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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?) (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.

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.

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.

Source Link
asv
asv
  • 21.8k
  • 6
  • 54
  • 121

Variations of Hodge structures over the line

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.