Relative Hodge base change over analytic Artin deformations

First, if $X$ has complex dimension zero, compactness and [F3] make it a finite disjoint union of charts $\mathrm{Specan}A$. Its only relative forms and cohomology occur at $(p,q)=(0,0)$, where they are copies of $A$; freeness, every Artin base change and degeneration follow directly. For the remaining proof assume ${\dim }_{\mathbb{C}}X\ge 1$, so the real dimension is at least two as required in [F2]. By [F3], relative holomorphic $p$-forms are resolved by a complex $(C_p\otimes A,D_p)$ with $C_p^q=A^{p,q}(X)$ and $D_p=\bar{\mathrm{\partial }}+{\delta }_p$, where ${\delta }_p$ increases the nilpotent maximal-ideal filtration. Its cohomology is exactly $H^q(X_A,{\mathrm{\Omega }}_{X_A\mathrm{/}A}^p)$. The same local chart construction proves that this deformation is locally smooth over $A$, rather than imposing relative smoothness as a new restriction. The total relative de Rham cohomology is the free $A$-module $H^k(X;\mathbb{C})\otimes A$ by [F3].

On the smooth central Dolbeault complex put $d=\bar{\mathrm{\partial }}$, take $i$ to be inclusion of harmonic forms, $p$ the harmonic projection, and $h=d^{\ast }G$. The identity $dh+hd=1-ip$ follows from the Green formula. The relations $h^2=hi=ph=0$ hold because $G$ commutes with $d,d^{\ast }$ and vanishes on harmonics. For precision, $\mathrm{\Delta }$ commutes with $d,d^{\ast }$ by $d^2=(d^{\ast }{)}^2=0$; applying this to $G$ and using its unique orthogonal solution and the energy identity proves the stated commutations. Thus [F4] applies to $D_p$ and transfers it to a bounded finite free complex $K_p^q={\mathcal{H}}^{p,q}\otimes A$ with differential $d_{H,p}$ in the maximal ideal. It follows that ${\dim }_{\mathbb{C}}{H}^q(X_A,{\mathrm{\Omega }}_{X_A\mathrm{/}A}^p)\le \mathrm{\ell }{h}^{p,q}$. This proves the upper bound using an explicit finite model, without an unproved proper coherent-cohomology semicontinuity theorem.

Filter the relative fine double resolution by holomorphic degree. By [F5] its relative Frölicher spectral sequence has these Hodge groups as $E_1^{p,q}$ and abuts to the free de Rham group in step1.1. For each total degree $k$, finite filtration and the subquotient property give $\mathrm{\ell }{b}_k={\sum }_{p+q=k}{\dim }_{\mathbb{C}}{E}_{\mathrm{\infty }}^{p,q}\le {\sum }_{p+q=k}{\dim }_{\mathbb{C}}{E}_1^{p,q}\le \mathrm{\ell }{\sum }_{p+q=k}{h}^{p,q}=\mathrm{\ell }{b}_k\mathrm{.}$ The last equality is the central $E_1$-degeneration in [F1] and the hypothesis. All inequalities are therefore equalities. Every deficit in the second inequality is nonnegative, so each individual relative Hodge group has dimension exactly $\mathrm{\ell }{h}^{p,q}$. All groups here are finite-dimensional over $\mathbb{C}$ by their finite models. Over a finite local complex Artin ring with residue $\mathbb{C}$, these dimensions also equal their module lengths: filter by powers of the maximal ideal and refine the resulting finite-dimensional residue-field quotients by vector-space flags.

The explicit equality formula in [F4] now forces every image of every differential $d_{H,p}$ to be zero, in every degree. Consequently $K_p$ has zero differential and its degree-$q$ cohomology is ${\mathcal{H}}^{p,q}\otimes A$, a free $A$-module of rank $h^{p,q}$. The transfer maps reduce to the original harmonic inclusion and projection, so the resulting natural restriction to the central fibre is exactly its Dolbeault cohomology, not merely a vector space with the correct dimension. The finite transfer formulas commute with any local Artin base change $A\to B$; since the transferred differential is zero, they give $K_p{\otimes }_{A}B$ with zero differential and identify the natural cohomology base-change map with the identity on ${\mathcal{H}}^{p,q}\otimes B$. This proves base change both to the central fibre in the Statement and to lower Artin orders required by its consumer.

Equality of the first inequality in step3.1 leaves no room for any later spectral-sequence differential: a nonzero incoming or outgoing image would strictly reduce the finite dimension of a page entry, hence that of its abutment entry and one of the nonnegative diagonal sums. Thus all differentials vanish and the relative sequence degenerates at $E_1$. If $X$ is zero-dimensional, compactness makes it a finite set of points and [F3]'s local charts are $\mathrm{Specan}A$; all cohomology is in degree zero and the same assertions are immediate. This proves the full approved freeness, restriction and relative-degeneration assertions without projectivity, simple connectivity or any Calabi metric hypothesis.