The Involution on the Kähler Operator

Introduction

On a compact Kähler manifold the Kähler form $\omega$ acts on the exterior algebra by the Kähler operator $$ L : \Omega^{\bullet}(M) \longrightarrow \Omega^{\bullet}(M), \qquad L\alpha = \omega\wedge\alpha, $$ the Lefschetz operator, with formal adjoint the contraction $\Lambda = L^{*}$; the pair $L,\Lambda$ is the operator face of the Kähler form, and the involution of the title is the pair of natural involutions of the exterior algebra that exchange $L$ with $\Lambda$. The first is the Hodge star $\star$, defined by the metric and the orientation, an isometry of the exterior algebra with $$ \star^{2} = (-1)^{k(2n-k)}\,\mathrm{id} \quad \text{on } \Omega^{k}, \qquad \star L = \Lambda\star , \qquad \star\Lambda = L\star ; $$ the second is the complex conjugation of forms, the map $\sigma(\alpha) = \bar\alpha$ exchanging the bidegrees $(p,q)$ and $(q,p)$, a conjugate-linear involution that commutes with $L$ and $\Lambda$ because the Kähler form is real. Their composite $C = \star\sigma$ (the "conjugate Hodge star") is the involution of the Kähler operator proper: it is the conjugation that carries the Lefschetz operator to the contraction, $$ C\,L\,C^{-1} = \Lambda, \qquad C\,\Lambda\,C^{-1} = L, $$ the operator form of the self-adjointness of the Lefschetz operator with respect to the Hermitian inner product $(\alpha,\beta) = \int_M\alpha\wedge\star\bar\beta$, and the reason the $\mathfrak{sl}_2$-structure of the cohomology is a symmetric structure.

The article has three sections: the Kähler operator and its adjoint; the Hodge star and the complex conjugation, and their composite, as the involution that exchanges $L$ and $\Lambda$; and the Hermitian inner product and the self-adjointness of the pair. The Kähler form, the Lefschetz operator, the $\mathfrak{sl}_2$-relations and the Hodge–Riemann relations are The Kähler Form Operator, earlier in this category, and Kähler Manifolds and the Hermitian Form; 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 type decomposition and the Cauchy–Riemann operators are Operators on a Complex Manifold; the Hermitian inner product and the self-adjointness are The Adjoint of a Hermitian Operator, the first article of this group. None of that is re-derived.

Throughout, $(M,\omega)$ is a compact Kähler manifold of complex dimension $n$ and real dimension $2n$, $L(\alpha) = \omega\wedge\alpha$, $\Lambda = L^{*}$ is the contraction, $\star$ is the Hodge star of the metric and the orientation, $\sigma(\alpha) = \bar\alpha$ is the complex conjugation of forms, and $(\alpha,\beta) = \int_M\alpha\wedge\star\bar\beta$ is the Hermitian inner product.

The Kähler Operator and Its Adjoint

Proposition (the Lefschetz operator is self-adjoint for the pairing). The contraction $\Lambda$ is the formal adjoint of $L$, $$ (L\alpha,\beta) = (\alpha,\Lambda\beta), $$ and with $H = [L,\Lambda]$ the three operators satisfy $[H,L] = 2L$, $[H,\Lambda] = -2\Lambda$, $[L,\Lambda] = H$, so that the exterior algebra is a finite-dimensional representation of the Lie algebra $\mathfrak{sl}_2$; on the $k$-forms $H = (k-n)\,\mathrm{id}$.

Proof. The metric gives the formal adjoint of exterior multiplication by the Kähler form as contraction with the metric-dual $(1,1)$-vector, which is $\Lambda$; the $\mathfrak{sl}_2$-relations are the standard commutator computation for the operators of exterior multiplication and contraction with a positive $(1,1)$-form, and the eigenvalue of the central element on $k$-forms is $k-n$. This is The Kähler Form Operator.

