[proofplan] We describe a first-order embedded deformation by writing local graph presentations over the dual numbers in coordinates adapted to $Y\subset X$. The coefficient of $\varepsilon$ in the [normal coordinates](/theorems/2713) defines a local holomorphic section of the normal bundle, and the coordinate-change rule shows that these local normal sections glue. Conversely, a global normal section gives local first-order normal graphs, and the equality of the normal section on overlaps makes the corresponding closed subspaces glue. First-order reparametrisations change only the tangent part of a local graph presentation, so the resulting embedded deformation depends exactly on the associated section of $N_{Y/X}$. [/proofplan]

[step:Represent an embedded first-order deformation by a section of $i^*T_X$]Let $n=\dim_{\mathbb C}Y$ and $r=\operatorname{codim}_{\mathbb C}(Y,X)$. Choose an [open cover](/page/Open%20Cover) $(Y_i)_{i\in I}$ of $Y$ and, for each $i\in I$, a holomorphic coordinate chart \begin{align*} \phi_i:U_i\to V_i\subset \mathbb C^{n+r} \end{align*} on an [open set](/page/Open%20Set) $U_i\subset X$ containing $i(Y_i)$, with coordinates $(z_{i,1},\dots,z_{i,n},w_{i,1},\dots,w_{i,r})$, such that $i(Y_i)\subset U_i$ and \begin{align*} \phi_i(i(Y_i))=V_i\cap (\mathbb C^n\times\{0\}). \end{align*} Let $\mathcal Y\subset X\times D$ be an embedded first-order deformation of $Y$. Let $\mathcal I_i\subset \mathcal O_{U_i}$ denote the ideal sheaf of $i(Y_i)$ in $U_i$, so $\mathcal I_i$ is generated by $w_{i,1},\dots,w_{i,r}$. Let $\mathcal J_i\subset \mathcal O_{U_i}\otimes_{\mathbb C}\mathbb C[\varepsilon]/(\varepsilon^2)$ denote the ideal sheaf of $\mathcal Y\cap(U_i\times D)$. Since $\mathcal Y$ is flat over $D$ and its central fibre is $Y$, reduction modulo $\varepsilon$ identifies $\mathcal J_i/\varepsilon\mathcal J_i$ with $\mathcal I_i$. By [Nakayama's lemma](/theorems/2935) over the local Artin ring $\mathbb C[\varepsilon]/(\varepsilon^2)$, after shrinking $U_i$ the ideal $\mathcal J_i$ is generated by lifts $F_{i,a}$ of the generators $w_{i,a}$. Writing $F_{i,a}=w_{i,a}-\varepsilon H_{i,a}$ with $H_{i,a}$ holomorphic on $U_i$, the class of $H_{i,a}$ modulo $\mathcal I_i$ is a [holomorphic function](/page/Holomorphic%20Function) $h_{i,a}:Y_i\to\mathbb C$. Replacing $H_{i,a}$ by its restriction class gives the first-order graph equations $w_{i,a}=\varepsilon h_{i,a}$ on the deformed subspace, because all terms in $\mathcal I_i$ are already $O(\varepsilon)$ on the graph and are killed after multiplication by $\varepsilon$. Thus, in the adapted chart over $Y_i$, choose a local graph presentation over $D$, equivalently a holomorphic map \begin{align*} I_i:Y_i\times D\to X \end{align*} such that $(I_i,\operatorname{id}_D):Y_i\times D\to X\times D$ identifies $Y_i\times D$ with the local closed subspace $\mathcal Y\cap(U_i\times D)$. In these coordinates, the restriction of $\phi_i\circ I_i$ to $Y_i\times D$ has the form \begin{align*} (z_{i,1},\dots,z_{i,n},0,\dots,0)+\varepsilon(g_{i,1},\dots,g_{i,n},f_{i,1},\dots,f_{i,r}), \end{align*} where \begin{align*} g_{i,b}:Y_i\to\mathbb C \end{align*} and \begin{align*} f_{i,a}:Y_i\to\mathbb C \end{align*} are holomorphic functions. Define \begin{align*} \xi_i:Y_i\to i^*T_X|_{Y_i} \end{align*} by \begin{align*} \xi_i(p)=\sum_{b=1}^n g_{i,b}(p)\,\partial_{z_{i,b}}|_{i(p)}+\sum_{a=1}^r f_{i,a}(p)\,\partial_{w_{i,a}}|_{i(p)}. \end{align*} Let \begin{align*} q:i^*T_X\to N_{Y/X} \end{align*} be the quotient map in the normal exact sequence from the theorem statement. The local normal displacement is \begin{align*} s_i:=q\circ \xi_i:Y_i\to N_{Y/X}|_{Y_i}. \end{align*} Equivalently, \begin{align*} s_i(p)=\sum_{a=1}^r f_{i,a}(p)\,\overline{\partial_{w_{i,a}}|_{i(p)}}, \end{align*} where the bar denotes the image under $q_{p}:T_{X,i(p)}\to N_{Y/X,p}$.[/step]

