Kähler identities

The Kähler identities were first proven by W. V. D. Hodge, appearing in his book on harmonic integrals in 1941.Hodge, W.V.D., 1989. The theory and applications of harmonic integrals. CUP Archive. The modern notation of $\backslash Lambda$ was introduced by André Weil in the first textbook on Kähler geometry, Introduction à L’Étude des Variétés Kähleriennes.Weil, A., 1958. Introduction à l'étude des variétés kählériennes{{Rp|page=42}}

A Kähler manifold $(X,\backslash omega,J)$ admits a large number of operators on its algebra of complex differential forms$$\backslash Omega(X)\; :=\; \backslash bigoplus\_\{k\; \backslash ge\; 0\}\; \backslash Omega^\{k\}(X,\backslash mathbb\{C\})\; =\; \backslash bigoplus\_\{p,q\backslash ge\; 0\}\; \backslash Omega^\{p,q\}(X)$$built out of the smooth structure (S), complex structure (C), and Riemannian structure (R) of $X$. The construction of these operators is standard in the literature on complex differential geometry. In the following the bold letters in brackets indicates which structures are needed to define the operator.

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

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

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

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

• $d^c\; =\; -\backslash frac\{i\}\{2\}\; (\backslash partial\; -\; \backslash bar\; \backslash partial):\; \backslash Omega^\{p,q\}(X)\; \backslash to\; \backslash Omega^\{p+1,q\}(X)\; \backslash oplus\; \backslash Omega^\{p,q+1\}(X)$.Some sources use the coefficients $-i/4\backslash pi$, $-i/2\backslash pi$, or just $-i$ in the definition of $d^c$ for notational convenience. With the first convention, the Ricci form of a Kähler metric has the local form $\backslash rho\; =\; dd^c\; \backslash log\; \backslash omega^n$. These conventions change the Kähler identities for $d^c$ by an appropriate constant. (C)

This operator is useful as the Kähler identities for $\backslash partial,\; \backslash bar\; \backslash partial$ can be deduced from the more succinctly phrased identities of $d^c$ by comparing bidegrees. It is also useful for the property that $dd^c\; =\; i\; \backslash partial\; \backslash bar\; \backslash partial$. It can be defined in terms of the complex structure operator $J$ by the formula$$d^c\; =\; J^\{-1\}\; \backslash circ\; d\; \backslash circ\; J\; .$$

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 $\backslash Lambda^\{p,q\}\_x\; :=\; \backslash Lambda^p\; T\_\{1,0\}^*\; X\_x\; \backslash otimes\; \backslash Lambda^q\; T\_\{0,1\}^*\; X\_x$ at each point $x\backslash in\; X$ individually.

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

• $\backslash Pi\_k:\; \backslash Omega(X)\; \backslash to\; \backslash Omega^k(X,\backslash mathbb\{C\})$, the projection onto the part of degree k. (S)
• $\backslash Pi\_\{p,q\}:\; \backslash Omega^k(X,\backslash mathbb\{C\})\; \backslash to\; \backslash Omega^\{p,q\}(X)$, the projection onto the part of bidegree (p,q). (C)
• $\backslash Pi\; =\; \backslash sum\_\{k=0\}^\{2n\}\; (k-n)\; \backslash Pi\_k:\; \backslash Omega(X)\; \backslash to\; \backslash Omega(X)$, known as the counting operator.Huybrechts, D., 2005. Complex geometry: an introduction (Vol. 78). Berlin: Springer.{{Rp|page=34}} (S)
• $J\; =\; \backslash sum\_\{p,q=0\}^n\; i^\{p-q\}\; \backslash Pi\_\{p,q\}$, the complex structure operator on the complex vector space $\backslash Omega(X)$. (C)

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

The Riemannian metric on $X$, 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 $\backslash star$.

To define the adjoints by integration, note that the Riemannian metric on $X$, defines an $L^2$-inner product on $\backslash Omega^\{p,q\}(X)$ according to the formula$$\backslash langle\; \backslash langle\; \backslash alpha,\backslash beta\; \backslash rangle\; \backslash rangle\_\{L^2\}\; =\; \backslash int\_X\; \backslash langle\; \backslash alpha,\; \backslash beta\; \backslash rangle\; \backslash frac\{\backslash omega^n\}\{n!\}$$

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

$$\backslash langle\; \backslash langle\; T\backslash alpha,\; \backslash beta\backslash rangle\; \backslash rangle\_\{L^2\}\; =\; \backslash langle\; \backslash langle\; \backslash alpha,\; T^*\; \backslash beta\backslash rangle\; \backslash rangle\_\{L^2\}.$$When the Kähler manifold is non-compact, the $L^2$-inner product makes formal sense provided at least one of $\backslash alpha,\; \backslash beta$ 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 $\backslash star$.Note that the sign $n(k+1)+1$ (see Codifferential) in front of the adjoint $d^*$ becomes $-1$ in all degrees since the dimension $n$ of the complex manifold $X$ is even.

