Differential operators

The following operators are differential operators and arise out of the smooth and complex structure of ${\displaystyle X}$:

• ${\displaystyle d:\Omega ^{k}(X,\mathbb {C} )\to \Omega ^{k+1}(X,\mathbb {C} )}$, the exterior derivative. (S)
• ${\displaystyle \partial$ :\Omega ^{p,q}(X)\to \Omega ^{p+1,q}(X)} , the ${\displaystyle (1,0)}$-Dolbeault operator. (C)
• ${\displaystyle {\bar {\partial }}:\Omega ^{p,q}(X)\to \Omega ^{p,q+1}(X)}$, the ${\displaystyle (0,1)}$-Dolbeault operator. (C)

The Dolbeault operators are related directly to the exterior derivative by the formula ${\displaystyle d=\partial +{\bar {\partial }}}$. The characteristic property of the exterior derivative that ${\displaystyle d^{2}=0}$ then implies ${\displaystyle \partial ^{2}={\bar {\partial }}^{2}=0}$ and ${\displaystyle \partial {\bar {\partial }}=-{\bar {\partial }}\partial }$.

Some sources make use of the following operator to phrase the Kähler identities.

• ${\displaystyle d^{c}=-{\frac {i}{2}}(\partial -{\bar {\partial }}):\Omega ^{p,q}(X)\to \Omega ^{p+1,q}(X)\oplus \Omega ^{p,q+1}(X)}$.^{[Note 1]} (C)

This operator is useful as the Kähler identities for ${\displaystyle \partial ,{\bar {\partial }}}$ can be deduced from the more succinctly phrased identities of ${\displaystyle d^{c}}$ by comparing bidegrees. It is also useful for the property that ${\displaystyle dd^{c}=i\partial {\bar {\partial }}}$. It can be defined in terms of the complex structure ${\displaystyle J}$ by the formula

$${\displaystyle d^{c}=J^{-1}\circ d\circ J=-J\circ d\circ J.}$$

Tensorial operators

The following operators are tensorial in nature, that is they are operators which only depend on the value of the complex differential form at a point. In particular they can each be defined as operators between vector spaces of forms ${\displaystyle \Lambda _{x}^{p,q}:=\Lambda ^{p}T_{1,0}^{*}X_{x}\otimes \Lambda ^{q}T_{0,1}^{*}X_{x}}$ at each point ${\displaystyle x\in X}$ individually.

• ${\displaystyle {\bar {\cdot }}:\Omega ^{p,q}(X)\to \Omega ^{q,p}(X)}$, the complex conjugate operator. (C)
• ${\displaystyle L:\Omega ^{p,q}(X)\to \Omega ^{p+1,q+1}(X)}$, the Lefschetz operator defined by ${\displaystyle L(\alpha ):=\omega \wedge \alpha }$ where ${\displaystyle \omega }$ is the Kähler form. (CR)
• ${\displaystyle \star$ :\Omega ^{p,q}(X)\to \Omega ^{n-q,n-p}(X)} , the Hodge star operator. (R)

The direct sum decomposition of the complex differential forms into those of bidegree (p,q) manifests a number of projection operators.

• ${\displaystyle \Pi _{k}:\Omega (X)\to \Omega ^{k}(X,\mathbb {C} )}$, the projection onto the part of degree k. (S)
• ${\displaystyle \Pi _{p,q}:\Omega ^{k}(X,\mathbb {C} )\to \Omega ^{p,q}(X)}$, the projection onto the part of bidegree (p,q). (C)
• ${\displaystyle \Pi =\sum _{k=0}^{2n}(k-n)\Pi _{k}:\Omega (X)\to \Omega (X)}$, known as the counting operator.^[3]: 34 (S)
• ${\displaystyle J=\sum _{p,q=0}^{n}i^{p-q}\Pi _{p,q}}$, the complex structure operator on the complex vector space ${\displaystyle \Omega (X)}$. (C)

Notice the last operator is the extension of the almost complex structure ${\displaystyle J}$ of the Kähler manifold to higher degree complex differential forms, where one recalls that ${\displaystyle J(\alpha )=i\alpha }$ for a ${\displaystyle (1,0)}$-form and ${\displaystyle J(\alpha )=-i\alpha }$ for a ${\displaystyle (0,1)}$-form, so ${\displaystyle J}$ acts with factor ${\displaystyle i^{p-q}}$ on a ${\displaystyle (p,q)}$-form.

Adjoints

The Riemannian metric on ${\displaystyle X}$, as well as its natural orientation arising from the complex structure can be used to define formal adjoints of the above differential and tensorial operators. These adjoints may be defined either through integration by parts or by explicit formulas using the Hodge star operator ${\displaystyle \star }$.

