[proofplan] We work with the Douady germ $D_{Y/X}$ at $[Y]$, which represents local embedded deformations of the compact complex submanifold $Y\subset X$. In adapted coordinates, lifting a deformation across a small extension of Artinian local $\mathbb C$-algebras is locally possible, and the only possible failure to glue is a Čech $1$-cocycle with values in the normal bundle $N_{Y/X}$. The vanishing of $H^1(Y,N_{Y/X})$ makes every such obstruction cocycle a coboundary, so the local liftings can be corrected across every small extension. The Kodaira-Spencer-Douady existence theorem for compact complex subspaces identifies this lifting property with smoothness of the representing Douady germ, and [citetheorem:9122] identifies the tangent space with $H^0(Y,N_{Y/X})$. [/proofplan]

[step:Choose adapted holomorphic charts and describe embedded deformations locally]Let $\{U_a\}_{a\in A}$ be a finite [open cover](/page/Open%20Cover) of $Y$ by relatively compact coordinate neighbourhoods in $X$ such that, for each $a\in A$, there is a holomorphic coordinate chart \begin{align*} \varphi_a:U_a\to V_a\subset\mathbb C^n \end{align*} with coordinates \begin{align*} (z_{a,1},\dots,z_{a,r},w_{a,1},\dots,w_{a,q}), \end{align*} where $r=\dim_{\mathbb C}Y$, $q=\operatorname{rank}_{\mathbb C}N_{Y/X}$, and \begin{align*} Y\cap U_a=\{p\in U_a:w_{a,1}(p)=\cdots=w_{a,q}(p)=0\}. \end{align*} This is the local normal form for embedded complex submanifolds. On $Y\cap U_a$, the classes of the coordinate vector fields \begin{align*} \left[\frac{\partial}{\partial w_{a,1}}\right],\dots,\left[\frac{\partial}{\partial w_{a,q}}\right] \end{align*} form a holomorphic frame of $N_{Y/X}|_{Y\cap U_a}$. Thus an infinitesimal embedded displacement over $Y\cap U_a$ is represented by a holomorphic map \begin{align*} s_a:Y\cap U_a&\to N_{Y/X}|_{Y\cap U_a} \end{align*} or, in this frame, by $q$ holomorphic functions on $Y\cap U_a$. For an Artinian local $\mathbb C$-algebra $R$ with maximal ideal $\mathfrak m_R$, an embedded deformation of $Y$ over $\operatorname{Spec}R$ means a closed complex subspace $\mathcal Y_R\subset X\times\operatorname{Spec}R$ that is flat over $\operatorname{Spec}R$ and whose special fibre over $R/\mathfrak m_R\cong\mathbb C$ is $Y$. The local Hilbert-Douady description of a flat deformation of a smooth graph says that, after possibly shrinking $U_a$, the projection of $\mathcal Y_R\cap(U_a\times\operatorname{Spec}R)$ to $(Y\cap U_a)\times\operatorname{Spec}R$ is an isomorphism reducing to the identity on the special fibre. Hence in the adapted chart $U_a$ such a deformation is locally given by equations \begin{align*} w_{a,\mu}=F_{a,\mu}(z_a) \end{align*} for $1\le \mu\le q$, where each $F_{a,\mu}:Y\cap U_a\to\mathfrak m_R$ is holomorphic. Compatibility on overlaps means that the closed subspaces of $(U_a\cap U_b)\times\operatorname{Spec}R$ defined by the two systems of graph equations agree after applying the coordinate transition maps of $X$.[/step]

