Compact Kahler Hodge Foundations

By [F1] there are a finite CW complex $K$ and maps $h:X\to K$, $g:K\to X$ with $g\circ h$ homotopic to ${\mathrm{i}\mathrm{d}}_X$ and $h\circ g$ homotopic to ${\mathrm{i}\mathrm{d}}_K$. Homotopy invariance [F2] gives $g_{\ast }{h}_{\ast }=(g\circ h{)}_{\ast }=\mathrm{i}\mathrm{d}$ and $h_{\ast }{g}_{\ast }=(h\circ g{)}_{\ast }=\mathrm{i}\mathrm{d}$ on singular homology with any coefficients, so $h_{\ast }:H_n(X;\kappa )\to H_n(K;\kappa )$ is an isomorphism for every $n$ and every coefficient field $\kappa$. For a field $\kappa$ each cellular chain group $C_n^{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}(K;\kappa )$ is a direct sum of one copy of $\kappa$ per $n$-cell [F3], and the finite CW complex $K$ has finitely many cells, so $C_n^{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}(K;\kappa )$ is a finite-dimensional $\kappa$-vector space and its subquotient $H_n^{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}(K;\kappa )=\ker d_n\mathrm{/}\mathrm{im}{d}_{n+1}$ is finite-dimensional as well [F3]; by cellular homology equals singular homology [F3], $H_n(K;\kappa )$ and hence $H_n(X;\kappa )$ are finite-dimensional. In particular $H_n(X;\mathbb{Q})$ and $H_n(X;\mathbb{R})$ are finite-dimensional for every $n$.

Let $k$ be a field, let $F$ be a $k$-vector space, and let $D_{\ast }$ be a chain complex of $k$-vector spaces with differential $\mathrm{\partial }$, cycles $Z_n=\ker {\mathrm{\partial }}_n$ and boundaries $B_n=\mathrm{im}{\mathrm{\partial }}_{n+1}$. Under AC choose $k$-linear complements $A_n,T_n$ with $Z_n=B_n\oplus A_n$ and $D_n=Z_n\oplus T_n$; then ${\mathrm{\partial }}_n$ maps $T_n$ isomorphically onto $B_{n-1}$, because it is injective ($T_n\cap Z_n=0$) and ${\mathrm{\partial }}_n(D_n)=B_{n-1}={\mathrm{\partial }}_n(T_n)$. Tensoring these supplied splittings preserves them: tensor the inclusions and projections, whose composites are identities or zero and whose sum is the identity. Thus $D_n{\otimes }_{k}F=(B_n{\otimes }_{k}F)\oplus (A_n{\otimes }_{k}F)\oplus (T_n{\otimes }_{k}F)$, and the boundary is zero on the first two summands and an isomorphism from the third onto $B_{n-1}{\otimes }_{k}F$. Consequently $\ker ({\mathrm{\partial }}_n\otimes 1)=Z_n{\otimes }_{k}F$, $\mathrm{im}({\mathrm{\partial }}_{n+1}\otimes 1)=B_n{\otimes }_{k}F$, and $H_n(D_{\ast }{\otimes }_{k}F)\cong A_n{\otimes }_{k}F\cong H_n(D_{\ast }){\otimes }_{k}F$, the isomorphism sending the class of $z\otimes \mu$ to $[z]\otimes \mu$. For $D_{\ast }=C_{\ast }(X;\mathbb{Q})$ and $k=\mathbb{Q}$ the chain-level identification $C_{\ast }(X;\mathbb{Q}){\otimes }_{\mathbb{Q}}F\cong C_{\ast }(X;F)$ is the associativity and unit isomorphism for $C_n(X;\mathbb{Z}){\otimes }_{\mathbb{Z}}\mathbb{Q}{\otimes }_{\mathbb{Q}}F\cong C_n(X;\mathbb{Z}){\otimes }_{\mathbb{Z}}F$ [F4], so $H_n(X;\mathbb{Q}){\otimes }_{\mathbb{Q}}F\cong H_n(X;F)$ canonically by the map induced by $c\mapsto c\otimes 1$.

