The Kähler Form Operator

Introduction

On a Kähler manifold the fundamental form is closed, and as a form it acts on the exterior algebra by exterior multiplication. The Lefschetz operator is $$ L : \Lambda^{k}(M)\otimes\mathbb C \longrightarrow \Lambda^{k+2}(M)\otimes\mathbb C, \qquad L(\alpha) = \Omega\wedge\alpha , $$ the operator of multiplication by the chosen Kähler form $\Omega$, of type $(1,1)$; its formal adjoint for the Kähler metric is the contraction operator $\Lambda = L^{*}$, which lowers the degree by two. These two operators, together with their commutator $H = [L,\Lambda]$, form a copy of the Lie algebra $\mathfrak{sl}_2$ acting on the forms, and this action is the whole content of the Lefschetz theory: the forms decompose into primitive pieces generated by powers of $L$, the operator $L$ is an isomorphism from degree $k$ to degree $2n-k$ in the middle range (hard Lefschetz), and the bilinear form of the manifold is definite on each primitive piece (the Hodge–Riemann relations). The operator is named for the form, and the form is the chosen structure: a different Kähler metric on the same complex manifold gives a different $L$, a different $\Lambda$ and a different decomposition, and it is in this sense that the Lefschetz operator is an object of the geometry.

The article has four sections: the Lefschetz and contraction operators and their adjointness; the $\mathfrak{sl}_2$-structure and the Kähler identities; the Lefschetz decomposition and the hard Lefschetz theorem; and the Hodge–Riemann relations. The Kähler manifold, the fundamental form, the Chern connection and the Hodge theory of the $\bar\partial$-complex are Kähler Geometry and Kähler Manifolds and the Hermitian Form, the latter later in this category; the Hodge star, the volume element and the codifferential are The Volume Element, Duality and the Hodge Star and Hermitian Metrics and the Codifferential; the representation theory of $\mathfrak{sl}_2$ and the primitive decomposition are quoted from Representations of Lie Algebras. The finiteness of the cohomology, the Hodge decomposition and the harmonic representatives are Part III, and the statements about cohomology below are made for harmonic forms and transported to cohomology by that theory, quoted.

Throughout, $(M,J,g)$ is a compact Kähler manifold of complex dimension $n$ with Kähler form $\Omega$, $L$ is the Lefschetz operator, $\Lambda = L^{*}$ is its adjoint for $g$, $H = [L,\Lambda]$, $\Lambda^k(M)\otimes\mathbb C$ is the complexified space of $k$-forms, $P^k \subseteq \Lambda^k$ is the space of primitive forms, and $Q$ is the bilinear form of the next-to-last section.

The Lefschetz Operator and its Adjoint

Definition. The Lefschetz operator of the Kähler form $\Omega$ is the operator of exterior multiplication by $\Omega$, $$ L(\alpha) = \Omega\wedge\alpha , $$ raising the total degree by two and the bidegree by $(1,1)$; the contraction operator is its formal adjoint $$ \Lambda = L^{*}, \qquad \langle L\alpha,\beta\rangle = \langle\alpha,\Lambda\beta\rangle , $$ for the $L^2$ inner product of the Kähler metric, lowering the total degree by two.

Proposition (the two operators are real and of bidegree $\pm(1,1)$). $L$ is complex-linear in the complexified forms and maps $\Lambda^{p,q}$ to $\Lambda^{p+1,q+1}$; $\Lambda$ maps $\Lambda^{p,q}$ to $\Lambda^{p-1,q-1}$ and is the transpose of $L$; both commute with complex conjugation, $\overline{L\alpha} = L\bar\alpha$ and $\overline{\Lambda\alpha} = \Lambda\bar\alpha$, because $\Omega$ is real. In local coordinates, with $\Omega = i\sum_{j,k}g_{j\bar k}\,dz^j\wedge d\bar z^k$, $$ \Lambda\alpha = \sum_{j,k} g^{j\bar k}\,\iota_{\partial_{\bar z^k}}\iota_{\partial_{z^j}}\,\alpha , $$ where $g^{j\bar k}$ is the inverse Hermitian matrix of the metric and $\iota$ is interior multiplication.

Proof. Exterior multiplication by a $(1,1)$-form is complex-linear and raises the bidegree by $(1,1)$; the adjoint of a wedge product with a real form is the interior contraction with the metric-dual $(1,1)$-vector, which is the displayed formula, and it lowers the bidegree by $(1,1)$ by the degree count; the reality assertion is that $\Omega$ is real and the metric pairing is conjugation-invariant. The identification of the formal adjoint with the contraction is The Volume Element, Duality and the Hodge Star, and the metric-dual contraction is Hermitian Metrics and the Codifferential.

Remark (the operator depends on the form). The operator $L$ is multiplication by the Kähler form, and it exists as soon as a form $\Omega$ is chosen; it is the closedness and positivity of $\Omega$ and its compatibility with $J$ and $g$ that make $\Lambda$ its adjoint and the pair a representation, so the chosen Kähler metric is what the operator layer records.

