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By definition, a morphism $f \colon X \to Y$ between smooth projective varieties is smooth if $f$ is flat and all fibres are smooth (in the scheme-theoretical sense). In particular, $f$ is a smooth submersion when it is considered as a differentiable map between real manifolds.

Then the answer to your question is yes, assuming that the fibres are connected. This remains true also replacing $\mathbb{P}^1$ with any smooth projective variety $Y$.

In fact, by a result due to Ehresmann (1951), if $ƒ \colon M \to N$ is a surjective submersion with $M$ and $N$ differentiable manifolds such that the preimage $ƒ^{-1}(x)$ is compact and connected for all $x \in N$, then $ƒ$ is differentiably locally trivial and admits a compatible fiber bundle structure. See http://en.wikipedia.org/wiki/Fiber_bundle.

In particular, all fibres of $f$ are diffeomorphic.

Remark 1. If one only requires that the fibres of $f \colon X \to Y$ are smooth in the set-theoretical sense, and not in the scheme-theoretical one (i.e. one allows multiple fibres) then in general $f$ is not a differentiable fibre bundle, see Jason Starr's comment.

Remark 2. Even if $f \colon X \to Y$ is differentiably locally trivial, in general it is not analitycally locally trivial. In fact, by a celebrated theorem of Grauert and Fischer, this happens if and only if all the fibres of $f$ are biholomorphic.

See for instance [Barth-Peters-Van de Ven, Compact Complex Surfaces, Chapter I].

By definition, a morphism $f \colon X \to Y$ between smooth projective varieties is smooth if $f$ is flat and all fibres are smooth (in the scheme-theoretical sense). In particular, $f$ is a smooth submersion when it is considered as a differentiable map between real manifolds.

Then the answer to your question is yes, assuming that the fibres are connected. This remains true also replacing $\mathbb{P}^1$ with any smooth projective variety $Y$.

In fact, by a result due to Ehresmann (1951), if $ƒ \colon M \to N$ is a surjective submersion with $M$ and $N$ differentiable manifolds such that the preimage $ƒ^{-1}(x)$ is compact and connected for all $x \in N$, then $ƒ$ is differentiably locally trivial and admits a compatible fiber bundle structure. See http://en.wikipedia.org/wiki/Fiber_bundle.

In particular, all fibres of $f$ are diffeomorphic.

Remark 1. If one only requires that the fibres of $f \colon X \to Y$ are smooth in the set-theoretical sense, and not in the scheme-theoretical one (i.e. one allows multiple fibres) then in general $f$ is not a differentiable fibre bundle, see Jason Starr's comment.

Remark 2. Even if $f \colon X \to Y$ is differentiably locally trivial, in general it is not analitycally locally trivial. In fact, by a celebrated theorem of Grauert and Fischer, this happens if and only if all the fibres of $f$ are biholomorphic. See for instance [Barth-Peters-Van de Ven, Compact Complex Surfaces, Chapter I].

By definition, a morphism $f \colon X \to Y$ between smooth projective varieties is smooth if $f$ is flat and all fibres are smooth (in the scheme-theoretical sense). In particular, $f$ is a smooth submersion when it is considered as a differentiable map between real manifolds.

Then the answer to your question is yes, assuming that the fibres are connected. This remains true also replacing $\mathbb{P}^1$ with any smooth projective variety $Y$.

In fact, by a result due to Ehresmann (1951), if $ƒ \colon M \to N$ is a surjective submersion with $M$ and $N$ differentiable manifolds such that the preimage $ƒ^{-1}(x)$ is compact and connected for all $x \in N$, then $ƒ$ is differentiably locally trivial and admits a compatible fiber bundle structure. See http://en.wikipedia.org/wiki/Fiber_bundle.

In particular, all fibres of $f$ are diffeomorphic.

Remark 1. If one only requires that the fibres of $f \colon X \to Y$ are smooth in the set-theoretical sense, and not in the scheme-theoretical one (i.e. one allows multiple fibres) then in general $f$ is not a differentiable fibre bundle, see Jason Starr's comment.

Remark 2. Even if $f \colon X \to Y$ is differentially locally trivial, in general it is not analitycally locally trivial. In fact, by a celebrated theorem of Grauert and Fischer, this happens if and only if all the fibres of $f$ are biholomorphic.

