Examples of complex manifolds satisfying the Kodaira-Spencer condition for unobstructed deformations The theorem of existence proved by Kodaira and Spencer shows that a complex manifold $M$ for which the second cohomology group $H^2(M, \Theta T^{1,0} M)$ is trivial, $\Theta T^{1,0} M$ being the sheaf of germs of holomorphic vector fields over $M$, has unobstructed deformations. In the hope that this result isn't effectively vacuous, what are some examples of complex manifolds satisfying this condition?
 A: For $M = \mathbb{CP}^1$, note that $T^{1,0}M = \mathcal{O}(2)$ so $H^2(M, \Theta T^{1,0}M) = H^2(\mathbb{CP}^1, \mathcal{O}(2))$ which is zero by the Kodaira Vanishing Theorem ($\mathcal{O}(k)$ is a positive line bundle for $k > 0$). Again by the Kodaira Vanishing Theorem, $H^1(M, \Theta T^{1,0}M) = H^1(\mathbb{CP}^1, \mathcal{O}(2)) = 0$ so $\mathbb{CP}^1$ is in fact rigid.
More generally, on $\mathbb{CP}^n$ we have the Euler sequence $0 \to \mathcal{O} \to \mathcal{O}(1)^{n+1} \to T^{1,0}\mathbb{CP}^n \to 0$. For $k \geq 1$, $H^k(\mathbb{CP}^n, \mathcal{O}(1)^{n+1}) = H^k(\mathbb{CP}^n, \mathcal{O}(1))^{n+1} = 0$ again by the Kodaira Vanishing Theorem, so from the long exact sequence in cohomology we see that for $k \geq 1$, 
$$H^k(\mathbb{CP}^n,\Theta T^{1,0}M) \cong H^{k+1}(\mathbb{CP}^n, \mathcal{O}) \cong H^{0,k+1}_{\bar{\partial}}(\mathbb{CP}^n) = 0.$$
In particular, $\mathbb{CP}^n$ has unobstructed deformations (and as before, is rigid).
For a more general class of examples, note that $(\Theta T^{1,0}M)^* \cong \Omega^{1,0}M$, so by Serre duality we have
$$H^2(M, \Theta T^{1,0}M) \cong H^{n-2}(M, \Omega^{1,0}M\otimes K_M)^*.$$
If $K_M$ is trivial, then this becomes
$$H^{n-2}(M, \Omega^{1,0}M)^* \cong H^{1,n-2}_{\bar{\partial}}(M)^*$$
via the Dolbeault isomorphism. So any compact complex manifold $M$ with $K_M$ trivial and $h^{1,n-2}(M) = 0$ will have unobstructed deformations. 
An example of a compact complex manifold satisfying the conditions above is a $K3$ surface. One way to obtain a $K3$ surface is as a smooth quartic hypersurface in $\mathbb{CP}^3$. More generally, a smooth hypersurface of degree $n+2$ in $\mathbb{CP}^{n+1}$ (which is a compact complex manifold of dimension $n$) has trivial canonical bundle, and if $n$ is even, $h^{1,n-2} = 0$ (because $b_{n-1} = 0$ by the Lefschetz Hyperplane Theorem). If $n$ is odd, then $b_{n-1} = 1$ so there is exactly one Hodge number which is non-zero. But we know which Hodge number this is: if $n = 2m+1$, then $1 = b_{n-1} = b_{2m} = h^{m,m}$. If $m \neq 1$ (i.e. $n \neq 3$), then $(m, m) \neq (1, n - 2)$ so $h^{1,n-2} = 0$. 
So, in conclusion, a smooth degree $n+2$ hypersurface in $\mathbb{CP}^{n+1}$ is an $n$-dimensional complex manifold with trivial canonical bundle, and if $n \neq 3$, it satisfies $h^{1,n-2} = 0$ and therefore has unobstructed deformations.
Tian and Todorov proved that if $M$ is a compact Kähler manifold with trivial canonical bundle (i.e. a Calabi-Yau manifold), then it has unobstructed deformations. That is, even if $H^2(M, \Theta T^{1,0}M) \cong H^{1,n-2}_{\bar{\partial}}(M)^* \neq 0$, the map $H^1(M, \Theta T^{1,0}M) \to H^2(M, \Theta T^{1,0}M)$ is the zero map. In particular, this applies to the case $n = 3$ in the previous paragraph.
All of the examples above are Kähler. A non-Kähler example with $H^2(M, \Theta T^{1,0}M) = 0$ is given by a secondary Kodaira surface (a compact complex surface which has trivial canonical bundle and $h^{1,0} = 0$, but is not Kähler as $b_1 = 1$).
