Lefschetz operator of a Kähler manifold ($L$)

The Lefschetz operator of a Kähler manifold is exterior multiplication by its Kähler form,Its formal adjoint is denoted by $\Lambda =L^{\ast }$ and lowers the bidegree of a differential form by $(1,1)$.

This wedge operator participates in the Kähler identities; the theorem on integral $(1,1)$ classes is a different result.

For a compact Kähler manifold of complex dimension $n$, wedging with the $(n-k)$th power of its Kähler class is the displayed isomorphism for $0\le k\le n$. The Hard Lefschetz isomorphism on Dolbeault cohomology and Hodge decomposition theorem for compact Kähler manifolds give the result by summing the bidegrees. In particular, $L:H^k\to H^{k+2}$ is injective when $k<n$: if $L\alpha =0$, then $L^{n-k}\alpha =0$. Hence $b_k\le b_{k+2}$ below the middle dimension.

On a compact Kähler manifold, the Lefschetz operator preserves harmonic forms, so the indicated power acts on harmonic representatives of Dolbeault cohomology. It is injective by injectivity of powers of the Lefschetz operator; Hodge symmetry and Hodge duality make its finite-dimensional source and target have equal dimension, so it is an isomorphism.

Wedge multiplication by a closed $(1,1)$ form induces maps on Dolbeault cohomology. If ${\omega }^{\prime }-\omega =\bar{\partial }\theta$, then $({\omega }^{\prime }{)}^k-{\omega }^k=\bar{\partial }(\theta \wedge {\sum }_{j=0}^{k-1}({\omega }^{\prime }{)}^j\wedge {\omega }^{k-1-j})$. Wedging with a closed representative shows that the maps agree. For a change of Kähler metric potential, take $\theta =-i\partial f$.

For degree $k\le n$, a form is primitive if $\Lambda \alpha =0$, equivalently $L^{n-k+1}\alpha =0$. In middle degree $k=n$ this is equivalent to $L\alpha =0$. On a compact Kähler manifold, the same definitions apply to cohomology via harmonic differential forms.

On a complex two-dimensional Kähler manifold, a primitive $(1,1)$ form satisfies $\ast \alpha =-\alpha$, making it an anti-self-dual differential form. For a nonzero real primitive harmonic form on a compact surface, $\int \alpha \wedge \alpha =-\Vert \alpha {\Vert }_{L^2}^2<0$.

In complex dimension $n$, the Lefschetz operator of a Kähler manifold and adjoint Lefschetz operator satisfy $[\Lambda ,L]=(n-k)\mathrm{id}$ on degree $k$. In middle degree, this yields $\Vert L\alpha {\Vert }^2=\Vert \Lambda \alpha {\Vert }^2$.

For the Hermitian Lefschetz operator on a complex $n$-manifold, the Lefschetz commutator and the commutator derivation identity imply this formula. When expanding a product commutator, one must evaluate $[L,\Lambda ]$ on the degree after the powers of $L$ to its right have acted.

The adjoint of $L=\omega \wedge -$ lowers real degree by two and bidegree by $(1,1)$. On a compact Kähler manifold it acts on cohomology through harmonic differential forms, because it commutes with the Hodge Laplacian. It does not generally preserve all closed forms.

For the Lefschetz operator of a Kähler manifold and its adjoint, the Kähler identities include

On a Kähler manifold the Chern connection components satisfy $[\Lambda ,{\bar{\partial }}_E]=-i(D_E^{\prime }{)}^{\ast }$ and $[\Lambda ,D_E^{\prime }]=i{\bar{\partial }}_E^{\ast }$. Holomorphic normal coordinates and a Chern connection in a normal holomorphic frame reduce them to the scalar identities pointwise. Closedness of the Kähler form also gives $[\Lambda ,{\bar{\partial }}_E^{\ast }]=0$.

For a Hermitian holomorphic bundle over a Kähler manifold, let $\Theta$ act by exterior multiplication. The bundle-valued Kähler identities give $[\Lambda ,{\Delta }_{\bar{\partial }}]=-i(\Theta \wedge -{)}^{\ast }$, and taking adjoints yields $[{\Delta }_{\bar{\partial }},L]=i\Theta \wedge -$. This measures the failure of the bundle-valued Dolbeault Laplacian to commute with the Lefschetz operator.

The Lefschetz operator commutes with the bundle-valued Dolbeault Laplacian on the entire smooth form space exactly when the Chern connection is flat. The Lefschetz-Dolbeault curvature commutator gives one direction. Conversely test its curvature-wedge operator on degree-zero sections with arbitrary prescribed values at any point to force every vector-bundle curvature endomorphism to vanish.

For the Chern connection of a Hermitian holomorphic vector bundle on a Kähler manifold, the two Dolbeault Laplacians differ by $[iF\wedge ,\Lambda ]$, where $F$ is its curvature form of a connection and $\Lambda$ is the adjoint Lefschetz operator. The curvature term controls vanishing estimates.

