[proofplan] Represent a first-order deformation of the complex structure by the first-order part of the deformed Dolbeault operator on vector-valued functions. The condition that the deformed operator square to zero forces this first-order term to be a $\bar\partial$-closed $T_X$-valued $(0,1)$-form. An infinitesimal coordinate change generated by a smooth vector field changes the representative by a $\bar\partial$-exact term. Therefore the quotient of first-order integrable deformations by infinitesimal coordinate changes is the Dolbeault cohomology group $H^{0,1}_{\bar\partial}(X,T_X)$, which is identified with the sheaf cohomology group $H^1(X,T_X)$ by the Dolbeault resolution. [/proofplan]

[step:Model a first-order deformation by a Beltrami differential] Let $n$ denote the complex dimension of $X$. Let $A^{0,0}(X)$ denote the [vector space](/page/Vector%20Space) of smooth maps $X\to\mathbb C$. For each integer $q\ge 0$, let $A^{0,q}(X,T_X)$ denote the vector space of smooth sections of $\Lambda^q(T^{0,1}X)^*\otimes T^{1,0}X$. A first-order variation of the almost-complex structure is represented by a Beltrami differential $\varphi\in A^{0,1}(X,T_X)$ as follows. Over $\mathbb C[\varepsilon]/(\varepsilon^2)$, the deformed antiholomorphic tangent bundle is locally the graph of a smooth bundle map $T^{0,1}X\to T^{1,0}X$ multiplied by $\varepsilon$. Such bundle maps are precisely smooth sections of $(T^{0,1}X)^*\otimes T^{1,0}X$, and these local graph maps agree on overlaps because the deformed subbundle is globally defined. Hence the first-order change is encoded by a global element $\varphi\in A^{0,1}(X,T_X)$. Equivalently, $\varphi$ acts on smooth complex-valued functions by contracting with their $(1,0)$ differential. More precisely, define the first-order operator \begin{align*} \varphi\cdot\partial:A^{0,0}(X)\to A^{0,1}(X) \end{align*} by the local formula below. In local holomorphic coordinates $(z_1,\dots,z_n)$ on an [open set](/page/Open%20Set) $U\subset X$, write \begin{align*} \varphi=\sum_{i=1}^n\sum_{j=1}^n \varphi_{i\bar j}\,d\bar z_j\otimes \frac{\partial}{\partial z_i}, \end{align*} where each coefficient $\varphi_{i\bar j}:U\to\mathbb C$ is smooth. For $f\in A^{0,0}(X)$, the restriction of $(\varphi\cdot\partial)f$ to $U$ is \begin{align*} ((\varphi\cdot\partial)f)|_U=\sum_{i=1}^n\sum_{j=1}^n \varphi_{i\bar j}\frac{\partial f}{\partial z_i}\,d\bar z_j. \end{align*} Thus the deformed antiholomorphic operator on functions has first-order form \begin{align*} \bar\partial_\varepsilon=\bar\partial+\varepsilon(\varphi\cdot\partial), \end{align*} where $\varepsilon^2=0$. [/step]

[step:Translate first-order integrability into $\bar\partial\varphi=0$]For a Beltrami differential $\varphi\in A^{0,1}(X,T_X)$, the integrability condition is the Maurer-Cartan equation \begin{align*} \bar\partial\varphi+\frac{1}{2}[\varphi,\varphi]=0, \end{align*} where $[\varphi,\varphi]\in A^{0,2}(X,T_X)$ is the Kodaira-Spencer bracket induced by the Lie bracket of local vector fields after wedging the $(0,1)$-form components. In local holomorphic coordinates, if $\varphi=\sum_{i,j}\varphi_{i\bar j}\,d\bar z_j\otimes \partial/\partial z_i$, then the coefficient of $d\bar z_j\wedge d\bar z_k\otimes \partial/\partial z_i$ in $[\varphi,\varphi]$ is obtained by skew-symmetrising the vector-field bracket expression $\sum_a \varphi_{a\bar j}\,\partial\varphi_{i\bar k}/\partial z_a-\varphi_{a\bar k}\,\partial\varphi_{i\bar j}/\partial z_a$. In a first-order deformation over $\mathbb C[\varepsilon]/(\varepsilon^2)$, the quadratic bracket term is multiplied by $\varepsilon^2$ and therefore vanishes. The coefficient of $\varepsilon$ in the Maurer-Cartan equation is exactly \begin{align*} \bar\partial\varphi=0. \end{align*} Hence an integrable first-order variation determines an element \begin{align*} \varphi\in \ker\left(\bar\partial:A^{0,1}(X,T_X)\to A^{0,2}(X,T_X)\right). \end{align*}[/step]

