Blowup formula for a morphism

Question

Let $f: X\to S$ be a smooth projective morphism between smooth schemes over $\mathbb C$, $i: Z \to X$ a closed subscheme of codimension $c$, also smooth over $S$, and let $g: Y\to S$ be the blowup along $Z$. In Voisin's book it is shown that for any geometric point $s$ of $S$, there is an isomorphism of Hodge structures $$ H^n(Y_s, \mathbb Z) = H^n(X_s,\mathbb Z) \oplus \bigoplus_{i=1}^{c-1}H^{n-2i}(Z_s, \mathbb Z), $$ so \begin{equation} H^q(Y_s, \Omega_{Y_s}^p) = H^q(X_s,\Omega_{X_s}^p) \oplus \bigoplus_{i=1}^{c-1}H^{q-i}(Z_s, \Omega_{Z_s}^{p-i}), \end{equation} and what I expect is that one has in fact an isomorphism of VHS \begin{equation} R^ig_* \mathbb Z \cong R^nf_* \mathbb Z \oplus \bigoplus_{i=1}^{c-1} R^{n-2i} (f\circ i)_* \mathbb Z, \end{equation} and that there are isomorphisms of Hodge sheaves

\begin{equation} R^q g_* \Omega_{Y/S}^p \cong R^q f_* \Omega_{X/S}^p \oplus \bigoplus_{i=1}^{c-1} R^{q-i}(f \circ i)_* \Omega^{p-i} _{Z/S} \end{equation} and that everything is compatible with the Hodge to De Rham spectral sequence.

In fact, I think I can prove that the VHS are isomorphic, but I have not idea about how to approach the isomorphism between the Hodge sheaves. On the other hand, I feel like this is something that someone must have done, but I cannot find any reference. Any help is appreciated.

Answer 1

This is not a full answer, but at least a reference with some comments. A good strategy for this type of result is presented in [Gros85, Ch. IV, Thm. 1.2.1]. The idea is that in a blowup square $$\begin{array}{ccc}E & \to & Y \\ \!\!\!\!{\scriptsize a}\downarrow & & \downarrow{\scriptsize b}\!\!\!\! \\ Z & \underset \iota\hookrightarrow & X\end{array}$$ with exceptional divisor $E$, there is a distinguished triangle $$\Omega_X^i \to Rb_*\Omega_Y^i \to \iota_*Ra_*\Omega_E \to \Omega_X^i[1].$$ This upgrades the computation of cohomology groups to one of sheaves on $X$. The advantage is that $Ra_*$ is easy to compute, as $a \colon E \to Z$ is just a projective bundle. Presumably the same type of computation can be done for the locally constant sheaf $\underline{\mathbf Z}$, and everything is compatible in the end.

That said, I don't know a reference fo the full Hodge structure statement, which I agree should exist somewhere.

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References.

[Gros85] M. Gros, Classes de Chern et classes de cycles en cohomologie de Hodge-Witt logarithmique. Mém. Soc. Math. Fr. (N. S.) 21 (1985). ZBL0615.14011.