[guided]The point of choosing adapted coordinates is to make the normal direction visible. Since $Y$ is an embedded complex submanifold, the local normal form theorem gives coordinates \begin{align*} (z_{a,1},\dots,z_{a,r},w_{a,1},\dots,w_{a,q}) \end{align*} on $U_a$ in which $Y$ is cut out by the $q$ equations $w_{a,\mu}=0$. The $z$-coordinates run along $Y$, while the $w$-coordinates measure displacement normal to $Y$. The holomorphic normal bundle is \begin{align*} N_{Y/X}=i^*T_X/T_Y. \end{align*} In the adapted chart, the tangent bundle $T_Y$ is spanned by the coordinate vector fields $\partial/\partial z_{a,j}$ along $Y$, while $i^*T_X$ is spanned by the $\partial/\partial z_{a,j}$ and $\partial/\partial w_{a,\mu}$. Therefore the quotient is locally spanned by the classes \begin{align*} \left[\frac{\partial}{\partial w_{a,1}}\right],\dots,\left[\frac{\partial}{\partial w_{a,q}}\right]. \end{align*} Thus a first-order normal displacement over $Y\cap U_a$ is exactly a holomorphic section of $N_{Y/X}$ on that set. For a general infinitesimal base $\operatorname{Spec}R$, where $R$ is an Artinian local $\mathbb C$-algebra with maximal ideal $\mathfrak m_R$, the deformation must reduce to the original submanifold modulo $\mathfrak m_R$. Hence the local graphing functions take values in $\mathfrak m_R$. In the adapted chart the deformation is written as \begin{align*} w_{a,\mu}=F_{a,\mu}(z_a), \end{align*} with $F_{a,\mu}$ holomorphic and $\mathfrak m_R$-valued. The problem is not local existence of such equations; locally graphs can always be written. The real issue is whether the local graphs define the same deformed submanifold on overlaps.[/guided]

[step:Identify first-order embedded deformations with sections of the normal bundle] Take $R=\mathbb C[\varepsilon]/(\varepsilon^2)$ and let $\mathfrak m_R=(\varepsilon)$. In the adapted chart $U_a$, every first-order deformation is locally of the form \begin{align*} w_{a,\mu}=\varepsilon f_{a,\mu}(z_a) \end{align*} for holomorphic functions $f_{a,\mu}:Y\cap U_a\to\mathbb C$. The tuple $(f_{a,1},\dots,f_{a,q})$ defines a local section $s_a\in H^0(Y\cap U_a,N_{Y/X})$. On overlaps $Y\cap U_a\cap U_b$, the coordinate transition law for normal vectors is precisely the transition law of $N_{Y/X}$, because the linear part in the [normal coordinates](/theorems/2713) is the induced map on the quotient $i^*T_X/T_Y$. Therefore the first-order local deformations glue if and only if the local sections $s_a$ glue to a global section of $N_{Y/X}$. By [citetheorem:9122], embedded first-order deformations of $Y$ in $X$, modulo first-order reparametrisations of $Y$, are naturally identified with \begin{align*} H^0(Y,N_{Y/X}). \end{align*} Thus the Zariski tangent space of the local embedded deformation space at $Y$ is $H^0(Y,N_{Y/X})$. [/step]

