Compact Kähler and rational Hodge polarization conventions

Throughout this page, assume the Axiom of Choice (The Axiom of Choice): the analytic and cohomological suppliers invoked by the results below assume full AC, and each item records its exact use. The definitional content below uses no choice.

(1) Compact Kähler manifolds. Let $n\ge 1$. A Kähler manifold is a complex manifold $X$ of complex dimension $n$ (Complex manifolds, holomorphic vector bundles and Hermitian metrics) with complex structure $J$ and a Hermitian metric $g$ — a Riemannian metric (Riemannian metric and riemannian manifold) satisfying $g(Jv,Jw)=g(v,w)$ for all tangent vectors $v,w$ — whose real fundamental form $\omega (v,w)=g(Jv,w)$ is

• a smooth real 2-form, the section of ${\mathrm{\Lambda }}^2{T}^{\ast }X$ obtained by alternating $g(J\cdot ,\cdot )$;
• positive: $\omega (v,Jv)=g(Jv,Jv)>0$ for every nonzero tangent vector $v$;
• closed: $d\omega =0$.

The manifold is compact Kähler when the topological space $X$ is compact.

Orientation. For a complex basis $(v_1,\dots ,v_n)$ of a tangent space $T_{x}X$, the ordered real basis $(v_1,J{v}_1,\dots ,v_n,J{v}_n)$ is declared positive. This is well defined: two complex bases differ by a complex-linear automorphism, whose real determinant is the square of the modulus of its complex determinant and so is positive, and the local orientations therefore glue along holomorphic charts to a global orientation of $X$. Let $v_1,\dots ,v_n$ be a $g$-orthonormal complex frame, so $g(v_j,v_k)={\delta }_{jk}$ and $g(v_j,J{v}_k)=0$, and let $e_j=g(v_j,\cdot )$ and $f_j=g(J{v}_j,\cdot )$ be the metric-dual covectors. Then $(e_1,f_1,\dots ,e_n,f_n)$ is a positively oriented $g$-orthonormal coframe, $f_j=-J{e}_j$ for the pullback action $(J\alpha )(v)=\alpha (Jv)$ of $J$ on covectors, and

$\omega ={\sum }_{j=1}^n{e}_j\wedge f_j\mathrm{.}$

The oriented volume form is the top form

$dV:=\frac{{\omega }^n}{n!}=e_1\wedge f_1\wedge \cdots \wedge e_n\wedge f_n,$

which evaluates to $1$ on each positively oriented $g$-orthonormal frame.

Complex dimension zero. For $n=0$ a complex manifold is a discrete space and compactness makes it a finite set of points with its unique complex structure; there are no tangent vectors and no forms of positive degree, $\omega =0$, and the operators $L,\mathrm{\Lambda },\mathrm{\partial },\bar{\mathrm{\partial }},d,{\mathrm{\partial }}^{\ast },{\bar{\mathrm{\partial }}}^{\ast },d^{\ast }$ and all Laplacians introduced below are zero. The Hodge star is the identity on degree zero, and $dV$ is the counting measure.

(2) Complex-valued forms and Kähler operators. Let $X$ be compact Kähler. Write ${\mathrm{\Omega }}^k(X;\mathbb{C})$ for the complex-valued smooth $k$-forms, the complex-linear extension of the real forms of A smooth differential $k$-form; the Hodge star $\ast$ of Riemannian hodge star extends complex-linearly, and for $\alpha ,\beta \in {\mathrm{\Omega }}^k(X;\mathbb{C})$ the integral inner product

$⟨\alpha ,\beta ⟩={\int }_X\alpha \wedge \ast \bar{\beta }$

is complex-linear in the first variable and conjugate-linear in the second; the integral is that of Integral of a compactly supported top form and the integrand is $\mid \alpha {\mid }^{2}dV$ when $\beta =\alpha$, so the pairing is a positive definite Hermitian form. In holomorphic charts

${\mathrm{\Omega }}^k(X;\mathbb{C})={⨁}_{p+q=k}{\mathrm{\Omega }}^{p,q}(X),d=\mathrm{\partial }+\bar{\mathrm{\partial }},$

with $\mathrm{\partial }$ and $\bar{\mathrm{\partial }}$ of bidegree $(1,0)$ and $(0,1)$ (Bigraded complex forms and the Dolbeault operators); the decomposition is independent of the holomorphic chart because holomorphic chart changes preserve bidegree.