Remark (the operator and its dual). The pair $(L,\Lambda)$ is the Kähler operator and its adjoint; the involution of the next sections is the transformation of the exterior algebra that exchanges the two, so that the Kähler geometry of a form is invariant under the exchange. This is the operator content of the symmetry $L\leftrightarrow\Lambda$ of the Lefschetz decomposition.

The Hodge Star, the Complex Conjugation and the Involution

Proposition (the Hodge star). The Hodge star is an isometry of the exterior algebra, $\star^{2} = (-1)^{k(2n-k)}\mathrm{id}$ on the $k$-forms, and it intertwines the Lefschetz operator and the contraction, $$ \star L = \Lambda\star, \qquad \star\Lambda = L\star . $$

Proof. The identity $\star L = \Lambda\star$ is the classical pairing of exterior multiplication with contraction through the star, with the sign convention of the Hodge star fixed as in The Volume Element, Duality and the Hodge Star; it holds on a Kähler manifold because $\omega$ is parallel and the star is built from the metric and the volume element. Applying $\star^{2}$ to both sides and cancelling the scalar $(-1)^{k(2n-k)}$ gives the second identity. The Hodge star and its properties are The Volume Element, Duality and the Hodge Star, and the Kähler intertwining is The Kähler Form Operator.

Proposition (the complex conjugation). The complex conjugation of forms $\sigma(\alpha) = \bar\alpha$ is a conjugate-linear involution of the exterior algebra, $\sigma^{2} = \mathrm{id}$, it exchanges the bidegrees, $\sigma(\Omega^{p,q}) = \Omega^{q,p}$, it is an isometry up to the conjugation of the scalar, $(\sigma\alpha,\sigma\beta) = \overline{(\alpha,\beta)}$, and it commutes with the Kähler operator and the contraction, $$ \sigma L = L\sigma, \qquad \sigma\Lambda = \Lambda\sigma, $$ because the Kähler form is real, $\bar\omega = \omega$ and $L$ commutes with $\sigma$ up to the reality of $\omega$.

Proof. The conjugation of a form is the conjugation of its coefficients in a real frame; it is conjugate-linear, involutive and exchanges the type; the reality of $\omega$ gives $\sigma(\omega\wedge\alpha) = \omega\wedge\sigma\alpha$, so $\sigma L = L\sigma$, and the conjugate-linearity with the metric (real) gives $\sigma\Lambda=\Lambda\sigma$. The type decomposition is Operators on a Complex Manifold.

Theorem (the involution on the Kähler operator). The composite $C = \star\sigma$ is an involution up to sign on each degree, $$ C^{2} = (-1)^{k(2n-k)}\,\mathrm{id}\quad\text{on }\Omega^{k}, $$ and it exchanges the Kähler operator and its adjoint, $$ C\,L\,C^{-1} = \Lambda, \qquad C\,\Lambda\,C^{-1} = L . $$

Proof. $C^{2} = \star\sigma\star\sigma$; the star is real, so it commutes with the conjugation $\sigma$, and $\sigma^{2}=\mathrm{id}$, giving $C^{2} = \star^{2}$, which is $(-1)^{k(2n-k)}$ on $\Omega^{k}$; and $C L C^{-1} = \star\sigma L\sigma^{-1}\star^{-1} = \star L\star^{-1} = \Lambda$ using $\sigma L = L\sigma$ and $\star L = \Lambda\star$. The inverse $\star^{-1}$ is $\pm\star$, absorbed in the sign. This is The Kähler Form Operator and The Volume Element, Duality and the Hodge Star.

The Hermitian Inner Product and the Self-Adjointness

Proposition (the Hermitian inner product of forms). The formula $$ (\alpha,\beta) = \int_{M}\alpha\wedge\star\bar\beta $$ is a positive-definite Hermitian inner product on the complex-valued forms, and it is invariant under the Hodge star and conjugate-invariant under the complex conjugation; the formal adjoints $d^{*}$, $\partial^{*}$, $\bar\partial^{*}$ are the adjoints of $d$, $\partial$, $\bar\partial$ for it.

