Compact Kähler Hodge decomposition from elliptic harmonic forms

Suppose $n=0$. Then compactness makes $X$ a finite discrete set of points; there are no forms of positive degree, every Laplacian is zero, and every function is smooth, so the de Rham cohomology is $H^0(X;\mathbb{C})={\mathbb{C}}^{{\pi }_0(X)}$ in degree zero and zero above, the harmonic forms are exactly the locally constant functions, and $H^{0,0}(X)=H^0(X;\mathbb{C})$ while $H^{p,q}(X)=0$ otherwise, with the statement's conjugation symmetry, uniqueness and metric-independence vacuous. Here ${\mathrm{\Omega }}_X^0$ is the kernel of $\bar{\mathrm{\partial }}$ on smooth functions, i.e. the sheaf ${\underset{‾}{\mathbb{C}}}_X$ of locally constant functions, and $H^0(X,{\underset{‾}{\mathbb{C}}}_X)=\mathrm{\Gamma }(X,{\underset{‾}{\mathbb{C}}}_X)={\mathbb{C}}^{{\pi }_0(X)}$ with $H^q(X,{\underset{‾}{\mathbb{C}}}_X)=0$ for $q>0$ because ${\underset{‾}{\mathbb{C}}}_X$ is a sheaf of modules over the sheaf of smooth functions [F6, F7]; the holomorphic-degree filtration is concentrated in degree zero, so it induces the trivial filtration and the spectral sequence degenerates at $E_1$ with $E_1^{p,q}=H^q(X,{\mathrm{\Omega }}_X^p)$ [F8]. Hence assume $n\ge 1$ from now on, so the real dimension of $X$ is $2n\ge 2$.

For each degree $k$ let $E$ be the complex Hermitian bundle of complex-valued $k$-forms and $P={\mathrm{\Delta }}_d=d{d}^{\ast }+d^{\ast }d$, whose bundles we are given by [F1]. In local coordinates $d$ has leading symbol $i\epsilon (\xi )$, exterior multiplication by $\xi$, and $d^{\ast }$ has leading symbol $-i{<mi\; mathvariant="normal">⌟</mi>}_{\xi }$, contraction against $\xi$; since $\epsilon (\xi ){<mi\; mathvariant="normal">⌟</mi>}_{\xi }+{<mi\; mathvariant="normal">⌟</mi>}_{\xi }\epsilon (\xi )=\mathrm{\mid }\xi {\mathrm{\mid }}^2\mathrm{i}\mathrm{d}$, the principal symbol of ${\mathrm{\Delta }}_d$ is ${\sigma }_2({\mathrm{\Delta }}_d)(x,\xi )=\mathrm{\mid }\xi {\mathrm{\mid }}^2\mathrm{i}\mathrm{d}$, a scalar positive-definite quadratic form, so ${\mathrm{\Delta }}_d$ is Laplace type; moreover $⟨{\mathrm{\Delta }}_{d}u,u⟩=⟨d^{\ast }u,d^{\ast }u⟩+⟨du,du⟩=\mathrm{\parallel }du{\mathrm{\parallel }}^2+\mathrm{\parallel }{d}^{\ast }u{\mathrm{\parallel }}^2$ for smooth $u$ by the defining adjoint property of $d^{\ast }$ [F1], so the energy is nonnegative, and the same computation read backwards gives $⟨{\mathrm{\Delta }}_{d}u,v⟩=⟨u,{\mathrm{\Delta }}_{d}v⟩$, so ${\mathrm{\Delta }}_d$ is formally symmetric. The bounds and elliptic hypotheses of [F3] are therefore met on the compact boundaryless manifold $X$ of real dimension $2n\ge 2$; let ${\mathcal{H}}^k=\ker ({\mathrm{\Delta }}_d)$ on $k$-forms, ${\mathrm{\Pi }}_k$ its orthogonal projection and $G_k$ the Green operator. For a smooth $d$-closed $k$-form $u$ one has $u={\mathrm{\Pi }}_{k}u+{\mathrm{\Delta }}_d{G}_{k}u$; since ${\mathrm{\Delta }}_d$ commutes with $d$ [F2], ${\mathrm{\Delta }}_d(d{G}_{k}u)=d({\mathrm{\Delta }}_d{G}_{k}u)=d(u-{\mathrm{\Pi }}_{k}u)=0$, so $d{G}_{k}u$ is harmonic, and it is exact, so $\mathrm{\parallel }d{G}_{k}u{\mathrm{\parallel }}^2=⟨G_{k}u,d^{\ast }(d{G}_{k}u)⟩=0$ because a harmonic form has vanishing adjoint by the energy identity above; hence $d{G}_{k}u=0$ and $u={\mathrm{\Pi }}_{k}u+d{d}^{\ast }{G}_{k}u$. Thus every de Rham class has the smooth harmonic representative ${\mathrm{\Pi }}_{k}u$, and it is unique: if $h$ is harmonic and $h=d\beta$ for a smooth $\beta$, then $\mathrm{\parallel }h{\mathrm{\parallel }}^2=⟨\beta ,d^{\ast }h⟩=0$, so $h=0$.