Fix a characteristic-zero field $F$ and a degree $n$, and put $V=H_n(X;\mathbb{Q})$, finite-dimensional by step 1.1. Evaluation gives natural isomorphisms $H^n(X;\kappa )\cong {\mathrm{Hom}}_{\kappa }(H_n(X;\kappa ),\kappa )$ for $\kappa =\mathbb{Q}$ and $\kappa =F$ [F5]; choosing a finite $\mathbb{Q}$-basis $(v_i)$ of the finite-dimensional space $V$ with dual basis $(f_i)$, the map $f_i\otimes \mu \mapsto (v_j\mapsto {\delta }_{ij}\mu )$ identifies ${\mathrm{Hom}}_{\mathbb{Q}}(V,\mathbb{Q}){\otimes }_{\mathbb{Q}}F$ with ${\mathrm{Hom}}_{\mathbb{Q}}(V,F)$, and $\psi \mapsto (v\otimes \mu \mapsto \psi (v)\mu )$ identifies ${\mathrm{Hom}}_{\mathbb{Q}}(V,F)$ with ${\mathrm{Hom}}_F(V{\otimes }_{\mathbb{Q}}F,F)$, with inverse $\chi \mapsto (v\mapsto \chi (v\otimes 1))$. Composing these isomorphisms with the identification $V{\otimes }_{\mathbb{Q}}F\cong H_n(X;F)$ of step 1.2 and the two evaluation isomorphisms gives a natural isomorphism $H^n(X;\mathbb{Q}){\otimes }_{\mathbb{Q}}F\cong H^n(X;F)$; on a cocycle $\phi$ and a chain $z\otimes \mu$ the image of $[\phi ]\otimes 1$ evaluates to $\phi (z)\mu$, exactly as the coefficient-change cochain $c\otimes \mu \mapsto \phi (c)\mu$ does, so the isomorphism is the natural coefficient map $[\phi ]\otimes 1\mapsto [\phi \otimes 1]$. Hence $H^n(X;\mathbb{Q}){\otimes }_{\mathbb{Q}}F\cong H^n(X;F)$ for every characteristic-zero field $F$, and $H^n(X;F)$ is finite-dimensional because $H_n(X;F)\cong V{\otimes }_{\mathbb{Q}}F$ is.

By [F6] the cup product on singular cochains with coefficients in a commutative unital ring is given by the front/back formula, and the coefficient-change map $\phi \mapsto \phi \otimes 1$ satisfies $((\phi ⌣\psi )\otimes 1)(\sigma )=(\phi \otimes 1)(\sigma [0,\dots ,p])\cdot (\psi \otimes 1)(\sigma [p,\dots ,p+q])$ on every singular simplex by multiplying the two coefficient values inside $F$, with $1\otimes 1=1$; hence coefficient change is a unital homomorphism of graded rings, and the isomorphism of step 2.1, being its $F$-linear extension, satisfies ${\mathrm{\Phi }}_F((a⌣b)\otimes \mu )={\mathrm{\Phi }}_F(a\otimes 1)⌣{\mathrm{\Phi }}_F(b\otimes \mu )$ for cohomology classes $a,b$ and scalars $\mu$, so the comparison isomorphisms preserve cup products. For a map $f:X\to Y$ of compact smooth manifolds the cochain identity $f^{\ast }(\phi \otimes 1)=(f^{\ast }\phi )\otimes 1$ holds because both sides send a simplex $\sigma$ to $\phi (f\circ \sigma )$, and pullback is a unital graded-ring homomorphism [F6]; hence the isomorphisms of step 2.1 commute with pullback and are natural for such maps, and multiplicativity is preserved as well.

