Let $f\colon X\to \mathbb{A}^1$ be a smooth projective morphism of complex algebraic manifolds, where the target $\mathbb{A}^1$ is the affine line. **Are there any restrictions on the Hodge structures on the cohomology groups of fibers of $f$ over different complex points of $\mathbb{A}^1$? (Say are there examples where these Hodge structures are not isomorphic to each other?)**


I apologize if this question is not of the research level; I am not an algebraic geometer. If there is a reference, it would be helpful.



Variations of Hodge structures over the line
See Theorem 11 on page 191 of [these notes](https://www-fourier.ujf-grenoble.fr/~peters/Articles/PubSteen2.pdf). A special case is as follows.


**Theorem of the fixed part.** Let $S$ be a smooth quasiprojective variety, and $V$ a variation of $\mathbb{Q}$-Hodge structures on $S$ (for example, $R^i\pi\_\*\underline{\mathbb{Q}}$, for $\pi: X\to S$ a smooth projective morphism). Then $H^0(S,V)$ naturally admits a Hodge structure, such that the map $H^0(S, V)\to V\_s$ is a morphism of Hodge structures for any $s\in S$.


In particular, if $S$ is simply connected (as it is in your case, where we take $S=\mathbb{A}^1\_{\mathbb{C}}$), then the map $H^0(S,V)\to V\_s$ is an isomorphism for any $S$. Hence all the $V\_s$ are isomorphic.