By [F2] ${\mathrm{\Delta }}_d=2{\mathrm{\Delta }}_{\bar{\mathrm{\partial }}}$ and ${\mathrm{\Delta }}_d$ preserves each bidegree, so ${\mathcal{H}}^k={⨁}_{p+q=k}{\mathcal{H}}^{p,q}$ with ${\mathcal{H}}^{p,q}=\ker ({\mathrm{\Delta }}_d)\cap {\mathrm{\Omega }}^{p,q}(X)$, an orthogonal sum for the pointwise metric; hence ${\mathrm{\Pi }}_k$ preserves each bidegree. The Green operator $G_k$ does too: if $u$ has pure type, decompose the smooth form $G_{k}u$ into types. Each component orthogonal to ${\mathcal{H}}^k$ solves the corresponding component of ${\mathrm{\Delta }}_d{G}_{k}u=u-{\mathrm{\Pi }}_{k}u$; all other-type components solve the homogeneous equation and are therefore both harmonic and orthogonal to the harmonic space, hence zero. Define $H^{p,q}(X)$ as the image in $H^{p+q}(X;\mathbb{C})$ of the $d$-closed forms of type $(p,q)$. A class in $H^{p,q}(X)$ has harmonic representative ${\mathrm{\Pi }}_{p+q}\alpha$ of type $(p,q)$ by bidegree preservation, and conversely a class whose harmonic representative has type $(p,q)$ lies in $H^{p,q}(X)$; since the harmonic representative is unique and additive, $H^k(X;\mathbb{C})={⨁}_{p+q=k}{H}^{p,q}(X)$ with $H^{p,q}(X)\cong {\mathcal{H}}^{p,q}$ via $h\mapsto [h]$. Complex conjugation of complex-valued forms is the conjugate-linear involution induced by the real forms [F1]; it commutes with $d$ and $d^{\ast }$, which have real coefficients, hence with ${\mathrm{\Delta }}_d$, and it exchanges $d{z}_j$ with $d{\bar{z}}_j$, hence ${\mathrm{\Omega }}^{p,q}(X)$ with ${\mathrm{\Omega }}^{q,p}(X)$ and ${\mathcal{H}}^{p,q}$ with ${\mathcal{H}}^{q,p}$; therefore $\overline{H^{p,q}(X)}=H^{q,p}(X)$. Finally $H^{p,q}(X)$ is defined by a closed representative of type $(p,q)$ alone, so it does not depend on the Kähler metric, and the decomposition just proved for one Kähler metric is a decomposition into these metric-free subspaces.

Let $f:X\to Y$ be a holomorphic map of compact Kähler manifolds. Pullback of forms is functorial, preserves wedge products and commutes with $d$ [F4], and in holomorphic coordinates $f^{\ast }(d{z}_j)$ and $f^{\ast }(d{\bar{z}}_j)$ are holomorphic respectively conjugate-holomorphic linear combinations of $d{z}_k,d{\bar{z}}_k$ [F4]; hence $f^{\ast }{\mathrm{\Omega }}^{p,q}(Y)\subseteq {\mathrm{\Omega }}^{p,q}(X)$ and $f^{\ast }$ carries $d$-closed $(p,q)$-forms to $d$-closed $(p,q)$-forms. Therefore $f^{\ast }{H}^{p,q}(Y)\subseteq H^{p,q}(X)$, so pullback preserves Hodge type; by the naturality in [F4] it is a morphism of the rational structures. For classes $a\in H^{p,q}(X)$, $b\in H^{r,s}(X)$ represented by $d$-closed forms $\alpha ,\beta$, the form $\alpha \wedge \beta$ is $d$-closed of type $(p+r,q+s)$ because $d$ is a graded derivation and $\mathrm{\partial },\bar{\mathrm{\partial }}$ have bidegrees $(1,0)$ and $(0,1)$ [F1], and it represents the cup product $a⌣b$ under the de Rham isomorphism [F4]; hence cup product preserves Hodge type. With $V=H^k(X;\mathbb{Q})$ finite-dimensional and $V_{\mathbb{C}}\cong H^k(X;\mathbb{C})$ as in [F4], the decomposition $V_{\mathbb{C}}={⨁}_{p+q=k}{H}^{p,q}(X)$ with conjugation symmetry of step 2.1 and the functoriality just proved make $H^k(X;\mathbb{Q})$ a functorial rational pure Hodge structure in the sense of [F1].