The de Rham theorem [F7] gives the natural isomorphism of graded real algebras $H_{\mathrm{d}\mathrm{R}}^{\ast }(X)\cong H_{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}^{\ast }(X;\mathbb{R})$, so composing with step 2.1 for $F=\mathbb{R}$ identifies $H^n(X;\mathbb{Q}){\otimes }_{\mathbb{Q}}\mathbb{R}$ with real de Rham cohomology; the real de Rham theorem assumes ${\mathrm{A}\mathrm{C}}_{\omega }$, which AC supplies through [F1]. For $F=\mathbb{C}$ the complex de Rham complex is ${\mathrm{\Omega }}_{\mathbb{R}}^{\ast }(X){\otimes }_{\mathbb{R}}\mathbb{C}$ with differential $d\otimes 1$ [F7], and the argument of step 1.2 applied degreewise to this complex over the field extension $\mathbb{R}\subseteq \mathbb{C}$ gives $H_{\mathrm{d}\mathrm{R}}^n(X;\mathbb{C})\cong H_{\mathrm{d}\mathrm{R}}^n(X;\mathbb{R}){\otimes }_{\mathbb{R}}\mathbb{C}$; likewise $H^n(X;\mathbb{C})\cong H^n(X;\mathbb{R}){\otimes }_{\mathbb{R}}\mathbb{C}$, either by the same base change and duality [F5] or by associativity from $H^n(X;\mathbb{Q}){\otimes }_{\mathbb{Q}}\mathbb{C}\cong H^n(X;\mathbb{Q}){\otimes }_{\mathbb{Q}}\mathbb{R}{\otimes }_{\mathbb{R}}\mathbb{C}$ via step 2.1 for $F=\mathbb{R}$ and $F=\mathbb{C}$. Hence the complex de Rham comparison is the complexification of the real one, and the identification with real or complex de Rham cohomology holds for $F=\mathbb{R},\mathbb{C}$ as asserted, with naturality and cup-product compatibility from steps 2.1 and 3.1.

Assume first that $\alpha$ is real and $m\ge 1$. By [F2] choose a smooth singular cycle $z={\sum }_{j=1}^N{a}_j{\sigma }_j$ representing the image of $[M]$ in $H_m(M;\mathbb{R})$, and choose a continuous cocycle $\phi$ representing the class $J_M[\alpha ]=r_M^{-1}[I_M\alpha ]$ with $r_M\phi -I_M\alpha =\delta \psi$ for a smooth cochain $\psi$; evaluating on the smooth cycle $z$ gives $⟨J_M[\alpha ],[M]⟩=\phi (z)=I_M\alpha (z)+\psi (\mathrm{\partial }z)=I_M\alpha (z)={\sum }_{j=1}^N{a}_j{\int }_{{\mathrm{\Delta }}^m}{\sigma }_j^{\ast }\alpha$ by [F1] and the definition of $I_M$. It therefore remains to prove ${\sum }_j{a}_j{\int }_{{\mathrm{\Delta }}^m}{\sigma }_j^{\ast }\alpha ={\int }_M\alpha$ for this smooth cycle.

Each ${\sigma }_j$ extends smoothly to an open neighbourhood of ${\mathrm{\Delta }}^m$. Applying [F4] in finitely many charts covering the compact images, the critical values of all ${\sigma }_j$ and the images of all faces of all ${\mathrm{\Delta }}^m$ form a set $N\subseteq M$ that is null in every smooth chart. For $y\mathrm{\notin }N$ every preimage $x\in {\sigma }_j^{-1}(y)$ is an interior regular point of ${\sigma }_j$, and there are only finitely many of them, because a regular point has a neighbourhood on which ${\sigma }_j$ is injective and ${\mathrm{\Delta }}^m$ is compact; put ${\mu }_j(y)={\sum }_{x\in {\sigma }_j^{-1}(y)}\mathrm{sgn}(d{\sigma }_j(x))$ and $\mu (y)={\sum }_j{a}_j{\mu }_j(y)$ for $y\mathrm{\notin }N$.