To define the adjoints by integration, note that the Riemannian metric on ${\displaystyle X}$, defines an ${\displaystyle L^{2}}$-inner product on ${\displaystyle \Omega ^{p,q}(X)}$ according to the formula

$${\displaystyle \langle \langle \alpha ,\beta \rangle \rangle _{L^{2}}=\int _{X}\langle \alpha ,\beta \rangle {\frac {\omega ^{n}}{n!}}}$$

where ${\displaystyle \langle \alpha ,\beta \rangle }$ is the inner product on the exterior products of the cotangent space of ${\displaystyle X}$ induced by the Riemannian metric. Using this ${\displaystyle L^{2}}$-inner product, formal adjoints of any of the above operators (denoted by ${\displaystyle T}$) can be defined by the formula

$${\displaystyle \langle \langle T\alpha ,\beta \rangle \rangle _{L^{2}}=\langle \langle \alpha ,T^{*}\beta \rangle \rangle _{L^{2}}.}$$

When the Kähler manifold is non-compact, the ${\displaystyle L^{2}}$-inner product makes formal sense provided at least one of ${\displaystyle \alpha ,\beta }$ are compactly supported differential forms.

In particular one obtains the following formal adjoint operators of the above differential and tensorial operators. Included is the explicit formulae for these adjoints in terms of the Hodge star operator ${\displaystyle \star }$.^{[Note 2]}

• ${\displaystyle d^{*}:\Omega ^{k}(X,\mathbb {C} )\to \Omega ^{k-1}(X,\mathbb {C} )}$ explicitly given by ${\displaystyle d^{*}=-\star \circ d\circ \star }$. (SR)
• ${\displaystyle \partial ^{*}:\Omega ^{p,q}(X)\to \Omega ^{p-1,q}(X)}$ explicitly given by ${\displaystyle \partial ^{*}=-\star \circ {\bar {\partial }}\circ \star }$. (CR)
• ${\displaystyle {\bar {\partial }}^{*}:\Omega ^{p,q}(X)\to \Omega ^{p,q-1}(X)}$ explicitly given by ${\displaystyle {\bar {\partial }}^{*}=-\star \circ \partial \circ \star }$. (CR)
• ${\displaystyle {d^{c}}^{*}:\Omega ^{k}(X,\mathbb {C} )\to \Omega ^{k+1}(X,\mathbb {C} )}$ explicitly given by ${\displaystyle {d^{c}}^{*}=-\star \circ d^{c}\circ \star }$. (CR)
• ${\displaystyle L^{*}=\Lambda$ :\Omega ^{p,q}(X)\to \Omega ^{p-1,q-1}(X)} explicitly given by ${\displaystyle \Lambda =\star ^{-1}\circ L\circ \star }$. (CR)

The last operator, the adjoint of the Lefschetz operator, is known as the contraction operator with the Kähler form ${\displaystyle \omega }$, and is commonly denoted by ${\displaystyle \Lambda }$.

Laplacians

Built out of the operators and their formal adjoints are a number of Laplace operators corresponding to ${\displaystyle d,\partial }$ and ${\displaystyle {\bar {\partial }}}$:

• ${\displaystyle \Delta _{d}:=dd^{*}+d^{*}d:\Omega ^{k}(X,\mathbb {C} )\to \Omega ^{k}(X,\mathbb {C} )}$, otherwise known as the Laplace–de Rham operator. (SR)
• ${\displaystyle \Delta _{\partial }:=\partial \partial ^{*}+\partial ^{*}\partial$ :\Omega ^{p,q}(X)\to \Omega ^{p,q}(X)} . (CR)
• ${\displaystyle \Delta _{\bar {\partial }}:={\bar {\partial }}{\bar {\partial }}^{*}+{\bar {\partial }}^{*}{\bar {\partial }}:\Omega ^{p,q}(X)\to \Omega ^{p,q}(X)}$. (CR)

Each of the above Laplacians are self-adjoint operators.

Real and complex operators

Even if the complex structure (C) is necessary to define the operators above, they may nevertheless be applied to real differential forms ${\displaystyle \alpha \in \Omega ^{k}(X,\mathbb {R} )\subset \Omega ^{k}(X,\mathbb {C} )}$. When the resulting form also has real coefficients, the operator is said to be a real operator. This can be further characterised in two ways: If the complex conjugate of the operator is itself, or if the operator commutes with the almost-complex structure ${\displaystyle J}$ acting on complex differential forms. The composition of two real operators is real.