[step:Construct the obstruction cocycle for a small extension]Let \begin{align*} 0\to I\to R'\to R\to 0 \end{align*} be a small extension of Artinian local $\mathbb C$-algebras, so $I\mathfrak m_{R'}=0$. Suppose an embedded deformation over $\operatorname{Spec}R$ has been constructed. Choose, on each $U_a$, holomorphic lifts \begin{align*} \widetilde F_{a,\mu}:Y\cap U_a\to\mathfrak m_{R'} \end{align*} of the local graphing functions $F_{a,\mu}:Y\cap U_a\to\mathfrak m_R$. On an overlap $Y\cap U_a\cap U_b$, let $G_{ab}$ denote the holomorphic coordinate transition map sending $a$-coordinates to $b$-coordinates on $U_a\cap U_b$. Thus $G_{ab}$ takes a point written as $(z_a,w_a)$ to its $b$-coordinate expression $(z_b,w_b)$. Because the two original graphs agree over $R$, the lifted graph from $U_a$, after applying $G_{ab}$, and the lifted graph from $U_b$ agree modulo $I$. Their discrepancy therefore has values in $I$. Since $I\mathfrak m_{R'}=0$, all nonlinear terms in the normal displacement vanish in the discrepancy, and the remaining linear normal part is exactly the transition action of the normal bundle $N_{Y/X}$. Written in the normal frame of $N_{Y/X}|_{Y\cap U_a\cap U_b}$ determined by the $b$-coordinates, this discrepancy defines a holomorphic section \begin{align*} \gamma_{ab}\in H^0(Y\cap U_a\cap U_b,N_{Y/X}\otimes_{\mathbb C} I). \end{align*} The family $\gamma=(\gamma_{ab})$ is a Čech $1$-cochain with values in $N_{Y/X}\otimes_{\mathbb C} I$. If the chosen lift on $U_a$ is changed by a local section $\eta_a\in H^0(Y\cap U_a,N_{Y/X}\otimes_{\mathbb C}I)$, then the overlap discrepancy changes by the Čech coboundary $\eta_b-\eta_a$. Thus the cohomology class of $\gamma$ is independent of the arbitrary local lifts. On a triple overlap $Y\cap U_a\cap U_b\cap U_c$, the identity $G_{ca}\circ G_{bc}\circ G_{ab}=\operatorname{id}$ for coordinate changes implies \begin{align*} \gamma_{ab}+\gamma_{bc}+\gamma_{ca}=0. \end{align*} Thus $\gamma$ is a Čech $1$-cocycle for the cover $\{Y\cap U_a\}_{a\in A}$ with coefficients in $N_{Y/X}\otimes_{\mathbb C} I$. Its cohomology class lies in \begin{align*} H^1(Y,N_{Y/X}\otimes_{\mathbb C} I)\cong H^1(Y,N_{Y/X})\otimes_{\mathbb C} I \end{align*} and is the obstruction to lifting the deformation from $R$ to $R'$. More precisely, the class vanishes exactly when there are local sections $\eta_a\in H^0(Y\cap U_a,N_{Y/X}\otimes_{\mathbb C}I)$ whose coboundary equals $\gamma$; subtracting these sections from the local lifted graphing functions makes the lifted equations agree on all overlaps.[/step]

[guided]We now make the obstruction construction explicit. Start with a small extension \begin{align*} 0\to I\to R'\to R\to 0 \end{align*} of Artinian local $\mathbb C$-algebras, with $I\mathfrak m_{R'}=0$. An embedded deformation over $\operatorname{Spec}R$ is already given, and on each adapted chart $U_a$ it is represented by graphing functions $F_{a,\mu}:Y\cap U_a\to\mathfrak m_R$. Choose arbitrary holomorphic lifts $\widetilde F_{a,\mu}:Y\cap U_a\to\mathfrak m_{R'}$ of these functions. On $Y\cap U_a\cap U_b$, let $G_{ab}$ be the holomorphic coordinate transition map sending $a$-coordinates to $b$-coordinates. Compare the lifted graph in the $a$-coordinates, after applying $G_{ab}$, with the lifted graph in the $b$-coordinates. Because the original graphs agree after reduction to $R$, their difference over $R'$ reduces to zero in $R$, so it has coefficients in $I$. The small-extension condition $I\mathfrak m_{R'}=0$ is what removes all products involving this discrepancy and any infinitesimal coordinate correction. Therefore, when the transition map is expanded along $Y$, only the linear normal part contributes to the discrepancy. The linear normal part of the coordinate transition is precisely the transition function of the quotient bundle $N_{Y/X}=i^*T_X/T_Y$: tangential terms lie in $T_Y$ and vanish after passing to the quotient, while the normal coordinates transform by the induced map on the quotient. Thus the overlap discrepancy is not just a tuple of functions depending on coordinates; it is a well-defined holomorphic section \begin{align*} \gamma_{ab}\in H^0(Y\cap U_a\cap U_b,N_{Y/X}\otimes_{\mathbb C} I). \end{align*} The family $\gamma=(\gamma_{ab})$ is a Čech $1$-cochain with coefficients in $N_{Y/X}\otimes_{\mathbb C}I$. On a triple overlap $Y\cap U_a\cap U_b\cap U_c$, the transition maps satisfy $G_{ca}\circ G_{bc}\circ G_{ab}=\operatorname{id}$. Taking the same linear normal part of this identity gives \begin{align*} \gamma_{ab}+\gamma_{bc}+\gamma_{ca}=0. \end{align*} Hence $\gamma$ is a Čech $1$-cocycle. Its class lies in \begin{align*} H^1(Y,N_{Y/X}\otimes_{\mathbb C} I)\cong H^1(Y,N_{Y/X})\otimes_{\mathbb C} I, \end{align*} and this is the obstruction class: it vanishes exactly when the chosen local lifts can be corrected by local normal sections so that they agree on overlaps.[/guided]

