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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 in general $H^1(X_0, T_{X_0})=0$ only ensures that there are no infinitesimal deformations (i.e. deformations over Artin rings), but the projectivity assumption ensures 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: 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 it follows that such surfaces are strongly 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 $T'$.

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: 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 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 $T'$.

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 in general $H^1(X_0, T_{X_0})=0$ only ensures that there are no infinitesimal deformations (i.e. deformations over Artin rings), but the projectivity assumption ensures 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: 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 it follows that such surfaces are strongly 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 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.

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 in general $H^1(X_0, T_{X_0})=0$ only ensures that there are no infinitesimal deformations (i.e. deformations over Artin rings), but the projectivity assumption ensures 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: 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 it follows that such surfaces are strongly 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 $T'$.

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 in general $H^1(X_0, T_{X_0})=0$ only ensures that there are no infinitesimal deformations (i.e. deformations over Artin rings), but the projectivity assumption ensures 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: 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 linehave no moduli, by general results of Catanese it follows that such surfaces are strongly rigid, and in fact their moduli space is given again by a finite collection of reduced points.

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 in general $H^1(X_0, T_{X_0})=0$ only ensures that there are no infinitesimal deformations (i.e. deformations over Artin rings), but the projectivity assumption ensures 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: 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 it follows that such surfaces are strongly 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 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.