The answer here is a resounding no. I think the most important point is that you're applying the wrong topological intuition here. A variety shouldn't be thought of as like a manifold, but as like a complex manifold, and the corresponding theorem to the "submersion=fiber bundle" theorem in smooth manifolds is just false for complex manifolds. Just as an example, all elliptic curves are topologically the same, so the solutions to $x(x-1)(x-a)=y$ are a smooth fiber bundle over (most of) $\mathbb{C}$ with coordinate $a$, but all the fibers which aren't in the same orbit of $SL_2(\mathbb Z)$ on $\mathbb{C}$ are not isomorphic as complex manifolds.

The other way of saying this is that complex structures can exist in families; they have moduli. Moduli spaces exactly measure how theorems like "submersion=fiber bundle" fail since they measure continuous variation of structure.

The answer here is a resounding no. I think the most important point is that you're applying the wrong topological intuition here. A variety shouldn't be thought of as like a manifold, but as like a complex manifold, and the corresponding theorem to the "submersion=fiber bundle" theorem in smooth manifolds is just false for complex manifolds. Just as an example, all elliptic curves are topologically the same, so the solutions to $x(x-1)(x-a)=y^2$ are a smooth fiber bundle over (most of) $\mathbb{C}$ with coordinate $a$, but all the fibers which aren't in the same orbit of $SL_2(\mathbb Z)$ on $\mathbb{C}$ are not isomorphic as complex manifolds.

The other way of saying this is that complex structures can exist in families; they have moduli. Moduli spaces exactly measure how theorems like "submersion=fiber bundle" fail since they measure continuous variation of structure.

The answer here is a resounding no. I think the most important point is that you're applying the wrong topological intuition here. A variety shouldn't be thought of as like a manifold, but as like a complex manifold, and the corresponding theorem to the "submersion=fiber bundle" theorem in smooth manifolds is just false for complex manifolds. Just as an example, all elliptic curves are topologically the same, so the solutions to $x(x-1)(x-a)=y$ are a smooth family over (most of) $\mathbb{C}$ with coordinate $a$, but all the fibers which aren't in the same orbit of $SL_2(\mathbb Z)$ on $\mathbb{C}$ are not isomorphic as complex manifolds.

The other way of saying this is that complex structures can exist in families; they have moduli. Moduli spaces exactly measure how theorems like "submersion=fiber bundle" since they measure continuous variation of structure.

The answer here is a resounding no. I think the most important point is that you're applying the wrong topological intuition here. A variety shouldn't be thought of as like a manifold, but as like a complex manifold, and the corresponding theorem to the "submersion=fiber bundle" theorem in smooth manifolds is just false for complex manifolds. Just as an example, all elliptic curves are topologically the same, so the solutions to $x(x-1)(x-a)=y$ are a smooth fiber bundle over (most of) $\mathbb{C}$ with coordinate $a$, but all the fibers which aren't in the same orbit of $SL_2(\mathbb Z)$ on $\mathbb{C}$ are not isomorphic as complex manifolds.

The other way of saying this is that complex structures can exist in families; they have moduli. Moduli spaces exactly measure how theorems like "submersion=fiber bundle" fail since they measure continuous variation of structure.