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Let $\pi:\mathcal X\to B$ be a holomorphic family of compact complex manifolds with $X_t:=\pi^{-1}(t),t\in B, X_0=X$.

It is known that if the central fiber $X_0$ is Kähler, then the Hodge filtration $F^pH^k(X_t,\mathbb C):=\frac{F^pA^k(X_t)\cap \ker d}{F^pA^k(X_t)\cap \text{im}d}$ varies smoothly (actually holomorphically) in $H^k(X,\mathbb C)$, I wonder if we relax the Kähler condition on $X$ to the condition that the Frölicher spectral sequence degenerates at $E_1$, then does the subspace $F^pH^k(X_t,\mathbb C)$ still vary smoothly in $H^k(X,\mathbb C)$?

My argument is as follows: it is known that if $X_0$ is $E_1$ (which means Frölicher spectral sequence degenerates at $E_1$), then $X_t$ is $E_1$ for $t$ small enough (see, for example, Voisin's book Hodge theory and complex algebraic geometry, I, Proposition 9.20). Then the dimension of $F^pH^k(X_t,\mathbb C)$ is independent of $t$, and $F^pH^k(X_t,\mathbb C)=H^{p,k-p}_{\bar\partial}(X_t)\oplus\cdots\oplus H^{k,0}_{\bar\partial}(X_t)$, then according to Proposition 9.22 of Voisin's book, since the Hodge number $h^{p,q}(X_t)$ is independent of $t$, the space $\mathbb H_{\bar\partial}^{p,q}(X_t)$ varies smoothly with $t$, then $F^pH^k(X_t,\mathbb C)$ as a sum of $H^{p,q}_{\bar\partial}(X_t)$ also varies smoothly in $H^k(X,\mathbb C)$. Is there any problem?

Tom
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1 Answers1

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Your result is correct, however there exists some confusion in your last paragraph statement.

"Then the dimension of $F^pH^k(X_t,\mathbb C)$ is independent of $t$, and $F^pH^k(X_t,\mathbb C)=H^{p,k-p}_{\bar\partial}(X_t)\oplus\cdots\oplus H^{k,0}_{\bar\partial}(X_t),$ then according to Proposition 9.22 of Voisin's book, since the Hodge number $h^{p,q}(X_t)$ is independent of $t$, the space $\mathbb H_{\bar\partial}^{p,q}(X_t)$ varies smoothly with $t$, then $F^pH^k(X_t,\mathbb C)$ as a sum of $H^{p,q}_{\bar\partial}(X_t)$ also varies smoothly in $H^k(X,\mathbb C)$."

The more clear statement of this paragraph should be:

Then the dimension of $F^pH^k(X_t,\mathbb C)$ is independent of $t$ since $$F^pH^k(X_t,\mathbb C)=H^{p,k-p}_{\bar\partial}(X_t)\oplus\cdots\oplus H^{k,0}_{\bar\partial}(X_t). \,\,\,\,\,\,\color{red}{(*)}$$ As we have known that the Hodge number $h^{p,q}(X_t)$ is independent of $t$, then the space $\mathbb H_{\bar\partial}^{p,q}(X_t)$ varies smoothly with $t$ thanks to Proposition 9.22 of Voisin's book. Therefore, $F^pH^k(X_t,\mathbb C)$ as a sum of $H^{p,q}_{\bar\partial}(X_t)$ also varies smoothly in $H^k(X,\mathbb C)$.

Invariance
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  • Thanks for your correction which seems more reasonable, and for your last paragraph, I don't see why (*) is a consequence of the terminology "pure de Rham cohomology", since in Popovici's book, the filtration is decomposed to pure type de Rham cohomologies $H_{dR}^{p,q}(X)$, but here in my case $H_{\bar\partial}^{p,q}(X)$, if $X$ is a $\partial\bar\partial$-manifold, they are isomorphic, but for a general compact complex manifold, it is not necessarily true, do I missed something? – Tom Jun 05 '23 at 12:47
  • Dear @Tom, the part of "pure de Rham cohomology" you have said is correct. Sorry for my careless, and I have edited the answer. Anyway, (*) is a consequence if $X$ degenerates at $E_1$-level. – Invariance Jun 05 '23 at 13:17