[step:Use the vanishing of $H^1(Y,N_{Y/X})$ to correct the local liftings]Refine the adapted coordinate cover, if necessary, to a finite Stein Leray cover of the compact complex manifold $Y$, meaning that every nonempty finite intersection of members of the cover is Stein; Čech cohomology on this cover computes sheaf cohomology for the coherent sheaf $N_{Y/X}\otimes_{\mathbb C}I$. By hypothesis, \begin{align*} H^1(Y,N_{Y/X})=0, \end{align*} so \begin{align*} H^1(Y,N_{Y/X}\otimes_{\mathbb C}I)=0. \end{align*} Since $\gamma$ is a Čech $1$-cocycle with values in $N_{Y/X}\otimes_{\mathbb C}I$, its cohomology class vanishes. Therefore there are holomorphic sections \begin{align*} \eta_a\in H^0(Y\cap U_a,N_{Y/X}\otimes_{\mathbb C}I) \end{align*} such that, on every overlap $Y\cap U_a\cap U_b$, \begin{align*} \gamma_{ab}=\eta_b-\eta_a. \end{align*} Modify the chosen lift on $U_a$ by subtracting $\eta_a$ in the normal direction. In coordinates, this means replacing the lifted graphing functions $\widetilde F_{a,\mu}$ by \begin{align*} \widetilde F'_{a,\mu}=\widetilde F_{a,\mu}-\eta_{a,\mu}, \end{align*} where $\eta_{a,\mu}:Y\cap U_a\to I$ is the $\mu$-th component of $\eta_a$ in the adapted normal frame. This correction does not change the deformation after reduction to $R$, and on overlaps the new discrepancy is \begin{align*} \gamma_{ab}-(\eta_b-\eta_a)=0. \end{align*} Hence the corrected local liftings glue to an embedded deformation over $\operatorname{Spec}R'$.[/step]

[guided]The obstruction cocycle measures the mismatch of arbitrary local lifts across a small extension $0\to I\to R'\to R\to 0$. Since the deformation has already been constructed over $R$, any mismatch between two lifts has values in $I$. The condition $I\mathfrak m_{R'}=0$ kills the higher-order terms in the coordinate transition formula, so the mismatch transforms by the linear normal transition functions. Therefore it is a section of $N_{Y/X}\otimes_{\mathbb C}I$ on the overlap. The equation \begin{align*} H^1(Y,N_{Y/X})=0 \end{align*} implies $H^1(Y,N_{Y/X}\otimes_{\mathbb C}I)=0$, because $I$ is a finite-dimensional $\mathbb C$-[vector space](/page/Vector%20Space). After refining to a finite Stein Leray cover, so that all nonempty finite intersections are Stein, Čech cohomology computes this sheaf cohomology group. Hence the cocycle $\gamma=(\gamma_{ab})$ can be written as \begin{align*} \gamma_{ab}=\eta_b-\eta_a \end{align*} for local holomorphic sections $\eta_a$ of $N_{Y/X}\otimes_{\mathbb C}I$. Why does subtracting $\eta_a$ fix the problem? The correction is invisible after reduction to $R$, so it preserves the deformation already constructed over $R$. But over $R'$, the discrepancy changes by the coboundary of $\eta=(\eta_a)$. Therefore the corrected discrepancy on $Y\cap U_a\cap U_b$ is \begin{align*} \gamma_{ab}-(\eta_b-\eta_a)=0. \end{align*} Thus the corrected local equations agree on every overlap, and they glue to a global embedded deformation over $\operatorname{Spec}R'$.[/guided]