See for instance [Barth-Peters-Van de Ven, Compact Complex Surfaces, Chapter I].

By definition, a morphism $f \colon X \to Y$ between smooth projective varieties is smooth if $f$ is flat and all fibres are smooth (in the scheme-theoretical sense). In particular, $f$ is a smooth submersion when it is considered as a differentiable map between real manifolds.

Then the answer to your question is yes, assuming that the fibres are connected. This remains true also replacing $\mathbb{P}^1$ with any smooth projective variety $Y$.

In fact, by a result due to Ehresmann (1951), if $ƒ \colon M \to N$ is a surjective submersion with $M$ and $N$ differentiable manifolds such that the preimage $ƒ^{-1}(x)$ is compact and connected for all $x \in N$, then $ƒ$ is differentiably locally trivial and admits a compatible fiber bundle structure. See http://en.wikipedia.org/wiki/Fiber_bundle.

In particular, all fibres of $f$ are diffeomorphic.

Remark 1. If one only requires that the fibres of $f \colon X \to Y$ are smooth in the set-theoretical sense, and not in the scheme-theoretical one (i.e. one allows multiple fibres) then in general $f$ is not a differentiable fibre bundle, see Jason Starr's comment.

Remark 2. Even if $f \colon X \to Y$ is differentiably locally trivial, in general it is not analitycally locally trivial. In fact, by a celebrated theorem of Grauert and Fischer, this happens if and only if all the fibres of $f$ are biholomorphic.

See for instance [Barth-Peters-Van de Ven, Compact Complex Surfaces, Chapter I].

By definition, a morphism $f \colon X \to Y$ between smooth projective varieties is smooth if $f$ is flat and all fibres are smooth.

Then the answer to your question is yes, assuming that the fibres are connected. This remains true also replacing $\mathbb{P}^1$ with any smooth projective variety $Y$.

In fact, by a result due to Ehresmann (1951), if $ƒ \colon M \to N$ is a surjective submersion with $M$ and $N$ differentiable manifolds such that the preimage $ƒ^{-1}(x)$ is compact and connected for all $x \in N$, then $ƒ$ is differentiably locally trivial and admits a compatible fiber bundle structure. See http://en.wikipedia.org/wiki/Fiber_bundle.

In particular, all fibres of $f$ are diffeomorphic.

In general, $f \colon X \to Y$ will be not analitycally locally trivial. In fact, by a celebrated theorem of Grauert and Fischer, this happens if and only if all the fibres of $f$ are biholomorphic.

See for instance [Barth-Peters-Van de Ven, Compact Complex Surfaces, Chapter I].

By definition, a morphism $f \colon X \to Y$ between smooth projective varieties is smooth if $f$ is flat and all fibres are smooth (in the scheme-theoretical sense). In particular, $f$ is a smooth submersion when it is considered as a differentiable map between real manifolds.

Then the answer to your question is yes, assuming that the fibres are connected. This remains true also replacing $\mathbb{P}^1$ with any smooth projective variety $Y$.

In fact, by a result due to Ehresmann (1951), if $ƒ \colon M \to N$ is a surjective submersion with $M$ and $N$ differentiable manifolds such that the preimage $ƒ^{-1}(x)$ is compact and connected for all $x \in N$, then $ƒ$ is differentiably locally trivial and admits a compatible fiber bundle structure. See http://en.wikipedia.org/wiki/Fiber_bundle.

In particular, all fibres of $f$ are diffeomorphic.

Remark 1. If one only requires that the fibres of $f \colon X \to Y$ are smooth in the set-theoretical sense, and not in the scheme-theoretical one (i.e. one allows multiple fibres) then in general $f$ is not a differentiable fibre bundle, see Jason Starr's comment.

Remark 2. Even if $f \colon X \to Y$ is differentially locally trivial, in general it is not analitycally locally trivial. In fact, by a celebrated theorem of Grauert and Fischer, this happens if and only if all the fibres of $f$ are biholomorphic.

See for instance [Barth-Peters-Van de Ven, Compact Complex Surfaces, Chapter I].