If $X_0$ is a smooth *projective* variety over $\mathbb{C}$ such that $H^1(X_0, T_{X_0})=0$, then any deformation over a disk $D_{\epsilon}:=\{z \in \mathbb{C} | |z| < \epsilon \}$ is trivial for $\epsilon$ small enough. 

Notice that, for a general scheme,  $H^1(X_0, T_{X_0})=0$ only ensures that there are no infinitesimal deformations (i.e. deformations over Artin rings); however, since $X_0$ is projective one shows that nearby fibers in *any* flat family of projective schemes are isomorphic to $X_0$ [see Hartshorne, Deformation theory, p. 39].

For instance, the complex projective space $\mathbb{P}^n$ is globally rigid.

Another class of example is given by [Fake Projective Planes][1]: since they are $2$-dimensional ball quotients, they are globally rigid by Mostow Rigidity Theorem. In fact, their moduli space consists of $100$ distinct isolated points.

Let me also mention *Beauville surfaces*: they are surfaces $S$ of general type with $p_g=q=0$, $K_S^2=8$ (i.e. they are *fake quadrics*) obtained as $$S=(C_1 \times C_2)/G,$$
where $G$ is a finite group acting faitfully on the smooth curves $C_i$ and freely on their product, in such a way that $C_i/G \cong \mathbb{P}^1$ and  the two maps $ C_i \longrightarrow C_i/G$ are both branched at *three* points. 

Since three points on the projective line have no moduli, by [general results of Catanese][2] it follows that such surfaces are globally rigid, and in fact their moduli space is given again by a finite collection of reduced points.   

Finally, notice that the $T=D_{\epsilon}$ is probably one of the most raisonable choices for checking global rigidity. In fact, if one choose as $T$ a compact complex curve, then even $X_0=\mathbb{P}^n$ fails to work: it is well known that there exist projective bundles over compact curves that are not trivial, think of Hirzebruch surfaces.  

However, by using [Hartshorne, Deformation Theory, ex. 24.7 p. 163] one still have the following result:

Let $X_0$ be a projective scheme such that  $H^1(X_0, T_{X_0})=0$ and let $\mathfrak{X} \to T$ be a flat family of projective schemes over any nonsingular curve such that the central fibre is isomorphic to $X_0$. Then there exist a dominant base change $T' \longrightarrow T$ such that the family $\mathfrak{X}'=\mathfrak{X} \times_T T'$ is trivial over $T'$.
  

[1]: http://en.wikipedia.org/wiki/Fake_projective_plane
  [2]: http://muse.jhu.edu/login?uri=/journals/american_journal_of_mathematics/v122/122.1catanese.pdf