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In the case of a family of compact complex manifolds we have the following:

Theorem. Let $f:X→B$ be a family of complex manifolds and assume that $X_0$ is Kähler for some $0\in B$. Then for $b$ in a neighborhood of $0$ in $B$, the Hodge numbers of $X_b$ are the same as the Hodge numbers of $X_0$.

Proof. See C. Voisin, Hodge Theory and Complex Algebraic Geometry I, p.235.

Now, suppose that we have a family of smooth affine complex manifolds such that the projection $\pi:\mathcal{U}\longrightarrow T$ is a locally trivial $C^{\infty}$ vibration.

Is there a similar result for the Hodge numbers of the mixed Hodge structure?

Hodge number--> $h^{p,q}(H)=dim_{\mathbb{C}} Gr_F^P Gr_{p+q}^W(H_\mathbb{C})$, where $H$ is the $\mathbb{Z}$-module with mixed Hodge structure.

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    $\begingroup$ I think this would be true if $\mathcal{U}$ was an open subvariety of a variety $\mathcal{X}$ which is smooth and proper over $T$ and the complement is a relative (to $T$) strict normal crossings divisor, but I am not sure whether your assumptions imply that such an $\mathcal{X}$ always exists. $\endgroup$
    – naf
    May 3, 2021 at 4:35
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    $\begingroup$ In general, i.e., without the assumption on the compactification or even affineness, this probably follows from the theory of Mixed Hodge Modules of Morihiko Saito. The point is that any mixed Hodge module whose underlying perverse sheaf is a local system is an admissible variation of mixed Hodge structure. $\endgroup$
    – naf
    May 3, 2021 at 8:53
  • $\begingroup$ @naf, If we are in the situation of your first comment, what would be the references for this, or could you explain it? Thanks for your attention $\endgroup$
    – Georgy
    May 3, 2021 at 15:03
  • $\begingroup$ Sorry, I don't know a reference. The proof is by inspecting the construction of the MHS in Deligne's Theorie de Hodge II, especially the spectral sequence in 3.2.4.1. This reduces to showing that if D is a smooth relative divisor in a smooth projective family $X \to S$ then the relative Gysin map induces a map of variations of Hodge structure (after a twist). $\endgroup$
    – naf
    May 4, 2021 at 5:40
  • $\begingroup$ @naf, thank you. $\endgroup$
    – Georgy
    May 10, 2021 at 15:11

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