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

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

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

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

Each of the above Laplacians are self-adjoint operators.

Even if the complex structure (C) is necessary to define the operators above, they may nevertheless be applied to real differential forms $\backslash alpha\; \backslash in\; \backslash Omega^k(X,\backslash mathbb\{R\})\; \backslash subset\; \backslash Omega^k(X,\backslash mathbb\{C\})$. 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 $J$ acting on complex differential forms. The composition of two real operators is real.

The complex conjugate of the above operators are as follows:

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

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

The Kähler identities are a list of commutator relationships between the above operators. Explicitly we denote by $[T,S]\; =\; T\backslash circ\; S\; -\; S\; \backslash circ\; T$ the operator in $\backslash Omega(X)\; =\; \backslash Omega^\{\backslash bullet\}(X,\backslash mathbb\{C\})$ obtained through composition of the above operators in various degrees.

The Kähler identities are essentially local identities on the Kähler manifold, and hold even in the non-compact case. Indeed they can be proven in the model case of a Kähler metric on $\backslash mathbb\{C\}^n$ and transferred to any Kähler manifold using the key property that the Kähler condition $d\backslash omega\; =\; 0$ implies that the Kähler metric takes the standard form up to second order. Since the Kähler identities are first order identities in the Kähler metric, the corresponding commutator relations on $\backslash mathbb\{C\}^n$ imply the Kähler identities locally on any Kähler manifold.Griffiths, P. and Harris, J., 2014. Principles of algebraic geometry. John Wiley & Sons.{{Rp|location=Ch 0 §7}}

When the Kähler manifold is compact the identities can be combined with Hodge theory to conclude many results about the cohomology of the manifold.

{{math theorem

| name = Kähler identities{{Rp|location=§3.1}}{{Rp|location=§5.1}}{{Rp|location=Ch V §4}}

| math_statement = Let $(X,\backslash omega,J)$ be a Kähler manifold. Then the following identities hold:

• $[\backslash bar\; \backslash partial,\; L]\; =\; 0$.
• $[\backslash partial,\; L]=0$.
• $[\backslash bar\; \backslash partial^*,\; \backslash Lambda]\; =\; 0$.
• $[\backslash partial^*,\; \backslash Lambda]\; =\; 0$.
• $[\backslash bar\; \backslash partial^*,\; L]\; =\; i\; \backslash partial$.
• $[\backslash partial^*,\; L]\; =\; -\; i\; \backslash bar\; \backslash partial$.
• $[\backslash Lambda,\; \backslash bar\; \backslash partial]\; =\; -i\backslash partial^*$.
• $[\backslash Lambda,\; \backslash partial]\; =\; i\; \backslash bar\; \backslash partial^*$.
• $\backslash Delta\_d\; =\; 2\; \backslash Delta\_\{\backslash partial\}\; =\; 2\; \backslash Delta\_\{\backslash bar\; \backslash partial\}$.
• $\backslash Delta\; =\; \backslash Delta\_d$ commutes with all of $\backslash star,\; \backslash partial,\; \backslash bar\; \backslash partial,\; \backslash partial^*,\; \backslash bar\; \backslash partial^*,\; L,$ and $\backslash Lambda$. It also commutes with $\backslash Pi^\{p,q\}$ and hence $\backslash Delta\_d$ preserves bidegree (p,q).

Furthermore the operators $d$ and $d^c$ satisfy the identities:

• $[\backslash Lambda,\; d]\; =\; -2\{d^c\}^*$.
• $[L,\; d]\; =\; 0$.
• $[\backslash Lambda,d^c]\; =\; 0$.
• $[L,\; d^*]\; =\; 2d^c$.

