Let $T\_{X}\rightarrow X$ be the tangent bundle over a complex manifold $X.$ Let $\pi:PT\_{X}\rightarrow X$ be a projectivization of that bundle. Let $L$ be the tautological line bundle of $PT\_{X}.$ 


1. How do we construct the isomorphism between the relative tangent bundle $T\_{\pi}$ and $R=L^{\*}\otimes \pi^{\*}T\_X/L$? For example, for every vector $v\in T\_{\pi}$ we know that $d\pi(v)=0,$ but how we can see that vector as an element of $R$?
2. What are the trivialization maps for both of these bundles $T\_{\pi}$ and $R$?



Relative tangent bundle and trivilization, tautological foliation
There is nothing special about the tangent bundle: for any vector bundle $E \rightarrow X$ with projectivisation $\pi : \mathbb{P}E \rightarrow X$, and tautological bundle $\mathcal{L} \rightarrow \mathbb{P}E$, we have $T\mathbb{P}E \cong (\pi^\*E / \mathcal{L}) \otimes \mathcal{L}^\*$. In fact the base $X$ is not really involved, so you may as well work with the case where $X$ is a point, and $E$ is just a vector space $V$. Then the result follows from the standard Euler sequence


$$ 0 \rightarrow \mathcal{O}\_{\mathbb{P}V} \rightarrow V \otimes \mathcal{O}\_{\mathbb{P}V}(1) \rightarrow T\mathbb{P}V \rightarrow 0, $$


where we regard $V$ as a constant sheaf on $\mathbb{P}V$.


***Added in response to comment:*** Explicitly, given a point $l \in \mathbb{P}V$, viewed as a line in $V$, and a vector $v \in T\_l \mathbb{P}V$, we can lift $v$ (under the projection $V \setminus \{0\}\rightarrow \mathbb{P}V$) to a vector in $V/l$ at each point of $l \setminus \{0\}$. Moreover, this lift is equivariant under the action of $k^\times$ on $l\setminus \{0\}$, where $k$ is our ground field (e.g. $\mathbb{C}$). In other words, it defines a **linear** $V/l$-valued function on $l$.


Linear functions on $l$ are precisely elements of $\mathcal{L}^\*\_l$, and $V/l$ is precisely $(\pi^\*V/\mathcal{L})\_l$, so $v$ defines an element of $((\pi^\*V/\mathcal{L}) \otimes \mathcal{L}^\*)\_l$.


***Added in response to further comments:*** Even more explicitly, if $e\_0, e\_1, \dots, e\_n$ is a basis for $V$, with corresponding coordinates $w\_0, w\_1, \dots, w\_n$, then the $w\_i$ form homogeneous coordinates on $\mathbb{P}V$. On the affine patch $w\_0 \neq 0$, we have local coordinates given by $z\_j = w\_j/w\_0$, for $j=1, \dots, n$. Given a point $l = [1: l\_1: \dots: l\_n] \in \mathbb{P}V$, the vector


$$ \sum\_{j=1}^n v\_j \frac{\partial}{\partial z\_j} $$


is tangent to the curve $z\_j(t) = l\_j + tv\_j$. This lifts to the curve in $V$ given by $w\_0(t)=f(t)$, $w\_j(t)=f(t)(l\_j+tv\_j)$ for $j\geq 1$, where $f$ is any smooth $\mathbb{C}^\times$-valued function. Its tangent at $t=0$, i.e. a lift of $v$ to $V$, is


$$ f'(0) \bigg( e\_0 + \sum\_{j=1}^n l\_je\_j \bigg) + \sum\_{j=1}^n v\_j e\_j. $$


The ambiguity coming from $f'(0)$ is exactly what is killed when we quotient by $l$, so the lift to $V/l$ is represented by $\sum v\_je\_j$.


***Explicit computation for $V=\mathbb{C}^2$:*** Take coordinates $w\_0, w\_1$ on $\mathbb{C}^2$, and let $U\_j = \{w\_j \neq 0\} \subset \mathbb{P}^1$. We have


$$(\pi^\*V / \mathcal{L})\_{[w\_0 : w\_1]} = \mathbb{C}^2 \Big/ \mathbb{C} \begin{pmatrix} w\_0 \\ w\_1 \end{pmatrix}.$$


On $U\_0$ a complement to $\mathcal{L}$ in $\pi^\*V$ is given by $\mathbb{C}e\_1$; we can therefore take $[e\_1]$ as a trivialising frame for the quotient bundle (where the square brackets denote the image in the quotient). Similarly on $U\_1$ a frame is given by $[e\_0]$. On $U\_0 \cap U\_1$ we have for all local holomorphic functions $f$ that


$$f e\_1 = \frac{f}{w\_1} \begin{pmatrix} w\_0 \\ w\_1 \end{pmatrix} - \frac{fw\_0}{w\_1} e\_0 $$


so


$$ f[e\_1] = -\frac{fw\_0}{w\_1}[e\_0].$$


The transition function $\psi\_{10}$ is therefore $-w\_0/w\_1$, which (up to the minus sign, which we could have eliminated by choosing $[-e\_0]$ as our frame on $U\_1$) is the transition function for $\mathcal{O}(1)$.