The Dolbeault Laplacian is ${\Delta }_{\bar{\partial }}=\bar{\partial }{\bar{\partial }}^{\ast }+{\bar{\partial }}^{\ast }\bar{\partial }$. On a Kähler manifold, the Kähler identities imply

On a compact Hermitian manifold without boundary, this operator is the inverse of the Dolbeault Laplacian on the orthogonal complement of its harmonic forms and is zero on the harmonic space. Elliptic theory gives ${\Delta }_{\bar{\partial }}G=G{\Delta }_{\bar{\partial }}=I-H$, with $H$ the harmonic projection, and ensures that $G$ sends smooth forms to smooth forms. It commutes with $\bar{\partial }$ and ${\bar{\partial }}^{\ast }$; their commutation with the Laplacian and orthogonality to its kernel give this by uniqueness of the inverse. If $\eta$ is $\bar{\partial }$-exact, then $\eta =\bar{\partial }{\bar{\partial }}^{\ast }G\eta$. On a Kähler manifold, the same operator serves the holomorphic Laplacian because ${\Delta }_{\partial }={\Delta }_{\bar{\partial }}$.

On a Kähler manifold, expanding $d=\partial +\bar{\partial }$ and using the Kähler identities makes the mixed anticommutators vanish and gives ${\Delta }_{\partial }={\Delta }_{\bar{\partial }}$. Therefore

The Kähler Laplacian identity ${\Delta }_d^{\mathrm{H}}=2{\Delta }_{\bar{\partial }}$ implies that every holomorphic function on a Kähler manifold is a harmonic function. On functions, $\bar{\partial }u=0$ and ${\bar{\partial }}^{\ast }u=0$ by degree, so both terms of the Dolbeault Laplacian vanish. Compactness is unnecessary. Changing the sign convention for the scalar Laplacian does not change this conclusion.

On a Kähler manifold, the Lefschetz operator of a Kähler manifold $L=\omega \wedge -$ commutes with the Dolbeault Laplacian:Consequently, if $\alpha$ is ${\Delta }_{\bar{\partial }}$-harmonic, then every $\alpha \wedge {\omega }^k=L^k\alpha$ is also harmonic.

For a compact Hermitian manifold, elliptic theory gives the orthogonal decompositionThe harmonic space ${\mathcal{H}}_{\bar{\partial }}^{p,q}=\ker {\Delta }_{\bar{\partial }}$ is finite-dimensional and represents Dolbeault cohomology.

On a compact Hermitian manifold of complex dimension $n$, $C\alpha =\ast \bar{\alpha }$, with complex-linear Hodge star, is a conjugate-linear bijection from Dolbeault Laplacian harmonic $(p,q)$-forms to harmonic $(n-p,n-q)$-forms. The identities ${\bar{\partial }}^{\ast }=-\ast \partial \ast$ and ${\ast }^2=(-1{)}^{p+q}$ show that $\bar{\partial }\alpha ={\bar{\partial }}^{\ast }\alpha =0$ implies the same two equations for $C\alpha$. Its square is $(-1{)}^{p+q}$. Thus these harmonic spaces have equal complex dimension even without the Kähler condition. The natural map is conjugate-linear, not complex-linear.

On a compact Kähler manifold, a pure-type differential form that is $d$-closed and is either $\partial$-exact or $\bar{\partial }$-exact is $\partial \bar{\partial }$-exact. In particular, if $\eta$ is $\bar{\partial }$-exact and $\partial \eta =0$, thenfor a form $\varphi$ of bidegree one lower in each component.

On a compact Kähler manifold, the difference of cohomologous Kähler forms is a real, de Rham cohomology exact $(1,1)$-form. The ddbar lemma gives $\tilde{\omega }-\omega =\partial \bar{\partial }h$. Reality implies $\partial \bar{\partial }\bar{h}=-(\tilde{\omega }-\omega )$, so $f=(h-\bar{h})/(2i)$ is real and has the displayed potential equation. The difference of two real potentials is a pluriharmonic function, hence constant on each connected component. On a connected manifold the only ambiguity is one additive constant.

If $\alpha$ is a $d$-closed $(p,q)$-form on a compact Kähler manifold, then $\alpha$ is ddbar lemma exact if and only if it is orthogonal to every harmonic differential form of type $(p,q)$. The forward implication follows by integration by parts. For the reverse implication, Dolbeault Hodge decomposition first makes $\alpha$ $\bar{\partial }$-exact, and the ddbar lemma then makes it $\partial \bar{\partial }$-exact.

For a $d$-closed pure-type form $\alpha$ orthogonal to harmonic forms on a compact Kähler manifold, let $G_{\bar{\partial }}$ be the inverse of the Dolbeault Laplacian on the harmonic complement, zero on the harmonic space. The Kähler Laplacian identity makes it commute with $\partial$ and $\bar{\partial }$. Thus $\alpha =\bar{\partial }{\bar{\partial }}^{\ast }{G}_{\bar{\partial }}\alpha$ and $G_{\bar{\partial }}\alpha =\partial {\partial }^{\ast }{G}_{\bar{\partial }}^2\alpha$. The mixed anticommutator ${\bar{\partial }}^{\ast }\partial +\partial {\bar{\partial }}^{\ast }=0$ gives $\alpha =\partial \bar{\partial }\beta$. This is an explicit potential for the ddbar lemma, with one lower degree of each type; using the inverse of the Hodge Laplacian instead multiplies the displayed expression by four.

On a compact Kähler manifold, if a differential form $\alpha$ is $d$-closed and $d^c$-exact, then it is $d{d}^c$-exact: there is a form $\beta$ two degrees lower such that $\alpha =d{d}^c\beta$. This real-form statement is equivalent, after decomposing by type, to the ddbar lemma.