Put $L=\omega \wedge$, so that $L:{\mathrm{\Omega }}^k(X;\mathbb{C})\to {\mathrm{\Omega }}^{k+2}(X;\mathbb{C})$, and let $\mathrm{\Lambda },{\mathrm{\partial }}^{\ast },{\bar{\mathrm{\partial }}}^{\ast },d^{\ast }$ be the formal adjoints of $L,\mathrm{\partial },\bar{\mathrm{\partial }},d$ with respect to $⟨,⟩$; existence and uniqueness are supplied by Formal adjoints of differential operators on metric vector bundles, applied to the bundle of complex-valued forms with the metric of Riemannian metrics induce metrics on dual tensor and exterior bundles and the density $dV$. The Laplacians are

${\mathrm{\Delta }}_d=d{d}^{\ast }+d^{\ast }d,{\mathrm{\Delta }}_{\mathrm{\partial }}=\mathrm{\partial }{\mathrm{\partial }}^{\ast }+{\mathrm{\partial }}^{\ast }\mathrm{\partial },{\mathrm{\Delta }}_{\bar{\mathrm{\partial }}}=\bar{\mathrm{\partial }}{\bar{\mathrm{\partial }}}^{\ast }+{\bar{\mathrm{\partial }}}^{\ast }\bar{\mathrm{\partial }},$

and $ϵ(k)=(-1{)}^{k(k-1)\mathrm{/}2}$.

(3) Rational pure Hodge structures. Let $k\in \mathbb{Z}$. A rational pure Hodge structure of weight $k$ is a finite-dimensional $\mathbb{Q}$-vector space $V$ together with a direct-sum decomposition by $\mathbb{C}$-subspaces

$V_{\mathbb{C}}=V{\otimes }_{\mathbb{Q}}\mathbb{C}={⨁}_{p+q=k}{V}^{p,q},\overline{V^{p,q}}=V^{q,p},$

where conjugation is the $\mathbb{C}$-antilinear involution $v\otimes \lambda \mapsto v\otimes \bar{\lambda }$. The sum is finite because $V_{\mathbb{C}}$ is finite dimensional, and it is direct by definition. The Weil operator $C$ is the $\mathbb{C}$-linear endomorphism of $V_{\mathbb{C}}$ acting as $i^{p-q}$ on $V^{p,q}$; the conjugation symmetry makes $C$ commute with conjugation, so $C$ is the complexification of a real-linear endomorphism of $V_{\mathbb{R}}=V{\otimes }_{\mathbb{Q}}\mathbb{R}$. It need not preserve $V$.

A polarization of such a Hodge structure is a nondegenerate $\mathbb{Q}$-bilinear form $Q:V\times V\to \mathbb{Q}$, extended $\mathbb{C}$-bilinearly to $V_{\mathbb{C}}$, such that

• $Q(y,x)=(-1{)}^{k}Q(x,y)$ for all $x,y$;
• $Q(V^{p,q},V^{a,b})=0$ unless $(a,b)=(q,p)$;
• $h(x,y)=Q(Cx,\bar{y})$ is a positive definite Hermitian form on $V_{\mathbb{C}}$.

A rational sub-Hodge structure is a $\mathbb{Q}$-subspace $U\subseteq V$ with $U_{\mathbb{C}}={⨁}_{p+q=k}(U_{\mathbb{C}}\cap V^{p,q})$. The Tate shift $V(-r)$, for $r\in \mathbb{Z}$, keeps the underlying $\mathbb{Q}$-vector space, the Weil operator and $Q$ as they are and moves the bidegrees to $(p+r,q+r)$, so that the weight becomes $k+2r$; equivalently, $(V(-r){)}^{p,q}=V^{p-r,q-r}$.

The orientation convention is exactly Deligne's ${\mathrm{o}\mathrm{r}}_i(X)$ for the choice of square root of $-1$ implicit in $J$; replacing $i$ by $-i$ reverses the orientation in odd complex dimension and preserves it in even dimension.

The normalization of $\omega$ above is the one in which a $g$-orthonormal complex frame has $\mathrm{\mid }{\zeta }_j{\mathrm{\mid }}^2=2$: at a point, write ${\zeta }_j=e_j+i{f}_j$ for the associated type-$(1,0)$ coframe. Then $\omega =\frac{i}{2}{\sum }_{j=1}^n{\zeta }_j\wedge {\bar{\zeta }}_j$. Some references instead write $\omega =i{\sum }_j{\zeta }_j\wedge {\bar{\zeta }}_j$ with $\mid {\zeta }_j\mid =1$; that is the same geometric structure with the rescaled metric $2g$, under which all four operator identities of the Kähler commutator lemma below are invariant, but the two normalizations must not be mixed inside one computation.

The declared The Axiom of Choice records the standing assumption of this page and not a use in the definitional content, which is choice-free.