[proofplan] We prove the result locally in holomorphic coordinates adapted to the embedded submanifold. In such coordinates, $T_Y$ is spanned by the coordinate vector fields tangent to $Y$, while $i^*T_X$ is spanned by all ambient coordinate vector fields restricted to $Y$. The quotient is therefore locally spanned by the normal coordinate directions. The only point requiring verification is that these local quotients glue holomorphically on overlaps; this follows because adapted coordinate changes preserve the ideal of functions vanishing on $Y$. [/proofplan]

[step:Choose adapted holomorphic coordinates near each point of $Y$]Let $n=\dim_{\mathbb C}X$ and $r=\dim_{\mathbb C}Y$. Fix a point $p\in Y$. Since $i:Y\hookrightarrow X$ is a holomorphic embedding, it identifies $Y$ locally with a closed complex coordinate submanifold of $X$. Thus there is an open neighbourhood $U\subset X$ of $p$ and a holomorphic coordinate map $z:U\to z(U)\subset \mathbb C^n$ with coordinate functions $z_1,\dots,z_n$ such that \begin{align*} Y\cap U=\{x\in U:z_{r+1}(x)=\cdots=z_n(x)=0\}. \end{align*} The restricted functions $z_1|_{Y\cap U},\dots,z_r|_{Y\cap U}$ form holomorphic coordinates on $Y\cap U$. In this chart, the holomorphic vector bundle $T_X|_{Y\cap U}$ has frame \begin{align*} \left.\frac{\partial}{\partial z_1}\right|_{Y\cap U},\dots,\left.\frac{\partial}{\partial z_n}\right|_{Y\cap U}, \end{align*} and the holomorphic vector bundle $T_Y|_{Y\cap U}$ has frame \begin{align*} \frac{\partial}{\partial z_1|_Y},\dots,\frac{\partial}{\partial z_r|_Y}. \end{align*} Under the differential $di:T_Y\to i^*T_X$, this frame maps to the first $r$ ambient coordinate vector fields restricted to $Y\cap U$: \begin{align*} di\left(\frac{\partial}{\partial z_a|_Y}\right)=\left.\frac{\partial}{\partial z_a}\right|_{Y\cap U} \end{align*} for every $a\in\{1,\dots,r\}$.[/step]

[guided]The purpose of adapted coordinates is to make the inclusion $Y\subset X$ look like the standard coordinate-plane inclusion \begin{align*} \mathbb C^r\hookrightarrow \mathbb C^n. \end{align*} Because $i$ is a holomorphic embedding, the holomorphic submanifold theorem gives, around each point $p\in Y$, a holomorphic coordinate map $z:U\to z(U)\subset \mathbb C^n$ with coordinate functions $z_1,\dots,z_n$ for which the submanifold is described by \begin{align*} Y\cap U=\{x\in U:z_{r+1}(x)=\cdots=z_n(x)=0\}. \end{align*} Thus the first $r$ coordinates are coordinates along $Y$, and the remaining $n-r$ coordinates are transverse coordinates. The ambient tangent bundle restricted to $Y\cap U$ is therefore freely generated, as a holomorphic vector bundle, by \begin{align*} \left.\frac{\partial}{\partial z_1}\right|_{Y\cap U},\dots,\left.\frac{\partial}{\partial z_n}\right|_{Y\cap U}. \end{align*} The tangent bundle of $Y$ itself is freely generated by the coordinate vector fields in the first $r$ directions: \begin{align*} \frac{\partial}{\partial z_1|_Y},\dots,\frac{\partial}{\partial z_r|_Y}. \end{align*} The differential \begin{align*} di:T_Y\to i^*T_X \end{align*} sends a tangent vector to $Y$ to the same derivation viewed as a tangent vector to $X$ along $Y$. In these coordinates this gives \begin{align*} di\left(\frac{\partial}{\partial z_a|_Y}\right)=\left.\frac{\partial}{\partial z_a}\right|_{Y\cap U} \end{align*} for every $a\in\{1,\dots,r\}$. Hence $di$ is locally the inclusion of the span of the first $r$ basis vectors into the span of all $n$ basis vectors.[/guided]

