Kähler identities explained
In complex geometry, the Kähler identities are a collection of identities between operators on a Kähler manifold relating the Dolbeault operators and their adjoints, contraction and wedge operators of the Kähler form, and the Laplacians of the Kähler metric. The Kähler identities combine with results of Hodge theory to produce a number of relations on de Rham and Dolbeault cohomology of compact Kähler manifolds, such as the Lefschetz hyperplane theorem, the hard Lefschetz theorem, the Hodge-Riemann bilinear relations, and the Hodge index theorem. They are also, again combined with Hodge theory, important in proving fundamental analytical results on Kähler manifolds, such as the
-lemma, the Nakano inequalities, and the Kodaira vanishing theorem.
History
The Kähler identities were first proven by W. V. D. Hodge, appearing in his book on harmonic integrals in 1941.[1] The modern notation of
was introduced by
André Weil in the first textbook on Kähler geometry,
Introduction à L’Étude des Variétés Kähleriennes.[2] The operators
A Kähler manifold
admits a large number of operators on its algebra of
complex differential formsbuilt out of the smooth structure (
S), complex structure (
C), and Riemannian structure (
R) of
. 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.
Differential operators
The following operators are differential operators and arise out of the smooth and complex structure of
:
d:\Omegak(X,C)\to\Omegak+1(X,C)
, the
exterior derivative. (
S)
\partial:\Omegap,q(X)\to\Omegap+1,q(X)
, the
-Dolbeault operator. (
C)
\bar\partial:\Omegap,q(X)\to\Omegap,q+1(X)
, the
-Dolbeault operator. (
C)
The Dolbeault operators are related directly to the exterior derivative by the formula
. The characteristic property of the exterior derivative that
then implies
\partial2=\bar\partial2=0
and
\partial\bar\partial=-\bar\partial\partial
.
Some sources make use of the following operator to phrase the Kähler identities.
dc=-
(\partial-\bar\partial):\Omegap,q(X)\to\Omegap+1,q(X) ⊕ \Omegap,q+1(X)
.
[3] (
C)
This operator is useful as the Kähler identities for
can be deduced from the more succinctly phrased identities of
by comparing bidegrees. It is also useful for the property that
ddc=i\partial\bar\partial
. It can be defined in terms of the complex structure operator
by the formula
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
at each point
individually.
\bar ⋅ :\Omegap,q(X)\to\Omegaq,p(X)
, the complex conjugate operator. (
C)
L:\Omegap,q(X)\to\Omegap+1,q+1(X)
, the
Lefschetz operator defined by
L(\alpha):=\omega\wedge\alpha
where
is the Kähler form. (
CR)
\star:\Omegap,q(X)\to\Omegan-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.
\Pik:\Omega(X)\to\Omegak(X,C)
, the projection onto the part of degree k. (
S)
\Pip,q:\Omegak(X,C)\to\Omegap,q(X)
, the projection onto the part of bidegree (p,q). (
C)
\Pi=
(k-n)\Pik:\Omega(X)\to\Omega(X)
, known as the
counting operator.
[4] (
S)
, the complex structure operator on the complex vector space
. (
C)Notice the last operator is the extension of the
almost complex structure
of the Kähler manifold to higher degree complex differential forms, where one recalls that
for a
-form and
for a
-form, so
acts with factor
on a
-form.
Adjoints
The Riemannian metric on
, 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
.
To define the adjoints by integration, note that the Riemannian metric on
, defines an
-
inner product on
according to the formula
where
\langle\alpha,\beta\rangle
is the inner product on the exterior products of the cotangent space of
induced by the Riemannian metric. Using this
-inner product, formal adjoints of any of the above operators (denoted by
) can be defined by the formula
When the Kähler manifold is non-compact, the
-inner product makes formal sense provided at least one of
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
.
[5] d*:\Omegak(X,C)\to\Omegak-1(X,C)
explicitly given by
d*=-\star\circd\circ\star
. (
SR)
\partial*:\Omegap,q(X)\to\Omegap-1,q(X)
explicitly given by
\partial*=-\star\circ\bar\partial\circ\star
. (
CR)
\bar\partial*:\Omegap,q(X)\to\Omegap,q-1(X)
explicitly given by
\bar\partial*=-\star\circ\partial\circ\star
. (
CR)
{dc}*:\Omegak(X,C)\to\Omegak+1(X,C)
explicitly given by
{dc}*=-\star\circdc\circ\star
. (
CR)
L*=Λ:\Omegap,q(X)\to\Omegap-1,q-1(X)
explicitly given by
Λ=\star-1\circL\circ\star
. (
CR)
The last operator, the adjoint of the Lefschetz operator, is known as the contraction operator with the Kähler form
, and is commonly denoted by
.
Laplacians
Built out of the operators and their formal adjoints are a number of Laplace operators corresponding to
and
:
\Deltad:=dd*+d*d:\Omegak(X,C)\to\Omegak(X,C)
, otherwise known as the
Laplace–de Rham operator. (
SR)
\Delta\partial:=\partial\partial*+\partial*\partial:\Omegap,q(X)\to\Omegap,q(X)
. (
CR)
\Delta\bar\partial:=\bar\partial\bar\partial*+\bar\partial*\bar\partial:\Omegap,q(X)\to\Omegap,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
\alpha\in\Omegak(X,R)\subset\Omegak(X,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
acting on complex differential forms. The composition of two real operators is real.
