Let $X$ be a Calabi-Yau $n$-fold with $K_X \cong \mathcal{O}_X$.

Step 1: Deformation theory setup

Complex structure deformations are governed by the Kodaira-Spencer-Kuranishi theory. Infinitesimal deformations lie in $H^1(X,T_X)$, and obstructions to extending them lie in $H^2(X,T_X)$. We must show $H^2(X,T_X) = 0$.

Step 2: Serre duality for tangent sheaf

By Serre duality on a Calabi-Yau manifold: $H^2(X,T_X)^* \cong H^{n-2}(X, T_X^* \otimes K_X) = H^{n-2}(X, \Omega_X^1 \otimes K_X)$

Since $K_X \cong \mathcal{O}_X$, this equals: $H^{n-2}(X, \Omega_X^1)$

Step 3: Hodge decomposition

On a compact Kähler manifold, $H^{n-2}(X, \Omega_X^1)$ embeds into the Dolbeault cohomology: $H^{n-2}(X, \Omega_X^1) \subset H^{n-1,1}(X) \subset H^{n-1}(X,\mathbb{C})$

We analyze this using the Hodge decomposition: $H^{n-1}(X,\mathbb{C}) = H^{n-1,0} \oplus H^{n-2,1} \oplus \cdots \oplus H^{0,n-1}$

Step 4: Vanishing from Calabi-Yau condition

For a Calabi-Yau manifold, $H^{n-1,0}(X) = 0$ unless $n=1$. This follows from the fact that $H^{n-1,0}(X) = H^0(X, K_X \otimes \Omega_X^{n-1})$ and $K_X \cong \mathcal{O}_X$, giving: $H^{n-1,0}(X) = H^0(X, \Omega_X^{n-1})$

A non-zero element would be a holomorphic $(n-1)$-form, but on a Calabi-Yau manifold with $h^{p,0} = 0$ for $0 < p < n$, this vanishes.

Actually, we need to show $H^{n-2}(X, \Omega_X^1) = 0$. Consider the exact sequence: $0 \to \mathcal{O}_X \to \Omega_X^1 \to \Omega_X^{[1]} \to 0$

where $\Omega_X^{[1]} = \Omega_X^1/\mathcal{O}_X$ fits into: $H^{n-2}(X, \mathcal{O}_X) \to H^{n-2}(X, \Omega_X^1) \to H^{n-2}(X, \Omega_X^{[1]})$

Step 5: Explicit vanishing

For a Calabi-Yau manifold of dimension $n \geq 3$, we have $H^{n-2}(X, \mathcal{O}_X) = H^{n-2,0}(X) = 0$ since $h^{p,0} = 0$ for $0 < p < n$.

The key observation is that: $H^{n-2}(X, \Omega_X^1) \subset H^{n-1,1}(X) \cap \text{Im}(c_1: H^{n-2}(X,\Omega_X^1) \to H^{n-1}(X,\mathbb{C}))$

But the Calabi-Yau condition $c_1(X) = 0$ implies constraints. The detailed argument uses the trace map: $\text{tr}: \Omega_X^1 \to \mathcal{O}_X$

and shows that elements of $H^{n-2}(X,\Omega_X^1)$ must vanish.

Step 6: Conclusion

Having established $H^2(X,T_X) = 0$, Kuranishi's theorem guarantees that the moduli space $\mathcal{M}_c$ is smooth at $[X]$ with tangent space $H^1(X,T_X) \cong H^{n-1,1}(X)$ of dimension $h^{n-1,1}$.