[step:Apply the Kodaira-Spencer-Douady criterion to obtain smoothness]The preceding two steps prove the lifting property for every small extension \begin{align*} 0\to I\to R'\to R\to 0 \end{align*} of Artinian local $\mathbb C$-algebras: every embedded deformation over $\operatorname{Spec}R$ lifts to one over $\operatorname{Spec}R'$. The Kodaira-Spencer-Douady smoothness criterion applies because $Y$ is compact and embedded as a closed complex submanifold of the complex manifold $X$; in this setting, the Douady germ $D_{Y/X}$ represents the local embedded deformation functor, the obstruction to lifting across a small extension lies in $H^1(Y,N_{Y/X})\otimes_{\mathbb C} I$, and if these obstruction classes vanish for every small extension then the representing germ is smooth at $[Y]$. Since $H^1(Y,N_{Y/X})=0$, all obstruction classes vanish for every finite-dimensional ideal $I$. Therefore $D_{Y/X}$ is smooth at the point $[Y]$ corresponding to $Y$.[/step]

[guided]We have proved the exact infinitesimal condition needed for smoothness of a deformation functor: lifting across every small extension of Artinian local $\mathbb C$-algebras. A small extension has the form $0\to I\to R'\to R\to 0$ with $I\mathfrak m_{R'}=0$, and the previous steps showed that the only obstruction to lifting an embedded deformation from $R$ to $R'$ is a class in $H^1(Y,N_{Y/X}\otimes_{\mathbb C}I)$. Because $H^1(Y,N_{Y/X})=0$ and $I$ is finite-dimensional over $\mathbb C$, this obstruction group is zero. Now we invoke the Kodaira-Spencer-Douady theorem for compact complex subspaces. Its hypotheses are satisfied: $X$ is a complex manifold, $Y\subset X$ is compact, and we are considering the Douady germ at the point $[Y]$ representing $Y$ as a compact complex subspace of $X$. The theorem provides a local analytic space $D_{Y/X}$ representing embedded deformations near $Y$, identifies the obstruction to a small-extension lifting problem with a class in $H^1(Y,N_{Y/X})\otimes_{\mathbb C}I$, and says that the germ is smooth at $[Y]$ when every such small-extension obstruction vanishes. This last condition is the analytic formal-smoothness criterion for the local ring of the germ: lifting over every small extension of Artinian local $\mathbb C$-algebras is equivalent to regularity of the completed local ring, hence to smoothness of the complex-space germ at $[Y]$. Since the obstruction group is zero for every finite-dimensional ideal $I$, every such obstruction vanishes. Hence $D_{Y/X}$ is smooth at $[Y]$.[/guided]

[step:Conclude smoothness and identify the tangent space]The previous step proves that the Douady germ $D_{Y/X}$ is smooth at $[Y]$. The Zariski tangent space at $[Y]$ is the space of embedded first-order deformations over $\operatorname{Spec}\mathbb C[\varepsilon]/(\varepsilon^2)$ modulo first-order reparametrisations of $Y$. By [citetheorem:9122], this tangent space is naturally \begin{align*} H^0(Y,N_{Y/X}). \end{align*} This proves the stated deformation criterion.[/step]

[guided]The smoothness statement has already been obtained from the small-extension [lifting criterion](/theorems/1897). It remains only to identify the tangent space at the point $[Y]$ of the Douady germ. For a local analytic deformation space, the Zariski tangent space at the base point is represented by deformations over the dual numbers $\mathbb C[\varepsilon]/(\varepsilon^2)$. In the embedded setting, such a first-order deformation is taken modulo infinitesimal reparametrisations of $Y$, because the Douady space records the embedded subspace of $X$ rather than a choice of parametrization of that subspace. The [infinitesimal embedded deformation theorem](/theorems/9122) [citetheorem:9122] applies to the compact complex submanifold $Y\subset X$ and identifies these first-order embedded deformations naturally with the global holomorphic sections of the normal bundle. Therefore \begin{align*} T_{[Y]}D_{Y/X}\cong H^0(Y,N_{Y/X}). \end{align*} Together with the smoothness of $D_{Y/X}$ at $[Y]$, this is exactly the claimed deformation criterion.[/guided]