In
complex geometry
In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and c ...
, the
lemma (pronounced ddbar lemma) is a
mathematical lemma about the
de Rham cohomology
In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapte ...
class of a
complex differential form
In mathematics, a complex differential form is a differential form on a manifold (usually a complex manifold) which is permitted to have complex coefficients.
Complex forms have broad applications in differential geometry. On complex manifol ...
. The
-lemma is a result of
Hodge theory
In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every cohom ...
and the
Kähler identities on a
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
Kähler manifold
In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnold ...
. Sometimes it is also known as the
-lemma, due to the use of a related operator
, with the relation between the two operators being
and so
.
Statement
The
lemma asserts that if
is a compact Kähler manifold and
is a complex differential form of bidegree (p,q) (with
) whose class
is zero in de Rham cohomology, then there exists a form
of bidegree (p-1,q-1) such that
where
and
are the
Dolbeault operator
In mathematics, a complex differential form is a differential form on a manifold (usually a complex manifold) which is permitted to have complex number, complex coefficients.
Complex forms have broad applications in differential geometry. On comp ...
s of the
complex manifold
In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic.
The term complex manifold is variously used to mean a com ...
.
ddbar potential
The form
is called the
-potential of
. The inclusion of the factor
ensures that
is a ''real'' differential operator, that is if
is a differential form with real coefficients, then so is
.
This lemma should be compared to the notion of an
exact differential form In mathematics, especially vector calculus and differential topology, a closed form is a differential form ''α'' whose exterior derivative is zero (), and an exact form is a differential form, ''α'', that is the exterior derivative of another ...
in de Rham cohomology. In particular if
is a
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
differential k-form (on any smooth manifold) whose class is zero in de Rham cohomology, then
for some differential (k-1)-form
called the
-potential (or just potential) of
, where
is the
exterior derivative
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The res ...
. Indeed, since the Dolbeault operators sum to give the exterior derivative
and square to give zero
, the
-lemma implies that
, refining the
-potential to the
-potential in the setting of compact Kähler manifolds.
Proof
The
-lemma is a consequence of
Hodge theory
In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every cohom ...
applied to a compact Kähler manifold.
The
Hodge theorem for an elliptic complex may be applied to any of the operators
and respectively to their Laplace operators
. To these operators one can define spaces of harmonic differential forms given by the kernels:
The Hodge decomposition theorem asserts that there are three orthogonal decompositions associated to these spaces of harmonic forms, given by
where
are the
formal adjoint
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...
s of
with respect to the
Riemannian metric
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T ...
of the Kähler manifold, respectively.
These decompositions hold separately on any compact complex manifold. The importance of the manifold being Kähler is that there is a relationship between the Laplacians of
and hence of the orthogonal decompositions above. In particular on a compact Kähler manifold
which implies an orthogonal decomposition
where there are the further relations
relating the spaces of
and
-harmonic forms.
As a result of the above decompositions, one can prove the following lemma.
The proof is as follows.
Let
be a closed (p,q)-form on a compact Kähler manifold
. It follows quickly that (d) implies (a), (b), and (c). Moreover, the orthogonal decompositions above imply that any of (a), (b), or (c) imply (e). Therefore, the main difficulty is to show that (e) implies (d).
To that end, suppose that
is orthogonal to the subspace
. Then
. Since
is
-closed and
, it is also
-closed (that is
). If
where
and
is contained in
then since this sum is from an orthogonal decomposition with respect to the inner product
induced by the Riemannian metric,
or in other words
and
. Thus it is the case that
. This allows us to write
for some differential form
. Applying the Hodge decomposition for
to
,
where
is
-harmonic,
and
. The equality
implies that
is also
-harmonic and therefore
. Thus
. However, since
is
-closed, it is also
-closed. Then using a similar trick to above,
also applying the
Kähler identity that
. Thus
and setting
produces the
-potential.
Local version
A local version of the
-lemma holds and can be proven without the need to appeal to the Hodge decomposition theorem.
It is the analogue of the
Poincaré lemma In mathematics, especially vector calculus and differential topology, a closed form is a differential form ''α'' whose exterior derivative is zero (), and an exact form is a differential form, ''α'', that is the exterior derivative of another diff ...
or
Dolbeault–Grothendieck lemma for the
operator. The local
-lemma holds over any domain on which the aforementioned lemmas hold.
The proof follows quickly from the aforementioned lemmas. Firstly observe that if
is locally of the form
for some
then
because
,
, and
. On the other hand, suppose
is
-closed. Then by the Poincaré lemma there exists an open neighbourhood
of any point
and a form
such that
. Now writing
for
and
note that
and comparing the bidegrees of the forms in
implies that
and
and that
. After possibly shrinking the size of the open neighbourhood
, the Dolbeault–Grothendieck lemma may be applied to
and
(the latter because
) to obtain local forms
such that
and
. Noting then that
this completes the proof as
where
.
Bott–Chern cohomology
The Bott–Chern cohomology is a cohomology theory for compact complex manifolds which depends on the operators
and
, and measures the extent to which the
-lemma fails to hold. In particular when a compact complex manifold is a Kähler manifold, the Bott–Chern cohomology is isomorphic to the
Dolbeault cohomology In mathematics, in particular in algebraic geometry and differential geometry, Dolbeault cohomology (named after Pierre Dolbeault) is an analog of de Rham cohomology for complex manifolds. Let ''M'' be a complex manifold. Then the Dolbeault cohom ...
, but in general it contains more information.
The Bott–Chern cohomology groups of a compact complex manifold
are defined by
Since a differential form which is both
and
-closed is
-closed, there is a natural map
from Bott–Chern cohomology groups to de Rham cohomology groups. There are also maps to the
and
Dolbeault cohomology groups
. When the manifold
satisfies the
-lemma, for example if it is a compact Kähler manifold, then the above maps from Bott–Chern cohomology to Dolbeault cohomology are isomorphisms, and furthermore the map from Bott–Chern cohomology to de Rham cohomology is injective.
As a consequence, there is an isomorphism
whenever
satisfies the
-lemma. In this way, the kernel of the maps above measure the failure of the manifold
to satisfy the lemma, and in particular measure the failure of
to be a Kähler manifold.
Consequences for bidegree (1,1)
The most significant consequence of the
-lemma occurs when the complex differential form has bidegree (1,1). In this case the lemma states that an
exact differential form In mathematics, especially vector calculus and differential topology, a closed form is a differential form ''α'' whose exterior derivative is zero (), and an exact form is a differential form, ''α'', that is the exterior derivative of another ...
has a
-potential given by a smooth function
:
In particular this occurs in the case where
is a Kähler form restricted to a small open subset
of a Kähler manifold (this case follows from the
local version of the lemma), where the aforementioned Poincaré lemma ensures that it is an exact differential form. This leads to the notion of a
Kähler potential Kähler may refer to:
;People
* Alexander Kähler (born 1960), German television journalist
* Birgit Kähler (born 1970), German high jumper
*Erich Kähler (1906–2000), German mathematician
*Heinz Kähler (1905–1974), German art historian and a ...
, a locally defined function which completely specifies the Kähler form. Another important case is when
is the difference of two Kähler forms which are in the same de Rham cohomology class