The complex conjugate of the above operators are as follows:
and
.
\overline{(\partial)}=\bar\partial
and
\overline{(\bar\partial)}=\partial
and similarly for
and
.
and
.
.
.
and
.
.
\bar\Delta\partial=\Delta\bar\partial
.
\bar\Delta\bar\partial=\Delta\partial
.
Thus
d,d*,dc,{dc}*,\star,L,Λ,\Deltad
are all real operators. Moreover, in Kähler case,
and
are real. In particular if any of these operators is denoted by
, then the commutator
where
is the complex structure operator above.
The identities
The Kähler identities are a list of commutator relationships between the above operators. Explicitly we denote by
the operator in
\Omega(X)=\Omega\bullet(X,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
and transferred to any Kähler manifold using the key property that the Kähler condition
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
imply the Kähler identities locally on any Kähler manifold.
[6] When the Kähler manifold is compact the identities can be combined with Hodge theory to conclude many results about the cohomology of the manifold.
The above Kähler identities can be upgraded in the case where the differential operators
are paired with a
Chern connection on a holomorphic vector bundle
. If
is a
Hermitian metric on
and
is a Dolbeault operator defining the holomorphic structure of
, then the unique compatible Chern connection
and its
-part
satisfy
DE=\partialE+\bar\partialE
. Denote the
curvature form of the Chern connection by
. The formal adjoints may be defined similarly to above using an
-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, hold without change, except for the following:
[7]
.
.
.
, known as the
Bochner–Kodaira–Nakano identity.
In particular note that when the Chern connection associated to
is a flat connection, so that the curvature
, one still obtains the relationship that
.
Primitive cohomology and representation of sl(2,C)
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
, the contraction operator
, and the counting operator
above. Then one can show the following commutation relations:
.
.
.Comparing with the
Lie algebra
, one sees that
form an
sl2-triple, and therefore the algebra
of complex differential forms on a Kähler manifold becomes a representation of
. The Kähler identities imply the operators
all commute with
and therefore preserve the harmonic forms inside
. In particular when the Kähler manifold is compact, by applying the
Hodge decomposition the triple of operators
descend to give an sl2-triple on the de Rham cohomology of X.
In the language of representation theory of
, the operator
is the
raising operator and
is the
lowering operator. When
is compact, it is a consequence of Hodge theory that the cohomology groups
are finite-dimensional. Therefore the cohomology
admits a direct sum decomposition into
irreducible finite-dimensional representations of
.
[8] Any such irreducible representation comes with a
primitive element, which is an element
such that
. The
primitive cohomology of
is given by
The primitive cohomology also admits a direct sum splitting
Hard Lefschetz decomposition
The representation theory of
describes completely an irreducible representation in terms of its primitive element. If
is a non-zero primitive element, then since differential forms vanish above dimension
, the chain
\alpha,L(\alpha),L2(\alpha),...
eventually terminates after finitely many powers of
. This defines a finite-dimensional vector space
which has an
-action induced from the triple
. This is the irreducible representation corresponding to
. Applying this simultaneously to each primitive cohomology group, the splitting of cohomology
into its irreducible representations becomes known as the
hard Lefschetz decomposition of the compact Kähler manifold.
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
and the
Dolbeault cohomology groups
. The triple
still acts as an sl2-triple on the bundle-valued cohomology, and the a version of the Hard Lefschetz decomposition holds in this case.
[9] Nakano inequalities
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
be a Hermitian holomorphic vector bundle over a compact Kähler manifold
, and let
denote the curvature of the associated Chern connection. The Nakano inequalities state that if
is harmonic, that is,
, then
i\langle\langleF(h)\wedgeΛ(\alpha),\alpha\rangle
\le0
, and
i\langle\langleΛ(F(h)\wedge\alpha),\alpha\rangle
\ge0
.These inequalities may be proven by applying the Kähler identities coupled to a holomorphic vector bundle as described above. In case where
is an
ample line bundle, the Chern curvature
is itself a Kähler metric on
. Applying the Nakano inequalities in this case proves the
Kodaira–Nakano vanishing theorem for compact Kähler manifolds.
Notes and References
- Hodge, W.V.D., 1989. The theory and applications of harmonic integrals. CUP Archive.
- Weil, A., 1958. Introduction à l'étude des variétés kählériennes
- Some sources use the coefficients
, , or just
in the definition of
for notational convenience. With the first convention, the Ricci form of a Kähler metric has the local form
. These conventions change the Kähler identities for
by an appropriate constant.
- Huybrechts, D., 2005. Complex geometry: an introduction (Vol. 78). Berlin: Springer.
- Note that the sign
(see Codifferential) in front of the adjoint
becomes
in all degrees since the dimension
of the complex manifold
is even.
- Griffiths, P. and Harris, J., 2014. Principles of algebraic geometry. John Wiley & Sons.
- Demailly, J.P., 2012. Analytic methods in algebraic geometry (Vol. 1). Somerville, MA: International Press.
- Wells, R.O.N. and García-Prada, O., 1980. Differential analysis on complex manifolds (Vol. 21980). New York: Springer.
- Ballmann, W., 2006. Lectures on Kähler manifolds (Vol. 2). European mathematical society.