The complex conjugate of the above operators are as follows:

• ${\displaystyle {\bar {d}}=d}$ and ${\displaystyle {\overline {d^{*}}}=d^{*}}$.
• ${\displaystyle {\overline {(\partial )}}={\bar {\partial }}}$ and ${\displaystyle {\overline {({\bar {\partial }})}}=\partial }$ and similarly for ${\displaystyle \partial ^{*}}$ and ${\displaystyle {\bar {\partial }}^{*}}$.
• ${\displaystyle {\overline {d^{c}}}=d^{c}}$ and ${\displaystyle {\overline {{d^{c}}^{*}}}={d^{c}}^{*}}$.
• ${\displaystyle {\bar {\star }}=\star }$.
• ${\displaystyle {\bar {J}}=-J}$.
• ${\displaystyle {\bar {L}}=L}$ and ${\displaystyle {\bar {\Lambda }}=\Lambda }$.
• ${\displaystyle {\bar {\Delta }}_{d}=\Delta _{d}}$.
• ${\displaystyle {\bar {\Delta }}_{\partial }=\Delta _{\partial }}$.
• ${\displaystyle {\bar {\Delta }}_{\bar {\partial }}=\Delta _{\bar {\partial }}}$.

Thus ${\displaystyle d,d^{*},d^{c},{d^{c}}^{*},\star ,L,\Lambda ,\Delta _{d},\Delta _{\partial },}$ and ${\displaystyle \Delta _{\bar {\partial }}}$ are all real operators. In particular if any of these operators is denoted by ${\displaystyle T}$, then the commutator ${\displaystyle [T,J]=0}$ where ${\displaystyle J}$ is the complex structure operator above.

Hard Lefschetz decomposition

The representation theory of ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$ describes completely an irreducible representation in terms of its primitive element. If ${\displaystyle \alpha \in P^{k}(X,\mathbb {C} )}$ is a non-zero primitive element, then since differential forms vanish above dimension ${\displaystyle 2n}$, the chain ${\displaystyle \alpha ,L(\alpha ),L^{2}(\alpha ),\dots }$ eventually terminates after finitely many powers of ${\displaystyle L}$. This defines a finite-dimensional vector space

$${\displaystyle V(\alpha )=\operatorname {span} \langle \alpha ,L(\alpha ),L^{2}(\alpha ),\dots \rangle }$$

which has an ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$-action induced from the triple ${\displaystyle \{\Pi ,L,\Lambda \}}$. This is the irreducible representation corresponding to ${\displaystyle \alpha }$. Applying this simultaneously to each primitive cohomology group, the splitting of cohomology ${\displaystyle H(X)}$ into its irreducible representations becomes known as the hard Lefschetz decomposition of the compact Kähler manifold.

Hard Lefschetz decomposition^{[6]: Thm 5.27 [3]: Prop 3.3.13 [5]: Ch VI Thm 8.17} — Let ${\displaystyle (X,\omega ,J)}$ be a compact Kähler manifold. Then the de Rham cohomology of ${\displaystyle X}$ admits an orthogonal direct sum decomposition

$${\displaystyle H^{k}(X,\mathbb {C} )=\bigoplus _{i\geq 0}L^{i}(P^{k-2i}(X,\mathbb {C} )).}$$

This decomposition is compatible with the Hodge decomposition into Dolbeault cohomology groups:

$${\displaystyle H^{p,q}(X)=\bigoplus _{i\geq 0}L^{i}(P^{p-i,q-i}(X)).}$$

In addition

• If ${\displaystyle k>n}$, then ${\displaystyle P^{k}(X,\mathbb {C} )=0}$.
• The map ${\displaystyle L^{n-k}:P^{k}(X,\mathbb {C} )\to H^{2n-k}(X,\mathbb {C} )}$ is injective for ${\displaystyle k\leq n}$, and restricts to give an injection ${\displaystyle L^{n-k}:P^{p,q}(X)\to H^{p+n-k,q+n-k}(X,\mathbb {C} )}$ for each (p,q) such that ${\displaystyle p+q=k}$.
• The map ${\displaystyle L^{n-k}:H^{k}(X,\mathbb {C} )\to H^{2n-k}(X,\mathbb {C} )}$ is bijective for ${\displaystyle k\leq n}$, and restricts to give a bijection ${\displaystyle L^{n-k}:H^{p,q}(X,\mathbb {C} )\to H^{p+n-k,q+n-k}(X,\mathbb {C} )}$ for each (p,q) such that ${\displaystyle p+q=k}$.
• If ${\displaystyle k\leq n}$, then ${\displaystyle P^{k}(X,\mathbb {C} )=\{\alpha \in H^{k}(X,\mathbb {C} )\mid L^{n-k+1}\alpha =0\}}$, and furthermore ${\displaystyle P^{p,q}(X)=\{\alpha \in H^{p,q}(X)\mid L^{n-p-q+1}\alpha =0\}}$.

By the Kähler identities paired with a holomorphic vector bundle, in the case where the holomorphic bundle is flat the Hodge decomposition extends to the twisted de Rham cohomology groups ${\displaystyle H_{dR}^{k}(X,E)}$ and the Dolbeault cohomology groups ${\displaystyle H^{p,q}(X,E)}$. The triple ${\displaystyle \{\Pi ,L,\Lambda \}}$ still acts as an sl2-triple on the bundle-valued cohomology, and the a version of the Hard Lefschetz decomposition holds in this case.^{[6]: Thm 5.31}