Define the sheaves ${\mathcal{A}}^{p,q}$ of smooth $(p,q)$-forms on $X$ and ${\mathrm{\Omega }}_X^p=\ker (\bar{\mathrm{\partial }}:{\mathcal{A}}^{p,0}\to {\mathcal{A}}^{p,1})$; the sections of ${\mathrm{\Omega }}_X^p$ over an open set are exactly the holomorphic $p$-forms, since $\bar{\mathrm{\partial }}f=0$ is the Cauchy-Riemann condition on the coefficients in degree zero. At each point, the polydisc lemma [F5] provides, for $q\ge 1$, a local primitive of every $\bar{\mathrm{\partial }}$-closed $(p,q)$-form on a smaller polydisc, so the stalk sequence $0\to {\mathrm{\Omega }}_X^p\to {\mathcal{A}}^{p,0}\stackrel{\bar{\mathrm{\partial }}}{\to }{\mathcal{A}}^{p,1}\stackrel{\bar{\mathrm{\partial }}}{\to }\cdots \stackrel{\bar{\mathrm{\partial }}}{\to }{\mathcal{A}}^{p,n}\to 0$ is exact: exactness at ${\mathcal{A}}^{p,0}$ holds by the definition of ${\mathrm{\Omega }}_X^p$, and exactness at ${\mathcal{A}}^{p,q}$ for $1\le q\le n$ is exactly the local solvability supplied by [F5]. Each ${\mathcal{A}}^{p,q}$ is a sheaf of modules over the sheaf of smooth functions (real-valued functions acting by multiplication), and its global sections are the smooth $(p,q)$-forms, so it is acyclic for global sections by [F7], and the acyclic resolution theorem [F7], whose DC hypothesis follows from AC [F7], therefore identifies $H^q(X,{\mathrm{\Omega }}_X^p)$ [F6] with the cohomology of the deleted complex of global sections ${\mathrm{\Omega }}^{p,0}(X)\stackrel{\bar{\mathrm{\partial }}}{\to }{\mathrm{\Omega }}^{p,1}(X)\stackrel{\bar{\mathrm{\partial }}}{\to }\cdots$, i.e. with the global Dolbeault cohomology $H_{\bar{\mathrm{\partial }}}^{p,q}(X)=\ker \bar{\mathrm{\partial }}\mathrm{/}\mathrm{im}\bar{\mathrm{\partial }}$ in degree $q$. For a $\bar{\mathrm{\partial }}$-closed smooth $(p,q)$-form $\alpha$ the identity $\alpha =\mathrm{\Pi }\alpha +2{\mathrm{\Delta }}_{\bar{\mathrm{\partial }}}G\alpha$ holds because ${\mathrm{\Delta }}_d=2{\mathrm{\Delta }}_{\bar{\mathrm{\partial }}}$ [F2]; here $\bar{\mathrm{\partial }}({\mathrm{\Delta }}_{d}G\alpha )={\mathrm{\Delta }}_d(\bar{\mathrm{\partial }}G\alpha )$ by [F2] and equals $\bar{\mathrm{\partial }}(\alpha -\mathrm{\Pi }\alpha )=0$, so $\bar{\mathrm{\partial }}G\alpha$ is harmonic, and for any harmonic $v$, $0=⟨{\mathrm{\Delta }}_{d}v,v⟩=2(\mathrm{\parallel }\bar{\mathrm{\partial }}v{\mathrm{\parallel }}^2+\mathrm{\parallel }{\bar{\mathrm{\partial }}}^{\ast }v{\mathrm{\parallel }}^2)$ by [F2] and adjunction. Thus ${\bar{\mathrm{\partial }}}^{\ast }\bar{\mathrm{\partial }}G\alpha =0$; hence $\alpha =\mathrm{\Pi }\alpha +2\bar{\mathrm{\partial }}({\bar{\mathrm{\partial }}}^{\ast }G\alpha )$ with $\mathrm{\Pi }\alpha$ harmonic of type $(p,q)$ by step 2.1, and every Dolbeault class has the harmonic representative $\mathrm{\Pi }\alpha$, unique because a harmonic form that is $\bar{\mathrm{\partial }}$-exact has zero norm. Therefore $H^q(X,{\mathrm{\Omega }}_X^p)\cong {\mathcal{H}}^{p,q}\cong H^{p,q}(X)$ by steps 1.2 and 2.1.