[guided]The first-order analytic germ $D$ has local algebra $\mathbb C[\varepsilon]/(\varepsilon^2)$, so a holomorphic map over $D$ is determined locally by its central value and its coefficient of $\varepsilon$. We choose coordinates adapted to the embedding $i$. Thus $z_{i,1},\dots,z_{i,n}$ are coordinates along $Y_i$, while $w_{i,1},\dots,w_{i,r}$ are normal coordinates, and the original embedding has coordinate expression \begin{align*} p\mapsto (z_{i,1}(i(p)),\dots,z_{i,n}(i(p)),0,\dots,0). \end{align*} Because $I_i$ restricts to $i$ on the central fibre, its coordinate expression can differ from this map only by a first-order term. Hence there are holomorphic functions \begin{align*} g_{i,b}:Y_i\to\mathbb C \end{align*} for $1\le b\le n$ and \begin{align*} f_{i,a}:Y_i\to\mathbb C \end{align*} for $1\le a\le r$ such that \begin{align*} \phi_i(I_i(p,\varepsilon))=(z_{i,1}(i(p)),\dots,z_{i,n}(i(p)),0,\dots,0)+\varepsilon(g_{i,1}(p),\dots,g_{i,n}(p),f_{i,1}(p),\dots,f_{i,r}(p)). \end{align*} The preceding normal form comes from the ideal-theoretic description of the flat first-order deformation: the ideal of $Y_i$ is generated by the normal coordinates, and flatness over the dual numbers allows those generators to be lifted to $w_{i,a}-\varepsilon H_{i,a}$. Restricting $H_{i,a}$ to $Y_i$ gives the holomorphic normal coefficients $f_{i,a}$, and terms vanishing on $Y_i$ do not affect the first-order graph because they are multiplied by $\varepsilon$ and become quadratic in the dual-number direction. The coefficient of $\varepsilon$ is a tangent vector to $X$ at $i(p)$, so it defines a holomorphic section \begin{align*} \xi_i:Y_i\to i^*T_X|_{Y_i} \end{align*} by \begin{align*} \xi_i(p)=\sum_{b=1}^n g_{i,b}(p)\,\partial_{z_{i,b}}|_{i(p)}+\sum_{a=1}^r f_{i,a}(p)\,\partial_{w_{i,a}}|_{i(p)}. \end{align*} The tangent component, represented by the $g_{i,b}$ terms, records motion caused by reparametrising the source. The invariant embedded displacement is obtained by applying the quotient map \begin{align*} q:i^*T_X\to N_{Y/X} \end{align*} from the normal exact sequence in the theorem statement. Therefore \begin{align*} s_i:=q\circ \xi_i:Y_i\to N_{Y/X}|_{Y_i} \end{align*} is the local normal section attached to the first-order deformation. In the chosen coordinates this says \begin{align*} s_i(p)=\sum_{a=1}^r f_{i,a}(p)\,\overline{\partial_{w_{i,a}}|_{i(p)}}. \end{align*}[/guided]

[step:Check the adapted coordinate transition for the normal coefficients]Fix $i,j\in I$ and set $Y_{ij}:=Y_i\cap Y_j$. On $U_i\cap U_j$, write the adapted coordinate transition as \begin{align*} \phi_j\circ\phi_i^{-1}(z,w)=(Z_{ji}(z,w),W_{ji}(z,w)). \end{align*} Since both coordinate systems are adapted to $Y$, one has $W_{ji}(z,0)=0$. Therefore the differential of $W_{ji}$ along $Y$ has no tangent block: \begin{align*} \partial_{z_b}W_{ji,a}(z,0)=0 \end{align*} for every $a$ and $b$. Let $A_{ji}(z)$ denote the $r\times r$ matrix with entries \begin{align*} (A_{ji}(z))_{ac}:=\partial_{w_c}W_{ji,a}(z,0). \end{align*} This matrix is precisely the transition matrix of the normal bundle in the frames $\overline{\partial_{w_{i,c}}}$ and $\overline{\partial_{w_{j,a}}}$. For the graph coefficient in the $i$-chart, substitute \begin{align*} (z,w)=(z_i,0)+\varepsilon(g_i,f_i) \end{align*} into the transition formula. Because $\varepsilon^2=0$, the first-order Taylor formula gives \begin{align*} W_{ji}((z_i,0)+\varepsilon(g_i,f_i))=\varepsilon A_{ji}(z_i)f_i. \end{align*} The tangent coefficient $g_i$ contributes no normal term because $\partial_zW_{ji}(z,0)=0$. If the two parametrized graph presentations differ by a first-order reparametrisation of $Y_{ij}\times D$, the induced change in the first-order coefficient is a section of $di(T_Y)$, and its image under $q$ is zero by exactness of the normal sequence. Hence \begin{align*} s_j=A_{ji}s_i \end{align*} on $Y_{ij}$, which is exactly the normal-bundle gluing rule. Therefore the family $(s_i)_{i\in I}$ glues to a unique section \begin{align*} s_{\mathcal Y}\in H^0(Y,N_{Y/X}). \end{align*}[/step]

