Recall the holomporphic line boundle $\mathcal{O}_{\mathbb{CP}_{1}}(-2):= \mathcal{O}_{\mathbb{CP}_{1}}(-1) \otimes \mathcal{O}_{\mathbb{CP}_{1}}(-1)$ where $\mathcal{O}_{\mathbb{CP}_{1}}(-1)$ is the "tautological line bundle" of $\mathbb{CP}_{1}$ , defined as $\mathcal{O}_{\mathbb{CP}_{1}}(-1) :=\{ \left( [z_0 ,z_1],(z_0 ,z_1) \right) \in \mathbb{CP}_{1}\times \mathbb{C}^{2}\setminus\{0\} \} $. Also $T^{*}\mathbb{CP}_{1}$ would be the holomorphic cotangent bundle of $\mathbb{CP}_{1}$.
Someone may know from invariants of the line bundles (degree, first Chern class, Euler class, .etc) that the bundles $\mathcal{O}_{\mathbb{CP}_{1}}(-2)$ and $T^{*}\mathbb{CP}_{1}$ are Isomorph (as smooth complex bundles).
I actually don't have a solid background on complex or algebraic geometry. I am looking for a directly writable Isomorphism map between these two line bundles that would be a holomorphic map between the total spaces (it should preserve the holomorphic structure too)
Terminology problem: what should I call this kind of isomorphism?
one possible approach that I have in mind :
both $\mathcal{O}_{\mathbb{CP}_{1}}(-2)$ and $T^{*}\mathbb{CP}_{1}$ can be regarded as Blow-up of $\mathbb{C}^{2}/\mathbb{Z}_2$ at the origin and we have a nice description of the Blow-down map in both cases: $$\pi : \mathcal{O}_{\mathbb{CP}_{1}}(-2) \mapsto \mathbb{C}^{2}/\mathbb{Z}_2 $$ $$\pi : \left( [z_0 ,z_1],\lambda^{2}(z_0 ,z_1)^2 \right) \mapsto \lambda(z_1,z_2) $$ and in here someone have described $T^{*}\mathbb{CP}_{1}$ as a resolution of $\mathbb{C}^{2}/\mathbb{Z}_2$. So, somebody may trace any line from one of these spaces to $\mathbb{C}^{2}/\mathbb{Z}_2$ and to the other one to find the isomorphism. but I can't figure out how exactly $T^{*}\mathbb{CP}_{1}$ and the blow-down map $T^{*}\mathbb{CP}_{1} \mapsto \mathbb{C}^{2}/\mathbb{Z}_2$ are described in the answer. Also, I guess my approach is overcomplicating a simple problem and someone may write the isomorphic map directly without thinking of these blow-downs.
Any help would be appreciated.