The orientation normalization can be computed on the standard simplex. For an interior point $x\in {\mathrm{\Delta }}^m$, radial projection from $x$ retracts ${\mathrm{\Delta }}^m\setminus \{x\}$ onto $\mathrm{\partial }{\mathrm{\Delta }}^m$. Since ${\mathrm{\Delta }}^m$ contracts, [F6] identifies $H_m({\mathrm{\Delta }}^m,{\mathrm{\Delta }}^m\setminus \{x\};\mathbb{Z})$ with ${\tilde{H}}_{m-1}(\mathrm{\partial }{\mathrm{\Delta }}^m;\mathbb{Z})$ and sends the identity simplex to its alternating boundary. In the simplicial boundary complex this alternating boundary generates top reduced homology: there are no $m$-simplices, and the cycle equations at each codimension-two face force the signed coefficients of adjacent facets to agree. For $m=1$ the generator is the endpoint difference. The simplicial comparison in [F6] therefore makes the identity simplex a local generator, whose sign fixes the positive Euclidean generator used in [F3]. Fix now $y\mathrm{\notin }N$ and $F_j={\sigma }_j^{-1}(y)$, a finite interior set by step 1.2. The identity simplex is a relative cycle for $({\mathrm{\Delta }}^m,{\mathrm{\Delta }}^m\setminus F_j)$, and ${\sigma }_j$ is a map from this pair to $(M,M\setminus \{y\})$. Excision and the disjoint-union formula [F3, F6], using disjoint small neighbourhoods of the points of $F_j$, identify its relative class with the tuple of positive local generators just computed. The local multiplier formula [F3] consequently sends its class to ${\mu }_j(y){\mu }_y$. If $F_j$ is empty its relative class is zero. Summing with $a_j$ shows that the point restriction of $[z]$ is $\mu (y){\mu }_y$. The restriction of $[M]$ is ${\mu }_y$, so their equality in the one-dimensional real local homology group forces $\mu (y)=1$.

Fix $j$ and let $R_j$ be the regular set in the interior of ${\mathrm{\Delta }}^m$, with its standard orientation. The pullback coefficient of $\alpha$ vanishes on the critical set because it contains the factor $\det D{\sigma }_j$, and the boundary of the simplex has measure zero, so ${\int }_{{\mathrm{\Delta }}^m}{\sigma }_j^{\ast }\alpha ={\int }_{R_j}{\sigma }_j^{\ast }\alpha$. Using a countable Euclidean basis and [F4], cover $R_j$ by connected relatively compact open inverse-branch neighbourhoods $V_i$ and choose a locally finite smooth partition $({\rho }_i)$ subordinate to them by [F5]. Each ${\rho }_i$ has compact support in $V_i$. Put $f_i={\rho }_i\circ ({\sigma }_j{\mathrm{\mid }}_{V_i}{)}^{-1}$ on ${\sigma }_j(V_i)$, extended by zero, and let ${\epsilon }_i$ be the constant differential sign on $V_i$. The change-of-variables formula gives ${\int }_{V_i}{\rho }_i{\sigma }_j^{\ast }\alpha ={\epsilon }_i{\int }_M{f}_i\alpha$. To bound the series, construct a smooth positive top form $\nu$ by patching positive chart volume forms with [F5], and choose $B$ with $\mathrm{\mid }\alpha \mathrm{/}\nu \mathrm{\mid }\le B$ on compact $M$. Put $\beta =B\nu$. Smooth signed change of variables gives ${\int }_M{f}_i\beta ={\epsilon }_i{\int }_{V_i}{\rho }_i{\sigma }_j^{\ast }\beta ={\int }_{V_i}{\rho }_i\mathrm{\mid }{\sigma }_j^{\ast }\beta \mathrm{\mid }$. Monotone convergence [F5] sums the latter nonnegative coefficients to ${\int }_{R_j}\mathrm{\mid }{\sigma }_j^{\ast }\beta \mathrm{\mid }<\mathrm{\infty }$, since the pullback coefficient is bounded on the compact simplex. Therefore ${\sum }_i{\int }_M\mathrm{\mid }{f}_i\alpha \mathrm{\mid }\le {\sum }_i{\int }_M{f}_i\beta <\mathrm{\infty }$. The Riemann–Lebesgue comparison in [F5] identifies the continuous compact-simplex coefficient integrals with these Lebesgue integrals. Thus positive and negative parts may be summed under the integral. For $y\mathrm{\notin }N$, the pointwise signed weight is ${\sum }_i{\epsilon }_i{f}_i(y)={\sum }_{x\in F_j}{\sum }_{i:x\in V_i}\mathrm{sgn}(d{\sigma }_j(x)){\rho }_i(x)={\mu }_j(y)$, since the partition sums to one at each preimage. Therefore ${\int }_{{\mathrm{\Delta }}^m}{\sigma }_j^{\ast }\alpha ={\int }_M{\mu }_j\alpha$, with arbitrary values on the null set $N$.