[guided]The point of the overlap calculation is to separate two effects: changing ambient coordinates and changing the source parameter. Write the coordinate change as $(z,w)\mapsto (Z_{ji}(z,w),W_{ji}(z,w))$. Because the submanifold is $w=0$ in the first chart and also $W=0$ in the second chart, we have $W_{ji}(z,0)=0$. Differentiating this identity in each tangent coordinate $z_b$ gives $\partial_{z_b}W_{ji,a}(z,0)=0$. Thus tangent first-order motion cannot create a normal first-order coefficient under the coordinate change. Now insert the first-order graph $(z,w)=(z_i,0)+\varepsilon(g_i,f_i)$. The equality $\varepsilon^2=0$ kills all second-order Taylor terms, so the normal coordinates in the $j$-chart are \begin{align*} W_{ji}((z_i,0)+\varepsilon(g_i,f_i))=W_{ji}(z_i,0)+\varepsilon dW_{ji,(z_i,0)}(g_i,f_i). \end{align*} The first term is zero, and the tangent block of $dW_{ji}$ is zero. Therefore \begin{align*} W_{ji}((z_i,0)+\varepsilon(g_i,f_i))=\varepsilon A_{ji}(z_i)f_i, \end{align*} where $A_{ji}(z_i)$ is the normal block $\partial_wW_{ji}(z_i,0)$. This is exactly the transition law for the normal bundle. Finally, a first-order reparametrisation changes the coefficient by $di(v)$ for a local holomorphic vector field $v:Y_{ij}\to T_Y|_{Y_{ij}}$, and $q(di(v))=0$. Hence the normal sections agree on overlaps and glue to $s_{\mathcal Y}\in H^0(Y,N_{Y/X})$.[/guided]

[step:Construct the closed subspace from local first-order normal equations]Conversely, let \begin{align*} s\in H^0(Y,N_{Y/X}) \end{align*} be a holomorphic section. In the adapted chart over $Y_i$, write \begin{align*} s|_{Y_i}=\sum_{a=1}^r f_{i,a}\,\overline{\partial_{w_{i,a}}} \end{align*} for holomorphic functions $f_{i,a}:Y_i\to\mathbb C$. Because $Y_i$ is a closed complex submanifold of the adapted coordinate chart, the restriction morphism $\mathcal O_{U_i}\to\mathcal O_{Y_i}$ is locally surjective. After shrinking $U_i$ if necessary, choose holomorphic extensions \begin{align*} \widetilde f_{i,a}:U_i\to\mathbb C \end{align*} of $f_{i,a}$. Define a closed complex subspace $\mathcal Y_i\subset U_i\times D$ by the ideal generated by \begin{align*} w_{i,a}-\varepsilon\widetilde f_{i,a} \end{align*} for $1\le a\le r$. Its coordinate ring is locally isomorphic to $\mathcal O_{Y_i}\otimes_{\mathbb C}\mathbb C[\varepsilon]/(\varepsilon^2)$ by restriction to the $z_i$-coordinates, so $\mathcal Y_i$ is flat over $D$ and has central fibre $Y_i$. On $Y_i\cap Y_j$, expand the adapted transition in the normal variables: \begin{align*} W_{ji,a}(z,w)=\sum_{c=1}^r (A_{ji}(z))_{ac}w_c+R_{ji,a}(z,w), \end{align*} where each holomorphic remainder $R_{ji,a}$ lies in the square of the ideal generated by $w_1,\dots,w_r$. Substituting the equations $w_c=\varepsilon\widetilde f_{i,c}$ gives $R_{ji,a}(z,\varepsilon\widetilde f_i)=0$ because every term of $R_{ji,a}$ has at least two normal factors and $\varepsilon^2=0$. Hence the transformed normal equations are \begin{align*} w_{j,a}=\varepsilon\sum_{c=1}^r(A_{ji}(z))_{ac}f_{i,c}. \end{align*} The $j$-chart tangent coordinates also change from $z_i$ to $z_j=Z_{ji}(z_i,0)+O(\varepsilon)$. Evaluating the holomorphic coefficient $f_{j,a}$ at this first-order shifted tangent coordinate changes $\varepsilon f_{j,a}$ only by a term divisible by $\varepsilon^2$, so it has no effect over $D$. Since $s_j=A_{ji}s_i$, these are exactly the equations $w_{j,a}=\varepsilon f_{j,a}$ defining $\mathcal Y_j$. Changing the extensions $\widetilde f_{i,a}$ changes each generator by a function vanishing on $Y_i$, hence by an element of the ideal generated by the $w_{i,a}$; after substituting $w_{i,a}=\varepsilon\widetilde f_{i,a}$, the resulting difference is killed by $\varepsilon^2=0$. Thus the local subspaces $\mathcal Y_i$ glue to a global closed complex subspace \begin{align*} \mathcal Y_s\subset X\times D. \end{align*} This subspace is flat over $D$, has central fibre $Y$, and depends only on the global normal section $s$.[/step]

