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Let $f:X\rightarrow Y$ be a locally trivial fibration with a variety $F$ as the fiber. Here $X, Y, F$ are smooth, projective varieties.

Does any automorphism of $F$ induce an automorphism of $X$?

In other words, does there exist an injective group homomorphism

$$Aut(F)\rightarrow Aut(X)$$

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Despite the negative answer to the question as stated, there is at least a natural injection from the center of $Aut(F)$ to $Aut(X)$. – John Pardon Mar 4 at 0:42

The answer, of course, is negative (there seems to be no reason for this to be true, moreover, there is no natural morphism from automorphisms of $F$ to those of $X$). For example, consider a simple vector bundle $E$ on a curve $C$ of genus $g > 1$. Then $P_C(E)$ (the projectivization of $E$) is a locally trivial fibration over $C$ with fiber a projective space, i.e. with an infinite group $Aut(F)$. However, $Aut(P_C(E))$ is finite.

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At least in dimension two for $\mathbb{P}^1$ bundles over a curve $C$, this only occurs if $X \simeq \mathbb{P}^1\times C$ and $C$ is a general curve without automorphisms. There is a classification of such automorphism groups due to Maruyama.

Take a look at Theorem 5.6 in http://arxiv.org/abs/1601.07945 and the references to the work of Maruyama.

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