[proofplan] We identify the local deformation space with the Kuranishi germ of $X$. Kuranishi theory gives a local analytic deformation space whose Zariski tangent space is $H^1(X,T_X)$ and whose possible singularities are governed by obstruction classes in $H^2(X,T_X)$. The [Bogomolov-Tian-Todorov theorem](/theorems/9140) says that, for compact Kähler Calabi-Yau manifolds, these obstructions vanish, so the Kuranishi germ is smooth with the stated tangent space. [/proofplan]

[step:Choose the Kuranishi germ as the local deformation space]Let $K$ denote the Kuranishi germ of the compact complex manifold $X$, and let $0\in K$ denote the point corresponding to the original complex structure on $X$. By [citetheorem:9119], there is a Kuranishi family \begin{align*} \pi:\mathcal X\to K \end{align*} whose base is an analytic germ at $0$ and whose Zariski tangent space at $0$ is naturally identified with $H^1(X,T_X)$. More precisely, the natural Kodaira-Spencer map of the Kuranishi family induces an isomorphism \begin{align*} T_0K \cong H^1(X,T_X), \end{align*} where $T_0K$ denotes the Zariski tangent space of the analytic germ $K$ at $0$. This identifies the tangent space to the local deformation space with the first cohomology of the holomorphic tangent bundle.[/step]

[guided]We first make precise what is meant by the local deformation space. Let $K$ be the Kuranishi germ of $X$, and let $0\in K$ be the point representing the given complex structure on $X$. The Kuranishi family is a holomorphic family \begin{align*} \pi:\mathcal X\to K \end{align*} whose fibre over $0$ is $X$. By [citetheorem:9119], this germ has Zariski tangent space naturally identified with $H^1(X,T_X)$. The relevant map is the Kodaira-Spencer map of the family, which sends a first-order tangent vector to the corresponding first-order deformation class of the complex structure. Thus \begin{align*} T_0K \cong H^1(X,T_X). \end{align*} This is the tangent-space part of the theorem. It does not yet prove smoothness: a germ can have the correct tangent space and still be singular. Smoothness requires the vanishing of the obstruction equations defining the Kuranishi germ.[/guided]

[step:Apply the Bogomolov-Tian-Todorov unobstructedness theorem]The formalized statement is a preview consequence of the full Bogomolov-Tian-Todorov theorem, not a proof of that theorem itself. Since $X$ is compact Kähler and has holomorphically trivial canonical bundle, the hypotheses of [citetheorem:9140] are satisfied. Therefore the Kuranishi space of complex structures on $X$ is smooth at the point corresponding to $X$, with tangent space $H^1(X,T_X)$. Equivalently, every first-order deformation class in $H^1(X,T_X)$ extends to an actual analytic deformation. Hence the Kuranishi germ $K$ is smooth at $0$.[/step]

[guided]This step uses the full Bogomolov-Tian-Todorov theorem as an external input. The hypotheses required by [citetheorem:9140] are exactly that the manifold be compact, Kähler, and Calabi-Yau in the sense that its canonical bundle is holomorphically trivial. These hypotheses hold for $X$ by the formalized statement. The conclusion of [citetheorem:9140] is that the Kuranishi space of complex structures on $X$ is smooth at the point corresponding to $X$, and that its tangent space is $H^1(X,T_X)$. Applied to the Kuranishi germ $K$ with base point $0$, this gives that $K$ is smooth at $0$. In deformation-theoretic language, the obstruction equations governed by the obstruction map from Kuranishi theory vanish for compact Kähler Calabi-Yau manifolds, so every class in $H^1(X,T_X)$ is represented by an actual local analytic deformation rather than only by a first-order deformation.[/guided]

[step:Conclude the stated form of the local deformation theorem] Combining the Kuranishi identification \begin{align*} T_0K \cong H^1(X,T_X) \end{align*} with the smoothness of $K$ from [citetheorem:9140], the local deformation space of $X$ is a smooth analytic germ whose tangent space at the base point is naturally $H^1(X,T_X)$. This proves the theorem. [/step]