The $\mathfrak{sl}_2$-Structure and the Kähler Identities

Proposition (the commutator). On the complexified $k$-forms, $$ [L, \Lambda] = (k - n)\,\mathrm{id} \qquad \text{on } \Lambda^{k}(M)\otimes\mathbb C , $$ and, writing $H = [L,\Lambda]$ for the operator that is $(k-n)\mathrm{id}$ on $k$-forms, $$ [L,\Lambda] = H, \qquad [H,L] = 2L, \qquad [H,\Lambda] = -2\Lambda . $$ Thus $L$, $\Lambda$ and $H$ span a subalgebra isomorphic to $\mathfrak{sl}_2$, and the complexified exterior algebra is a finite-dimensional $\mathfrak{sl}_2$-module.

Proof. The commutator is computed locally by the elementary relation for the contraction and the wedge with the metric-dual form; on a Kähler manifold the metric form is parallel, which removes the derivative terms and leaves the pure degree count $[L,\Lambda]\alpha = (k-n)\alpha$ on a $k$-form (the computation on $\mathbb C^n$ with a constant Kähler form, transported to the manifold by the Kähler condition). The remaining brackets $[H,L]=2L$ and $[H,\Lambda]=-2\Lambda$ are the degree statement, since $H$ acts by $k-n$ and $L,\Lambda$ shift $k$ by $\pm2$. This is the $\mathfrak{sl}_2$-structure of Representations of Lie Algebras read on the exterior algebra.

Proposition (the Kähler identities for the operators). With $\partial^{*},\bar\partial^{*}$ the formal adjoints of the Cauchy–Riemann operators, the contraction operator satisfies $$ [\Lambda, \partial] = -i\,\bar\partial^{*}, \qquad [\Lambda, \bar\partial] = i\,\partial^{*}, \qquad [\Lambda, d] = d^{*}, $$ and consequently the Laplacians agree, $\Delta_{\partial} = \Delta_{\bar\partial} = \tfrac12\Delta_d$.

Proof. The identities are the Kähler identities, in which the operator $L$ and its adjoint $\Lambda$ mediate between the Cauchy–Riemann operators and their adjoints; their proof uses $d\Omega = 0$ and the primitivity of $\Omega$, and it is Kähler Manifolds and the Hermitian Form, where the analytic input is given. The Laplacian identity is the corollary already used in Operators on a Complex Manifold.

The Lefschetz Decomposition and Hard Lefschetz

Definition. A form $\alpha \in \Lambda^k$ is primitive when $$ L^{n-k+1}\alpha = 0 \qquad (k \le n), $$ that is when its power of $L$ beyond the middle degree vanishes; the space of primitive forms of degree $k$ is written $P^k \subseteq \Lambda^k$.

Theorem (the Lefschetz decomposition). Every form decomposes uniquely as $$ \alpha = \sum_{r \ge 0} \frac{1}{r!}\, L^{r}\alpha_{r}, \qquad \alpha_{r} \in P^{k-2r}, $$ and the summands are the images of the primitive projections; the decomposition is compatible with the bidegree, so that the primitive part of type $(p,q)$ generates the $(p+r,q+r)$ term by $L^{r}$.

Proof. On a finite-dimensional $\mathfrak{sl}_2$-module the primitive vectors — those killed by the lowering operator $\Lambda$ in the standard labelling — generate the module by the raising operator, and the multiplicity-free degree structure forces the stated unique decomposition; the translation to $L,\Lambda$ uses $[L,\Lambda]=(k-n)\mathrm{id}$ from the previous section. This is the representation-theoretic decomposition of Representations of Lie Algebras.

Theorem (hard Lefschetz). For $0 \le k \le n$ the map $$ L^{n-k} : \Lambda^{k}(M)\otimes\mathbb C \longrightarrow \Lambda^{2n-k}(M)\otimes\mathbb C $$ is an isomorphism on the harmonic forms, and therefore on cohomology, $$ L^{n-k} : H^{k}(M;\mathbb C) \xrightarrow{\ \sim\ } H^{2n-k}(M;\mathbb C) . $$

Proof. The map is bijective on each primitive piece by the $\mathfrak{sl}_2$-theory: on the module generated by a primitive $\alpha_{r}\in P^{k-2r}$ the operators $L^{r'}$ and $\Lambda^{r'}$ pair the degrees symmetrically, and $L^{n-k}$ carries the degree-$k$ part onto the degree-$2n-k$ part isomorphically; the passage from forms to cohomology is Hodge's theorem, which represents each class by a unique harmonic form, Part III. The consequence that $b_k = b_{2n-k}$ and that the odd Betti numbers are even over $\mathbb C$ is the numerical shadow.

