Let $X$ be a compact complex manifold of complex dimension $n$, let $T_X$ denote its holomorphic tangent sheaf, let $T^{1,0}X$ and $T^{0,1}X$ denote the smooth complex subbundles of $TX\otimes_{\mathbb R}\mathbb C$ determined by the complex structure, and let $A^{0,0}(X)$ denote the [vector space](/page/Vector%20Space) of smooth complex-valued functions on $X$. For $q\ge 0$, 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 deformation of the complex structure of $X$ means a deformation over $\operatorname{Spec}\mathbb C[\varepsilon]/(\varepsilon^2)$ whose deformed antiholomorphic tangent bundle is locally the graph of a smooth bundle map $T^{0,1}X\to T^{1,0}X$ modulo $\varepsilon^2$. An infinitesimal change of coordinates means the first-order pullback action induced by a smooth vector field $\xi\in A^0(X,T_X)$. The Kodaira-Spencer map induces a natural isomorphism from first-order deformations of the complex structure of $X$, modulo infinitesimal changes of coordinates, onto

\begin{align*} H^1(X,T_X). \end{align*}

Equivalently, in the Beltrami differential model, first-order variations are represented by elements $\varphi\in A^{0,1}(X,T_X)$; first-order integrability is the condition $\bar\partial\varphi=0$; infinitesimal coordinate changes identify $\varphi$ with $\varphi+\bar\partial\xi$ for $\xi\in A^0(X,T_X)$; and the resulting quotient is naturally

\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)} \cong H^1(X,T_X). \end{align*}