}P^1 = \{ w \in \mathbb{C}^2 | \langle z,w\rangle =0 \} $$ where the inner product is just the usual inner product of $\mathbb{C}^2$. However, i am not able to make sense of element of this tangent bundle as derivations. I think one choice of path having $w \in T_{[z]}\mathbb{C}P^1$ as tangent vector is $$ (-\epsilon,\epsilon) \stackrel{\gamma}{\rightarrow} \mathbb{C}P^1 \\ \ \ \ \ \ \ \ \ \ \ \ \ t \mapsto [z + t w] $$ but if one tries to use this to undestand what $w$ looks like in the coordinate basis given by the chart $$ \{ [z^1:z^2] |z^1 \neq 0 \} =: U \ni [z^1:z^2] \mapsto \xi := \frac{z^2}{z^1} $$ the resulting expression does not only depend on the coordinate $\xi$. To be more specific, if $w = (w^1,w^2)$ then the derivation associated to it should act as $$ (wf)([z]) = \xi \frac{\partial{f}}{\partial \xi}(1+|\xi|^2) \frac{w^2}{z^2} + \text{anti-holomorphic part} $$ Note that i used that $w$ is orhtogonal to $z$ and that $f$ depends only on the equivalence class of $z$ to eliminate some derivatives. I can also give more details on the calculation, but i have the impression that there is something i am not seeing, so help would be much appreciated.
1 Answers
Locally, if you want to differentiate a map $f\colon (-\epsilon,\epsilon)\to\Bbb P^1$ (omitting the $\Bbb C$'s), you can take any lifting $\tilde f\colon (-\epsilon,\epsilon)\to\Bbb C^2-\{0\}$ and compute the derivative $\tilde f{}'(t)$ mod the subspace spanned by $f(t)$. This is in effect what you're doing when you use orthogonal complements to the subspace. However, the danger in doing that is that you depart from the holomorphic category when you use the hermitian inner product. Note, for example, that $w = (-\bar\xi,1)$ in your case. So I'm not sure where your computation even comes from. Oh, so you're using instead $w=(1,-\xi/|\xi|^2)$?
By the way, you need the Euler sequence to see the global twisting of the tangent bundle. See, for example, my answer to this question.
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