The Hodge–Riemann Relations

Definition. For forms $\alpha,\beta$ of the same degree $k \le n$, the Hodge–Riemann bilinear form is $$ Q(\alpha, \beta) = \int_M \alpha\wedge\beta\wedge\frac{\Omega^{\,n-k}}{(n-k)!} , $$ complex-bilinear in $\alpha,\beta$; it is the intersection form of the manifold, written with the chosen Kähler form.

Theorem (the Hodge–Riemann relations). For a nonzero primitive form $\alpha$ of type $(p,q)$, with $k = p+q$, the complex number $$ i^{\,p-q}\,Q(\alpha, \bar\alpha) $$ is a positive real number; equivalently, the Hermitian form $i^{p-q}Q(\alpha,\bar\beta)$ is positive definite on each primitive space $P^{p,q}$. In particular, on the primitive $(p,p)$-forms the form $Q$ is positive definite, and the intersection form of $M$ is positive definite on the primitive part of middle cohomology and of that fixed sign on each primitive piece.

Proof. The relation is proved by comparing $Q$ with the $L^2$ inner product through the Lefschetz decomposition: for a primitive form the two differ by a nonzero constant of modulus one that depends only on $p$, $q$ and the normalisation of the $L^2$ form, and the constant is computed from $[L,\Lambda]$ and the eigenform identities of the $\mathfrak{sl}_2$-action; the computation is the Hodge–Riemann calculation of Kähler Manifolds and the Hermitian Form, quoted here, and the sign $i^{p-q}$ displayed is the one it yields for the $L^2$ normalisation of this article. The Hodge decomposition that transports the definite form to cohomology is Part III.

Remark (the operator content). The Hodge–Riemann relations say that the $\mathfrak{sl}_2$-module of the forms is not merely a formal representation but carries a definite form: the raising operator $L$ is adjoint to $\Lambda$ for the $L^2$ form, the primitive pieces are the subspaces on which the metric pairing is definite, and the signs $i^{p-q}$ are the signatures of the pieces. The Hodge index theorem $Q(\alpha,\alpha)>0$ for a real $(1,1)$-class is the case $p=q=1$ and is the most used consequence.

Summary

On a compact Kähler manifold the Kähler form acts by exterior multiplication as the Lefschetz operator $L(\alpha) = \Omega\wedge\alpha$, raising the degree by two and the bidegree by $(1,1)$, with formal adjoint the contraction $\Lambda = L^{*}$, lowering it, given in coordinates by contraction with the metric-dual $(1,1)$-vector. The two are real, and with $H = [L,\Lambda] = (k-n)\mathrm{id}$ on $k$-forms they satisfy $[L,\Lambda]=H$, $[H,L]=2L$, $[H,\Lambda]=-2\Lambda$, so the complexified exterior algebra is a finite-dimensional $\mathfrak{sl}_2$-module. The Kähler identities $[\Lambda,\partial] = -i\bar\partial^{*}$, $[\Lambda,\bar\partial] = i\partial^{*}$ and $[\Lambda,d] = d^{*}$ express the Cauchy–Riemann adjoints through $\Lambda$ and give $\Delta_{\partial} = \Delta_{\bar\partial} = \tfrac12\Delta_d$. Every form decomposes uniquely as $\alpha = \sum_r \tfrac1{r!}L^{r}\alpha_{r}$ through the primitive forms $\alpha_r$ with $L^{n-k+1}\alpha_r = 0$, the hard Lefschetz map $L^{n-k}:H^{k}\to H^{2n-k}$ is an isomorphism, and the Hodge–Riemann form $Q(\alpha,\beta) = \int_M\alpha\wedge\beta\wedge\Omega^{n-k}/(n-k)!$ satisfies $i^{p-q}Q(\alpha,\bar\alpha)>0$ on nonzero primitive forms of type $(p,q)$, so it is definite of that sign on each primitive piece. The Hodge theory and the cohomological transport are Part III; the Kähler structure is Kähler Geometry and Kähler Manifolds and the Hermitian Form.

Summary of Notation

Symbol	Meaning
$\Omega$, $L(\alpha)=\Omega\wedge\alpha$	the Kähler form and the Lefschetz operator
$\Lambda = L^{*}$	the contraction operator, the metric adjoint of $L$
$H = [L,\Lambda]$	the degree operator, $(k-n)\mathrm{id}$ on $k$-forms
$[L,\Lambda]=H$, $[H,L]=2L$, $[H,\Lambda]=-2\Lambda$	the $\mathfrak{sl}_2$-relations
$[\Lambda,\partial]=-i\bar\partial^{}$, $[\Lambda,\bar\partial]=i\partial^{}$	the Kähler identities
$P^{k}$	primitive forms, $L^{n-k+1}\alpha = 0$
$L^{n-k}:H^{k}\to H^{2n-k}$	hard Lefschetz
$Q(\alpha,\beta)$	the Hodge–Riemann form, $\int_M\alpha\wedge\beta\wedge\Omega^{n-k}/(n-k)!$