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I'm noticing I know positive results to the following question in a number of special cases, but I don't know the general situation.

Let $S$ be a complete trait, to make sure I'm not misspeaking assume both residue fields are perfect, and let $X/S$ be a flat projective scheme whose generic fiber is smooth. Denote by $X_s$, $X_\eta$ the special and generic fibers respectively. Given only $X_s$, can we determine the Betti numbers of $X_\eta$? In other words, given flat projective families $X,Y/S$ with $X_s=Y_s$, do $X_\eta$ an $Y_\eta$ have the same Betti numbers?

My feeling is in this generality, the answer is yes if the relative dimension is 1 and no if the relative dimension is any higher, there are too many different families that degenerate to the same variety; I would appreciate a counterexample. A few results of this form:

  1. If $X/S$ is smooth, the answer is very well-known to be yes: this follows from étale smooth base change in general, the crystalline/de Rham comparison theorem if $S$ is the integers in a finite extension of $\mathbb{Q}_p$, and the infinitesimal/de Rham comparison theorem or even just plain homotopy invariance of singular cohomology in the case $k[[t]]$, $k$ of characteristic zero.
  2. If $S$ is the integers in a finite extension of $\mathbb{Q}_p$ and $X/S$ is semistable, then the semistable comparison theorem of Kato, Tsuji, and Faltings implies the Betti numbers of $X_s$ in log-crystalline cohomology suffice. I do not know about the semistable case for other base traits, but there is probably a known positive answer. I would greatly appreciate a reference or sketch in this case.
  3. If our model is not semistable, then I think under some fairly strong assumptions that are automatic in relative dimension 1, $\ell$-adic intersection cohomology should work: $H^{*}(X_{\eta},\mathbb{Q}_{\ell})=IH^{*}(X_{s})$. I want to derive this from de Jong's results, but haven't been able to get the details to work out. However, if we cannot find an appropriate alteration that is finite over $X_s$, i.e. some fibers are more substantially blown-up, then this sketch breaks down.
  4. I vaguely remember hearing something to the effect of the Euler characteristics of the fibers in $\ell$-adic étale cohomology agree, though this may have pertained only to curves. What I was remembering was: the Euler characteristics of the structure sheaves $\mathcal{O}_{X_s}$ and $\mathcal{O}_{X_\eta}$ in a flat family agree, proven somewhere in EGA or SGA. This is about very different sheaves, but otherwise enough the right flavor of result to maybe be encouraging.
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    $\begingroup$ math.stackexchange.com/questions/3641101/… is related, but I don't think answers this. $\endgroup$
    – Curious
    Mar 19, 2023 at 15:16
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    $\begingroup$ Observation: we can instead ask this question about a flat proper formal scheme over Spf of the base DVR, with its adic generic fiber. I don't expect this formalism to change anything. $\endgroup$
    – Curious
    Mar 19, 2023 at 15:51
  • $\begingroup$ I think this is a question of a level better suited for Mathoverflow $\endgroup$ Mar 20, 2023 at 11:45
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    $\begingroup$ @IDC I considered asking on MO, but the underlying question "can two families with different Betti numbers degenerate to the same variety" felt basic enough to maybe be worth asking here first. However I spoke with my advisor earlier and the answer is "yes, you can compute the cohomology of the generic fiber from the special fiber," albeit I believe up to computing the differentials of a spectral sequence; I'll track down a reference and write up the details as an answer soon. $\endgroup$
    – Curious
    Mar 20, 2023 at 19:18

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