}}

The above Kähler identities can be upgraded in the case where the differential operators $d,\; \backslash partial,\; \backslash bar\; \backslash partial$ are paired with a Chern connection on a holomorphic vector bundle $E\; \backslash to\; X$. If $h$ is a Hermitian metric on $E$ and $\backslash bar\; \backslash partial\_E$ is a Dolbeault operator defining the holomorphic structure of $E$, then the unique compatible Chern connection $D\_E$ and its $(1,0)$-part $\backslash partial\_E$ satisfy $D\_E\; =\; \backslash partial\_E\; +\; \backslash bar\backslash partial\_E$. Denote the curvature form of the Chern connection by $F$. The formal adjoints may be defined similarly to above using an $L^2$-inner product where the Hermitian metric is combined with the inner product on forms. In this case all the Kähler identities, sometimes called the Nakano identities,{{Rp|location=Lem 5.2.3}} hold without change, except for the following:Demailly, J.P., 2012. Analytic methods in algebraic geometry (Vol. 1). Somerville, MA: International Press.{{Rp|location=Ch VII §1}}{{Rp|location=§5.1}}

• $[L,\; \backslash Delta\_\{\backslash bar\; \backslash partial\_E\}]\; =\; -\; i\; F\; \backslash wedge\; -$.
• $[L,\; \backslash Delta\_\{\backslash partial\_E\}]\; =\; i\; F\; \backslash wedge\; -$.
• $\backslash Delta\_\{\backslash bar\; \backslash partial\_E\}\; +\; \backslash Delta\_\{\backslash partial\_E\}\; =\; \backslash Delta\_\{D\_E\}$.
• $\backslash Delta\_\{\backslash bar\; \backslash partial\_E\}\; -\; \backslash Delta\_\{\backslash partial\_E\}\; =\; [iF\backslash wedge\; -,\; \backslash Lambda]$, known as the Bochner–Kodaira–Nakano identity.{{Rp|location=Ch VII § 1}}

In particular note that when the Chern connection associated to $(h,\backslash bar\; \backslash partial\_E)$ is a flat connection, so that the curvature $F=0$, one still obtains the relationship that $\backslash Delta\_\{D\_E\}\; =\; 2\; \backslash Delta\_\{\backslash partial\_E\}\; =\; 2\; \backslash Delta\_\{\backslash bar\; \backslash partial\_E\}$.

In addition to the commutation relations contained in the Kähler identities, some of the above operators satisfy other interesting commutation relations. In particular recall the Lefschetz operator $L$, the contraction operator $\backslash Lambda$, and the counting operator $\backslash Pi$ above. Then one can show the following commutation relations:{{Rp|location=Prop 1.2.26}}

• $[\backslash Pi,\; L]\; =\; 2L$.
• $[\backslash Pi,\; \backslash Lambda]\; =\; -2\backslash Lambda$.
• $[L,\; \backslash Lambda]\; =\; \backslash Pi$.

Comparing with the Lie algebra $\backslash mathfrak\{sl\}(2,\backslash mathbb\{C\})$, one sees that $\backslash \{\backslash Pi,\; L,\; \backslash Lambda\backslash \}$ form an sl2-triple, and therefore the algebra $\backslash Omega(X)$ of complex differential forms on a Kähler manifold becomes a representation of $\backslash mathfrak\{sl\}(2,\backslash mathbb\{C\})$. The Kähler identities imply the operators $\backslash Pi,\; L,\; \backslash Lambda$ all commute with $\backslash Delta\_d$ and therefore preserve the harmonic forms inside $\backslash Omega(X)$. In particular when the Kähler manifold is compact, by applying the Hodge decomposition the triple of operators $\backslash \{\backslash Pi,\; L,\; \backslash Lambda\backslash \}$ descend to give an sl2-triple on the de Rham cohomology of X.

In the language of representation theory of $\backslash mathfrak\{sl\}(2,\backslash mathbb\{C\})$, the operator $L$ is the raising operator and $\backslash Lambda$ is the lowering operator. When $X$ is compact, it is a consequence of Hodge theory that the cohomology groups $H^i(X,\backslash mathbb\{C\})$ are finite-dimensional. Therefore the cohomology$$H(X)\; =\; \backslash bigoplus\_\{k=0\}^\{2n\}\; H^i(X,\backslash mathbb\{C\})\; =\; \backslash bigoplus\_\{p,q\backslash ge\; 0\}\; H^\{p,q\}(X)$$admits a direct sum decomposition into irreducible finite-dimensional representations of $\backslash mathfrak\{sl\}(2,\backslash mathbb\{C\})$.Wells, R.O.N. and García-Prada, O., 1980. Differential analysis on complex manifolds (Vol. 21980). New York: Springer.{{Rp|location=Ch V §3}} Any such irreducible representation comes with a primitive element, which is an element $\backslash alpha$ such that $\backslash Lambda\; \backslash alpha\; =\; 0$. The primitive cohomology of $X$ is given by $$P^k(X,\backslash mathbb\{C\})\; =\; \backslash \{\; \backslash alpha\; \backslash in\; H^k(X,\backslash mathbb\{C\})\; \backslash mid\; \backslash Lambda\; \backslash alpha\; =\; 0\backslash \},\; \backslash quad\; P^\{p,q\}(X)\; =\; P^k(X,\backslash mathbb\{C\})\; \backslash cap\; H^\{p,q\}(X).$$The primitive cohomology also admits a direct sum splitting$$P^k(X,\backslash mathbb\{C\})\; =\; \backslash bigoplus\_\{p+q=k\}\; P^\{p,q\}(X).$$