Summing the identity of step 2.2 with the coefficients $a_j$ and using $\mu =1$ outside $N$ from step 2.1 gives, for real $\alpha$ and $m\ge 1$, ${\sum }_{j=1}^N{a}_j{\int }_{{\mathrm{\Delta }}^m}{\sigma }_j^{\ast }\alpha ={\int }_M\mu (y)\alpha (y)={\int }_M\alpha (y),$ which combined with step 1.1 proves $⟨J_M[\alpha ],[M]⟩={\int }_M\alpha$. For complex $\alpha$ apply the real case to $\mathrm{Re}\alpha$ and $\mathrm{Im}\alpha$ and use the complex-linear extension of $J_M$, of the Kronecker pairing and of the integral. For $m=0$ the manifold is a finite set of points with orientation signs $\epsilon (p)$, the fundamental class is $[M]={\sum }_p\epsilon (p)[p]$, the degree-zero comparison sends the class of a function $f$ to the evaluation classes $[f]={\sum }_{p}f(p)[p{]}^{\ast }$, and both sides of the identity are ${\sum }_p\epsilon (p)f(p)$ by the zero-dimensional convention in [F5]; the empty manifold gives $0=0$. The multiplicative consequence follows from the algebra isomorphism property of [F1] applied to the closed representatives. AC enters only through the ${\mathrm{A}\mathrm{C}}_{\omega }$ recorded in the Given, which the de Rham comparison, smooth partitions of unity, smooth-chain comparison, manifold integration and Riemann–Lebesgue comparison require; Sard, excision, change of variables, monotone convergence and the local multiplier computation are choice-free.

Suppose $n=1$. Choose a unit vector $e_1$ and put $e_2=-J{e}_1$; then $g(e_1,e_2)=0$, $g(e_2,e_2)=1$, $J{e}_1=-e_2$, $J{e}_2=e_1$, and $\omega (e_1,e_2)=-g(J{e}_1,e_2)=g(e_2,e_2)=1$, so $\omega =e_1\wedge e_2$ is the positive unit volume form and $(e_1,e_2)$ is positively oriented. The basis $1,e_1,e_2,\omega$ of ${\bigwedge }^{\bullet }{V}_{\mathbb{C}}$ is orthonormal for $h$, and [F1] gives the star table $\star 1=\omega$, $\star e_1=e_2$, $\star e_2=-e_1$, $\star \omega =1$. Hence $L(1)=\omega$ and $L$ kills $e_1,e_2,\omega$, while the adjoint $\mathrm{\Lambda }$ satisfies $\mathrm{\Lambda }\omega =1$ and kills $1,e_1,e_2$. Therefore $[L,\mathrm{\Lambda }]=H$ is $-\mathrm{i}\mathrm{d}$, $0$, $0$, $\mathrm{i}\mathrm{d}$ on degrees $0,1,1,2$, that is $H=(k-1)\mathrm{i}\mathrm{d}$ on ${\bigwedge }^k$, and evaluating $HL-LH$ and $H\mathrm{\Lambda }-\mathrm{\Lambda }H$ on the four basis elements gives $[H,L]=2L$ and $[H,\mathrm{\Lambda }]=-2\mathrm{\Lambda }$. Evaluating the exponentials gives $W(1)=\omega$, $W(e_1)=e_1$, $W(e_2)=e_2$, $W(\omega )=-1$, while $J(1)=1$, $J{e}_1=-e_2$, $J{e}_2=e_1$, $J\omega =\omega$ give $ϵ(0)J\star 1=\omega$, $ϵ(1)J\star e_1=e_1$, $ϵ(1)J\star e_2=e_2$ and $ϵ(2)J\star \omega =-1$; since both sides are complex linear, $W=ϵ(k)J\star$ on ${\bigwedge }^k{V}_{\mathbb{C}}$ for $n=1$.