Filter the de Rham complex by holomorphic degree, $F^a{\mathrm{\Omega }}^k(X;\mathbb{C})={⨁}_{r\ge a}{\mathrm{\Omega }}^{r,k-r}(X)$; both $\mathrm{\partial }$ and $\bar{\mathrm{\partial }}$ map $F^a$ into $F^a$ by their bidegrees [F1], so the filtration is compatible with $d$ and finite on each degree ($F^a=0$ for $a>k$, $F^a={\mathrm{\Omega }}^k$ for $a\le 0$). Reindex to the homology convention of [F8] by setting $C_m={\mathrm{\Omega }}^{-m}(X;\mathbb{C})$ and the increasing filtration $G_p{C}_m=F^{-p}{\mathrm{\Omega }}^{-m}(X;\mathbb{C})$; [F8] then supplies a spectral sequence with pages ${\tilde{E}}_{p,q}^r$ and differentials ${\tilde{d}}^r:{\tilde{E}}_{p,q}^r\to {\tilde{E}}_{p-r,q+r-1}^r$, stabilizing pointwise and abutting to $H_m(C_{\bullet })=H^{-m}(X;\mathbb{C})$ with the image filtration. Writing $E_r^{a,q}={\tilde{E}}_{-a,-q}^r$ for the cohomological indexing, we get $E_1^{a,q}=H^{a+q}(F^a\mathrm{/}{F}^{a+1})=H^q({\mathrm{\Omega }}^{a,\bullet },\bar{\mathrm{\partial }})=H_{\bar{\mathrm{\partial }}}^{a,q}(X)\cong H^{a,q}(X)$ by step 3.2, and $d_r:E_r^{a,q}\to E_r^{a+r,q-r+1}$ is induced by $d$. Each $E_1^{a,q}$ is finite-dimensional by steps 2.1 and 3.2. For a class in $E_1^{a,q}$ represented by the harmonic form $h$ of step 3.2, $dh=\mathrm{\partial }h+\bar{\mathrm{\partial }}h=0$ with $\bar{\mathrm{\partial }}h=0$ gives $\mathrm{\partial }h=0$, so $d_1=0$. By [F8] each page is naturally the homology of the preceding one, so $E_r$ is a subquotient of $E_1$ and the dimensions are non-increasing in $r$; explicitly $\dim E_{r+1}^{a,q}=\dim E_r^{a,q}-\mathrm{rank}(d_r^{a,q})-\mathrm{rank}(d_r^{a-r,q+r-1})$. The abutment identifies ${\mathrm{gr}}^a{H}^k(X;\mathbb{C})\cong E_{\mathrm{\infty }}^{a,k-a}$, so ${\sum }_{a+q=k}\dim E_{\mathrm{\infty }}^{a,q}=\dim H^k(X;\mathbb{C})$, while steps 2.1 and 3.2 give ${\sum }_{a+q=k}\dim E_1^{a,q}={\sum }_{a+q=k}\dim H^{a,q}(X)=\dim H^k(X;\mathbb{C})$; since $\dim E_{\mathrm{\infty }}^{a,q}\le \dim E_1^{a,q}$ for every $(a,q)$, the diagonal sums being equal forces $\dim E_{\mathrm{\infty }}^{a,q}=\dim E_1^{a,q}$ termwise, hence every rank vanishes and $d_r=0$ for all $r\ge 1$ with $E_{\mathrm{\infty }}=E_1$, whence ${\mathrm{gr}}^a{H}^k(X;\mathbb{C})\cong H^{a,k-a}(X)$ and the spectral sequence degenerates at $E_1$ with $E_1^{a,q}=H^q(X,{\mathrm{\Omega }}_X^a)$. Finally the image filtration is $F^a{H}^k(X;\mathbb{C})=\mathrm{im}(H^k(F^a{\mathrm{\Omega }}^{\bullet })\to H^k(X;\mathbb{C}))$ [F8], which is the set of classes with a $d$-closed representative in ${⨁}_{r\ge a}{\mathrm{\Omega }}^{r,k-r}(X)$; it contains ${⨁}_{r\ge a}{H}^{r,k-r}(X)$ because such a class has a closed representative of pure type, and its successive graded quotients are ${\mathrm{gr}}^r{H}^k\cong H^{r,k-r}(X)$, so the finite filtrations have equal dimensions and $F^a{H}^k(X;\mathbb{C})={⨁}_{r\ge a}{H}^{r,k-r}(X)$, as asserted.