Proof. The wedge with the star of the conjugate is the metric pairing against the volume element, positive definite because the metric is positive; the star is an isometry, and the conjugation is an antiunitary involution by the previous proposition. The formal adjoints are The Codifferential and Hermitian Metrics and the Codifferential.

Corollary (the symmetry of the Kähler structure). The map $C = \star\sigma$ is the involution under which the Kähler operator and the contraction are exchanged, and it is self-adjoint for $(\cdot,\cdot)$ up to sign; the Lefschetz decomposition and the Hodge–Riemann form are invariant under $C$, which is why the Kähler cohomology is symmetric in the passage from $L$ to $\Lambda$ and from $(p,q)$ to $(n-p,n-q)$.

Proof. The first assertion is the theorem; the invariance of the decomposition follows from the commutation relations and the definition of primitive forms, and the symmetry of the Hodge–Riemann form is the $\mathfrak{sl}_2$-symmetry. This is The Kähler Form Operator.

Example (the Riemann surface). For $n = 1$ the exterior algebra of a Riemann surface has the forms $1, dz, d\bar z, dz\wedge d\bar z$; the star satisfies $\star 1 = dz\wedge d\bar z$ and acts on the $(1,0)$- and $(0,1)$-forms by $\pm i$, and $C = \star\sigma$ exchanges $L$ (multiplication by the area form) with $\Lambda$ (contraction), so the involution is the Poincaré duality of the surface read through the complex conjugation. On $\mathbb{CP}^n$ the same involution exchanges the powers $L^{k}$ and $\Lambda^{k}$ of the Lefschetz and contraction operators, giving the symmetry of the Hodge numbers and the hard Lefschetz theorem.

Summary

On a compact Kähler manifold the Kähler operator $L(\alpha) = \omega\wedge\alpha$ has the contraction $\Lambda = L^{*}$ for its adjoint, and the pair satisfies the $\mathfrak{sl}_{2}$-relations $[H,L]=2L$, $[H,\Lambda]=-2\Lambda$ with $H = [L,\Lambda] = (k-n)\mathrm{id}$ on $k$-forms. The Hodge star is an isometry with $\star^{2} = (-1)^{k(2n-k)}$ on $\Omega^{k}$ and intertwines $\star L = \Lambda\star$, $\star\Lambda = L\star$; the complex conjugation $\sigma(\alpha) = \bar\alpha$ is a conjugate-linear involution exchanging the bidegrees and commuting with $L$ and $\Lambda$. Their composite $C = \star\sigma$, an involution up to sign on each degree, exchanges the Kähler operator and its adjoint, $CLC^{-1} = \Lambda$ and $C\Lambda C^{-1} = L$, and it is the symmetry of the Hermitian inner product $(\alpha,\beta) = \int\alpha\wedge\star\bar\beta$ under which the Lefschetz decomposition and the Hodge–Riemann form are invariant. The Kähler operator and the $\mathfrak{sl}_2$-relations are The Kähler Form Operator and Kähler Manifolds and the Hermitian Form; the Hodge star is The Volume Element, Duality and the Hodge Star; the type decomposition is Operators on a Complex Manifold; the self-adjointness is The Adjoint of a Hermitian Operator.

Summary of Notation

Symbol	Meaning
$L(\alpha)=\omega\wedge\alpha$	the Kähler (Lefschetz) operator
$\Lambda=L^{*}$	the contraction
$[H,L]=2L$, $[H,\Lambda]=-2\Lambda$	the $\mathfrak{sl}_2$-relations, $H=(k-n)\mathrm{id}$ on $k$-forms
$\star^{2}=(-1)^{k(2n-k)}$ on $\Omega^{k}$	the Hodge star, an isometry
$\star L=\Lambda\star$, $\star\Lambda=L\star$	the intertwining
$\sigma(\alpha)=\bar\alpha$	the complex conjugation, exchanging $(p,q)$
$C=\star\sigma$	the involution, $CLC^{-1}=\Lambda$