[step:Identify the local quotient with the normal coordinate directions] On $Y\cap U$, define the local quotient bundle \begin{align*} Q_U:=\left(i^*T_X|_{Y\cap U}\right)\big/\left(di(T_Y|_{Y\cap U})\right). \end{align*} Let \begin{align*} q_U:i^*T_X|_{Y\cap U}\to Q_U \end{align*} be the quotient map. Since $di(T_Y|_{Y\cap U})$ is generated by the first $r$ ambient coordinate vector fields, the classes \begin{align*} q_U\left(\left.\frac{\partial}{\partial z_{r+1}}\right|_{Y\cap U}\right),\dots,q_U\left(\left.\frac{\partial}{\partial z_n}\right|_{Y\cap U}\right) \end{align*} form a holomorphic frame for $Q_U$. Therefore $Q_U$ is a holomorphic vector bundle of rank $n-r$ on $Y\cap U$. [/step]

[step:Verify that the local quotient frames glue holomorphically]Let $U$ and $V$ be two adapted coordinate neighbourhoods with coordinates $z_1,\dots,z_n$ on $U$ and $w_1,\dots,w_n$ on $V$. On $Y\cap U\cap V$, write the coordinate change as holomorphic functions \begin{align*} w_k=w_k(z_1,\dots,z_n) \end{align*} for $k\in\{1,\dots,n\}$. Because both coordinate systems are adapted to $Y$, the functions $w_\mu$ vanish on $Y\cap U\cap V$ for every $\mu\in\{r+1,\dots,n\}$. Therefore, for every tangent index $a\in\{1,\dots,r\}$ and every normal index $\mu\in\{r+1,\dots,n\}$, \begin{align*} \left.\frac{\partial w_\mu}{\partial z_a}\right|_{Y\cap U\cap V}=0. \end{align*} The holomorphic chain rule gives, along $Y\cap U\cap V$, \begin{align*} \left.\frac{\partial}{\partial z_\nu}\right|_Y=\sum_{k=1}^{n}\left.\frac{\partial w_k}{\partial z_\nu}\right|_Y\left.\frac{\partial}{\partial w_k}\right|_Y \end{align*} for every $\nu\in\{r+1,\dots,n\}$. Passing to the quotient by $T_Y$, the terms with $k\le r$ disappear. Hence \begin{align*} q_U\left(\left.\frac{\partial}{\partial z_\nu}\right|_Y\right)=\sum_{\mu=r+1}^{n}\left.\frac{\partial w_\mu}{\partial z_\nu}\right|_Y q_V\left(\left.\frac{\partial}{\partial w_\mu}\right|_Y\right). \end{align*} The transition matrix \begin{align*} \left(\left.\frac{\partial w_\mu}{\partial z_\nu}\right|_Y\right)_{\mu,\nu=r+1}^{n} \end{align*} has holomorphic entries. Since the full [Jacobian matrix](/page/Jacobian%20Matrix) of the coordinate change is invertible and the tangent-to-normal block vanishes along $Y$, this normal block is invertible. Thus the local quotient frames glue by holomorphic invertible transition functions.[/step]