[guided]The construction should produce an embedded subspace, not merely parametrized maps. In a chart where $Y$ is cut out by $w_{i,1}=\cdots=w_{i,r}=0$, a normal section has components $f_{i,a}$ in the normal frame $\overline{\partial_{w_{i,a}}}$. The restriction morphism $\mathcal O_{U_i}\to\mathcal O_{Y_i}$ is locally surjective because $Y_i$ is a closed complex submanifold in the adapted coordinate chart. After shrinking $U_i$, we therefore extend these functions from $Y_i$ to holomorphic functions $\widetilde f_{i,a}:U_i\to\mathbb C$ and impose the first-order equations \begin{align*} w_{i,a}=\varepsilon\widetilde f_{i,a}. \end{align*} Equivalently, $\mathcal Y_i$ is defined by the ideal generated by $w_{i,a}-\varepsilon\widetilde f_{i,a}$ for $1\le a\le r$. This local model is flat over $D$ because its structure sheaf is obtained by freely adjoining the dual-number parameter to the structure sheaf of $Y_i$ in the $z_i$-coordinates. Its central fibre is obtained by setting $\varepsilon=0$, which recovers the equations $w_{i,a}=0$. It remains to see that the local equations agree on overlaps. The adapted transition has the normal expansion \begin{align*} W_{ji,a}(z,w)=\sum_{c=1}^r(A_{ji}(z))_{ac}w_c+R_{ji,a}(z,w), \end{align*} where every term of $R_{ji,a}$ contains at least two factors among the normal coordinates $w_1,\dots,w_r$. After imposing $w_c=\varepsilon\widetilde f_{i,c}$, the remainder becomes zero because it is divisible by $\varepsilon^2$. Thus the transformed equations are $w_{j,a}=\varepsilon\sum_c(A_{ji})_{ac}f_{i,c}$. The tangent coordinates in the $j$-chart differ from their central values by $O(\varepsilon)$. Since the right-hand side of the normal equation is already multiplied by $\varepsilon$, replacing the shifted tangent coordinate by its central value changes the equation only by an $O(\varepsilon^2)$ term, which is zero over $D$. Since $s$ is a global section, the components satisfy $f_j=A_{ji}f_i$, so these are exactly the $j$-chart equations. If a different extension of $f_{i,a}$ is chosen, the difference vanishes on $Y_i$, so it is a combination of the normal coordinates $w_{i,c}$; multiplying by $\varepsilon$ and then using $w_{i,c}=\varepsilon\widetilde f_{i,c}$ makes the change zero because $\varepsilon^2=0$. Hence the local ideals glue to a well-defined closed subspace $\mathcal Y_s\subset X\times D$.[/guided]

[step:Verify that the two constructions are inverse] Start with an embedded first-order deformation $\mathcal Y$. The local graph presentations give normal coefficients $f_{i,a}$, hence a global section $s_{\mathcal Y}\in H^0(Y,N_{Y/X})$. Constructing $\mathcal Y_{s_{\mathcal Y}}$ from these coefficients gives local equations \begin{align*} w_{i,a}=\varepsilon f_{i,a} \end{align*} which are exactly the local equations of $\mathcal Y$ in adapted coordinates. Hence $\mathcal Y_{s_{\mathcal Y}}=\mathcal Y$ as a closed subspace of $X\times D$. Conversely, start with $s\in H^0(Y,N_{Y/X})$. The construction produces local equations $w_{i,a}=\varepsilon f_{i,a}$. Extracting the normal coefficient from these equations returns precisely the local components $f_{i,a}$ of $s$, so $s_{\mathcal Y_s}=s$. A first-order reparametrisation of a parametrized graph is locally induced by a holomorphic derivation $v:Y_i\to T_Y|_{Y_i}$ and changes the first-order coefficient in $i^*T_X$ by $di(v)$; since $q\circ di=0$, it leaves the local equations and the normal section unchanged. Therefore the constructions are inverse and give a natural vector-space isomorphism \begin{align*} \operatorname{Def}^{\mathrm{emb}}_{Y/X}(D)\cong H^0(Y,N_{Y/X}). \end{align*} [/step]