Let $V=V_1\oplus V_2$ with $V_1,V_2$ $g$-orthogonal and $J$-stable, oriented so that the positive volume form is ${\mathrm{vol}}_1\wedge {\mathrm{vol}}_2$; write ${\omega }_i=\omega {\mathrm{\mid }}_{V_i}$, so $\omega ={\omega }_1+{\omega }_2$, and let $L_i,{\mathrm{\Lambda }}_i,W_i,J_i$ be the corresponding operators on the factors. Under the linear isomorphism ${\bigwedge }^{\bullet }{V}_{1,\mathbb{C}}\otimes {\bigwedge }^{\bullet }{V}_{2,\mathbb{C}}\to {\bigwedge }^{\bullet }{V}_{\mathbb{C}}$, $a\otimes b\mapsto a\wedge b$, a check on orthonormal wedge bases gives the tensor rules $h(a\wedge b,a^{\mathrm{\prime }}\wedge b^{\mathrm{\prime }})=h_1(a,a^{\mathrm{\prime }})h_2(b,b^{\mathrm{\prime }})$, $\star (a\wedge b)=(-1{)}^{\mathrm{deg}a\mathrm{deg}b}({\star }_{1}a)\wedge ({\star }_{2}b)$ and $J(a\wedge b)=J_{1}a\wedge J_{2}b$ for homogeneous $a,a^{\mathrm{\prime }}$ and $b,b^{\mathrm{\prime }}$, the middle one being forced by the characterizing property [F1] and consistent with the isometry property [F2]. Consequently $L=L_1+L_2$, $\mathrm{\Lambda }={\mathrm{\Lambda }}_1+{\mathrm{\Lambda }}_2$ (the adjoint of a sum of operators acting on the two tensor factors is the sum of the adjoints), $H=H_1+H_2$ where $H_i=(\mathrm{deg}-n_i)\mathrm{i}\mathrm{d}$ on the $i$-th factor with $n_i=\dim V_i\mathrm{/}2$, and $W=W_1{W}_2$, $J=J_1{J}_2$, because operators on the two factors commute. If the three commutator relations and the identity $W=ϵ(k)J\star$ hold on both factors, then they hold on $V$: the commutator relations split termwise, and for homogeneous $a,b$ the identity gives $W(a\wedge b)=ϵ(\mathrm{deg}a)ϵ(\mathrm{deg}b)(J_1{\star }_{1}a)\wedge (J_2{\star }_{2}b)$, which equals $ϵ(\mathrm{deg}a+\mathrm{deg}b)J\star (a\wedge b)$ because $ϵ(m+\mathrm{\ell })=(-1{)}^{m\mathrm{\ell }}ϵ(m)ϵ(\mathrm{\ell })$ and the sign $(-1{)}^{\mathrm{deg}a\mathrm{deg}b}$ of the star rule cancels against $(-1{)}^{m\mathrm{\ell }}$. Splitting off a two-plane $W=\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}(e_1,-J{e}_1)$ with $V_2=W^{\perp }$ and iterating the base case 1.1 proves the three relations of part 1 and the identity of part 4 for every $n\ge 0$: the degenerate factor $V_2=0$ has ${\bigwedge }^{\bullet }{V}_2=\mathbb{C}$ in degree zero, where all operators are the identity or zero and the relations are immediate, and for $n=0$ both sides of $W=ϵ(k)J\star$ are the identity on this one-dimensional space.