[guided]We must check that the quotient constructed in one adapted chart is the same holomorphic vector bundle as the quotient constructed in another adapted chart. Let $U$ and $V$ be adapted coordinate neighbourhoods, with coordinates $z_1,\dots,z_n$ on $U$ and $w_1,\dots,w_n$ on $V$. On the overlap $U\cap V$, each $w_k$ is a [holomorphic function](/page/Holomorphic%20Function) of the $z$-coordinates: \begin{align*} w_k=w_k(z_1,\dots,z_n). \end{align*} The adapted condition says that $Y$ is cut out in the $w$-chart by \begin{align*} w_{r+1}=\cdots=w_n=0. \end{align*} Thus, for every normal index $\mu\in\{r+1,\dots,n\}$, the function $w_\mu$ vanishes identically on $Y\cap U\cap V$. If we differentiate this identity along a tangent coordinate direction $z_a$ with $a\in\{1,\dots,r\}$, we obtain \begin{align*} \left.\frac{\partial w_\mu}{\partial z_a}\right|_{Y\cap U\cap V}=0. \end{align*} This is the key block-triangularity statement: an adapted coordinate change cannot send a tangent direction of $Y$ into a normal direction to first order along $Y$. Now apply the holomorphic chain rule to the ambient coordinate vector fields. For every normal $z$-direction $\nu\in\{r+1,\dots,n\}$, along $Y\cap U\cap V$ we have \begin{align*} \left.\frac{\partial}{\partial z_\nu}\right|_Y=\sum_{k=1}^{n}\left.\frac{\partial w_k}{\partial z_\nu}\right|_Y\left.\frac{\partial}{\partial w_k}\right|_Y. \end{align*} When we pass to the quotient by $T_Y$, every term involving $\partial/\partial w_k$ with $k\le r$ becomes zero, because these are precisely tangent vector fields to $Y$. Therefore the class of the $z_\nu$ normal vector is \begin{align*} q_U\left(\left.\frac{\partial}{\partial z_\nu}\right|_Y\right)=\sum_{\mu=r+1}^{n}\left.\frac{\partial w_\mu}{\partial z_\nu}\right|_Y q_V\left(\left.\frac{\partial}{\partial w_\mu}\right|_Y\right). \end{align*} The coefficients are holomorphic functions on $Y\cap U\cap V$ because the coordinate change is holomorphic. It remains to justify invertibility of this transition matrix. The full Jacobian matrix \begin{align*} \left(\frac{\partial w_k}{\partial z_j}\right)_{k,j=1}^{n} \end{align*} is invertible because $z$ and $w$ are coordinate systems. The adapted condition gives $w_\mu|_Y=0$ for every $\mu\in\{r+1,\dots,n\}$, so differentiating along each tangent coordinate direction $z_a$ with $a\in\{1,\dots,r\}$ yields \begin{align*} \left.\frac{\partial w_\mu}{\partial z_a}\right|_Y=0. \end{align*} Hence the full Jacobian is block upper triangular along $Y$, with one diagonal block acting on tangent directions and the other diagonal block \begin{align*} \left(\left.\frac{\partial w_\mu}{\partial z_\nu}\right|_Y\right)_{\mu,\nu=r+1}^{n} \end{align*} acting on normal directions. Since an invertible block upper triangular matrix has invertible diagonal blocks, the normal block is invertible. Thus the quotient frames glue by holomorphic invertible transition functions, so they define a holomorphic vector bundle.[/guided]

[step:Assemble the quotient bundle and prove exactness] The preceding gluing construction defines a holomorphic vector bundle on $Y$, denoted $N_{Y/X}$, whose local frames are the quotient classes of the normal coordinate vector fields. The maps \begin{align*} di:T_Y\to i^*T_X \end{align*} and \begin{align*} q:i^*T_X\to N_{Y/X} \end{align*} are holomorphic bundle morphisms because, in adapted coordinates, $di$ is the inclusion of the first $r$ frame vectors and $q$ is projection onto the last $n-r$ quotient frame vectors. Let $\mathcal O_{Y\cap U}$ denote the sheaf of holomorphic functions on the complex manifold $Y\cap U$. In the adapted local frame, the sequence becomes \begin{align*} 0\longrightarrow \mathcal O_{Y\cap U}^{\,r}\longrightarrow \mathcal O_{Y\cap U}^{\,n}\longrightarrow \mathcal O_{Y\cap U}^{\,n-r}\longrightarrow 0, \end{align*} where the first map includes the first $r$ factors and the second map projects to the last $n-r$ factors. This sequence is exact at every point of $Y\cap U$. Since exactness of vector bundle morphisms can be checked in local frames, the global sequence \begin{align*} 0 \longrightarrow T_Y \xrightarrow{\,di\,} i^*T_X \xrightarrow{\,q\,} N_{Y/X} \longrightarrow 0 \end{align*} is short exact. [/step]