The representation theory of $\backslash mathfrak\{sl\}(2,\backslash mathbb\{C\})$ describes completely an irreducible representation in terms of its primitive element. If $\backslash alpha\backslash in\; P^k(X,\backslash mathbb\{C\})$ is a non-zero primitive element, then since differential forms vanish above dimension $2n$, the chain $\backslash alpha,\; L(\backslash alpha),\; L^2(\backslash alpha),\; \backslash dots$ eventually terminates after finitely many powers of $L$. This defines a finite-dimensional vector space $$V(\backslash alpha)\; =\; \backslash operatorname\{span\}\; \backslash langle\backslash alpha,\; L(\backslash alpha),\; L^2(\backslash alpha),\; \backslash dots\; \backslash rangle$$which has an $\backslash mathfrak\{sl\}(2,\backslash mathbb\{C\})$-action induced from the triple $\backslash \{\backslash Pi,\; L,\; \backslash Lambda\backslash \}$. This is the irreducible representation corresponding to $\backslash alpha$. Applying this simultaneously to each primitive cohomology group, the splitting of cohomology $H(X)$ into its irreducible representations becomes known as the hard Lefschetz decomposition of the compact Kähler manifold.

{{math theorem

| name = Hard Lefschetz decomposition{{rp|location=Thm 5.27}}{{Rp|location=Prop 3.3.13}}{{Rp|location=Ch VI Thm 8.17}}

| math_statement = Let $(X,\backslash omega,J)$ be a compact Kähler manifold. Then the de Rham cohomology of $X$ admits an orthogonal direct sum decomposition

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

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

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

In addition

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

}}

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 $H\_\{dR\}^k(X,E)$ and the Dolbeault cohomology groups $H^\{p,q\}(X,E)$. The triple $\backslash \{\backslash Pi,\; L,\; \backslash Lambda\backslash \}$ still acts as an sl2-triple on the bundle-valued cohomology, and the a version of the Hard Lefschetz decomposition holds in this case.Ballmann, W., 2006. Lectures on Kähler manifolds (Vol. 2). European mathematical society.{{Rp|location=Thm 5.31}}

The Nakano inequalities are a pair of inequalities associated to inner products of harmonic differential forms with the curvature of a Chern connection on a holomorphic vector bundle over a compact Kähler manifold. In particular let $(E,h)$ be a Hermitian holomorphic vector bundle over a compact Kähler manifold $(X,\backslash omega)$, and let $F(h)$ denote the curvature of the associated Chern connection. The Nakano inequalities state that if $\backslash alpha\; \backslash in\; \backslash Omega^\{p,q\}(X)$ is harmonic, that is, $\backslash Delta\_\{\backslash bar\; \backslash partial\}\; \backslash alpha\; =\; 0$, then{{Rp|location=Ch VI Prop 2.5}}

• $i\backslash langle\; \backslash langle\; F(h)\; \backslash wedge\; \backslash Lambda(\backslash alpha),\; \backslash alpha\; \backslash rangle\; \backslash rangle\_\{L^2\}\; \backslash le\; 0$, and
• $i\backslash langle\; \backslash langle\; \backslash Lambda(F(h)\backslash wedge\; \backslash alpha),\; \backslash alpha\; \backslash rangle\; \backslash rangle\_\{L^2\}\; \backslash ge\; 0$.

These inequalities may be proven by applying the Kähler identities coupled to a holomorphic vector bundle as described above. In case where $E=L$ is an ample line bundle, the Chern curvature $iF(h)$ is itself a Kähler metric on $X$. Applying the Nakano inequalities in this case proves the Kodaira–Nakano vanishing theorem for compact Kähler manifolds.

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Category:Complex manifolds

Category:Differential geometry