[guided]A Beltrami differential does not satisfy integrability by an ordinary square calculation alone; the nonlinear condition is the Maurer-Cartan equation. For $\varphi\in A^{0,1}(X,T_X)$, this equation is \begin{align*} \bar\partial\varphi+\frac{1}{2}[\varphi,\varphi]=0, \end{align*} where the Kodaira-Spencer bracket $[\varphi,\varphi]$ is a smooth $T_X$-valued $(0,2)$-form. The first term is linear in $\varphi$, while the bracket term is quadratic in $\varphi$. In a first-order deformation the actual perturbation is $\varepsilon\varphi$ with $\varepsilon^2=0$. Substituting $\varepsilon\varphi$ into the Maurer-Cartan equation gives \begin{align*} \varepsilon\bar\partial\varphi+\frac{1}{2}\varepsilon^2[\varphi,\varphi]=0. \end{align*} The second term is zero in the dual-number algebra because $\varepsilon^2=0$. Therefore the remaining first-order condition is \begin{align*} \bar\partial\varphi=0. \end{align*} Thus first-order integrable deformations are represented precisely by $\bar\partial$-closed elements of $A^{0,1}(X,T_X)$.[/guided]

[step:Compute the effect of an infinitesimal coordinate change] Let $\xi\in A^0(X,T_X)$ be a smooth section of $T^{1,0}X$, interpreted as the infinitesimal generator of a first-order change of coordinates. The infinitesimal pullback action on the antiholomorphic operator is the commutator with the first-order derivation generated by $\xi$. In local holomorphic coordinates on $U\subset X$, write $\xi=\sum_i \xi_i\,\partial/\partial z_i$ with smooth coefficient functions $\xi_i:U\to\mathbb C$. For a local smooth function $f:U\to\mathbb C$, the commutator of $\bar\partial$ with the derivation $\xi$ is \begin{align*} [\bar\partial,\xi]f=\bar\partial(\xi f)-\xi(\bar\partial f)=\sum_{i=1}^n\sum_{j=1}^n \frac{\partial \xi_i}{\partial \bar z_j}\frac{\partial f}{\partial z_i}\,d\bar z_j. \end{align*} This is exactly the action on functions of the Beltrami differential \begin{align*} \bar\partial\xi=\sum_{i=1}^n\sum_{j=1}^n \frac{\partial \xi_i}{\partial \bar z_j}\,d\bar z_j\otimes \frac{\partial}{\partial z_i}\in A^{0,1}(X,T_X). \end{align*} With the pullback convention used in the statement, the corresponding gauge action on Beltrami representatives is \begin{align*} \varphi\longmapsto \varphi+\bar\partial\xi. \end{align*} Thus two first-order integrable representatives $\varphi_1,\varphi_2\in A^{0,1}(X,T_X)$ define equivalent deformations modulo infinitesimal coordinate changes if and only if there exists $\xi\in A^0(X,T_X)$ such that \begin{align*} \varphi_2-\varphi_1=\bar\partial\xi. \end{align*} Hence the quotient by infinitesimal coordinate changes is \begin{align*} \frac{\ker\left(\bar\partial:A^{0,1}(X,T_X)\to A^{0,2}(X,T_X)\right)} {\operatorname{im}\left(\bar\partial:A^0(X,T_X)\to A^{0,1}(X,T_X)\right)}. \end{align*} [/step]

[step:Identify the quotient with Dolbeault cohomology] By definition, the first Dolbeault cohomology group with coefficients in $T_X$ is \begin{align*} H^{0,1}_{\bar\partial}(X,T_X) = \frac{\ker\left(\bar\partial:A^{0,1}(X,T_X)\to A^{0,2}(X,T_X)\right)} {\operatorname{im}\left(\bar\partial:A^0(X,T_X)\to A^{0,1}(X,T_X)\right)}. \end{align*} The previous step identifies the set of first-order deformations modulo infinitesimal coordinate changes with this quotient. Therefore the desired deformation space is naturally \begin{align*} H^{0,1}_{\bar\partial}(X,T_X). \end{align*} [/step]

[step:Pass from Dolbeault cohomology to sheaf cohomology] Let $\mathcal A^{0,q}(T_X)$ denote the sheaf of smooth $T_X$-valued $(0,q)$-forms on $X$. The Dolbeault complex of sheaves \begin{align*} 0\longrightarrow T_X\longrightarrow \mathcal A^{0,0}(T_X)\xrightarrow{\bar\partial}\mathcal A^{0,1}(T_X)\xrightarrow{\bar\partial}\mathcal A^{0,2}(T_X)\longrightarrow\cdots \end{align*} is a resolution of the holomorphic tangent sheaf $T_X$: the degree-zero kernel consists exactly of holomorphic vector fields, and the higher local exactness is the Dolbeault lemma for vector bundles. Each sheaf $\mathcal A^{0,q}(T_X)$ is fine, because smooth partitions of unity act on smooth vector-valued forms, hence it is acyclic for sheaf cohomology. Therefore the Dolbeault theorem for holomorphic vector bundles identifies the cohomology of global sections with sheaf cohomology: \begin{align*} H^{0,1}_{\bar\partial}(X,T_X)\cong H^1(X,T_X). \end{align*} Combining this identification with the quotient computed above gives the natural isomorphism between first-order deformations of the complex structure modulo infinitesimal coordinate changes and $H^1(X,T_X)$. Compactness is not needed for the formal Dolbeault identification, but in the deformation-theoretic setting it ensures the resulting cohomology and local deformation spaces are finite-dimensional. This proves the Kodaira-Spencer correspondence. [/step]