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 cohomology groups
depend on a pair of integers ''p'' and ''q'' and are realized as a subquotient of the space of
complex differential forms of degree (''p'',''q'').
Construction of the cohomology groups
Let Ω
''p'',''q'' be the
vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
of complex differential forms of degree (''p'',''q''). In the article on
complex forms, the Dolbeault operator is defined as a differential operator on smooth sections
:
Since
:
this operator has some associated
cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
. Specifically, define the cohomology to be the
quotient space
:
Dolbeault cohomology of vector bundles
If ''E'' is a
holomorphic vector bundle In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold such that the total space is a complex manifold and the projection map is holomorphic. Fundamental examples are the holomorphic tangent bundle of a ...
on a complex manifold ''X'', then one can define likewise a fine
resolution of the sheaf
of holomorphic sections of ''E'', using the
Dolbeault operator of ''E''. This is therefore a resolution of the
sheaf cohomology of
.
In particular associated to the holomorphic structure of
is a Dolbeault operator
taking sections of
to
-forms with values in
. This satisfies the characteristic Leibniz rule with respect to the Dolbeault operator
on differential forms, and is therefore sometimes known as a
-connection on
, Therefore, in the same way that a connection on a vector bundle can be extended to the
exterior covariant derivative
In the mathematical field of differential geometry, the exterior covariant derivative is an extension of the notion of exterior derivative to the setting of a differentiable principal bundle or vector bundle with a connection.
Definition
Let ''G' ...
, the Dolbeault operator of
can be extended to an operator
which acts on a section
by
and is extended linearly to any section in
. The Dolbeault operator satisfies the integrability condition
and so Dolbeault cohomology with coefficients in
can be defined as above:
The Dolbeault cohomology groups do not depend on the choice of Dolbeault operator
compatible with the holomorphic structure of
, so are typically denoted by
dropping the dependence on
.
Dolbeault–Grothendieck lemma
In order to establish the Dolbeault isomorphism we need to prove the Dolbeault–Grothendieck lemma (or
-Poincaré lemma). First we prove a one-dimensional version of the
-Poincaré lemma; we shall use the following generalised form of the
Cauchy integral representation for smooth functions:
Proposition: Let
the open ball centered in
of radius
open and
, then
:
Lemma (
-Poincaré lemma on the complex plane): Let
be as before and
a smooth form, then
:
satisfies
on
''Proof.'' Our claim is that
defined above is a well-defined smooth function and
. To show this we choose a point
and an open neighbourhood
, then we can find a smooth function
whose support is compact and lies in
and
Then we can write
:
and define
:
Since
in
then
is clearly well-defined and smooth; we note that
:
which is indeed well-defined and smooth, therefore the same is true for
. Now we show that
on
.
:
since
is holomorphic in
.
:
applying the generalised Cauchy formula to
we find
:
since
, but then
on
. Since
was arbitrary, the lemma is now proved.
Proof of Dolbeault–Grothendieck lemma
Now are ready to prove the Dolbeault–Grothendieck lemma; the proof presented here is due to
Grothendieck. We denote with
the open
polydisc
In the theory of functions of several complex variables, a branch of mathematics, a polydisc is a Cartesian product of discs.
More specifically, if we denote by D(z,r) the open disc of center ''z'' and radius ''r'' in the complex plane, then a ...
centered in
with radius
.
Lemma (Dolbeault–Grothendieck): Let
where
open and
such that
, then there exists
which satisfies:
on
Before starting the proof we note that any
-form can be written as
:
for multi-indices
, therefore we can reduce the proof to the case
.
''Proof.'' Let
be the smallest index such that
in the sheaf of
-modules, we proceed by induction on
. For
we have
since
; next we suppose that if
then there exists
such that
on
. Then suppose
and observe that we can write
:
Since
is
-closed it follows that
are holomorphic in variables
and smooth in the remaining ones on the polydisc
. Moreover we can apply the
-Poincaré lemma to the smooth functions
on the open ball
, hence there exist a family of smooth functions
which satisfy
:
are also holomorphic in
. Define
:
then
:
therefore we can apply the induction hypothesis to it, there exists
such that
:
and
ends the induction step. QED
:The previous lemma can be generalised by admitting polydiscs with
for some of the components of the polyradius.
Lemma (extended Dolbeault-Grothendieck). If
is an open polydisc with
and
, then
''Proof.'' We consider two cases:
and
.
Case 1. Let
, and we cover
with polydiscs
, then by the Dolbeault–Grothendieck lemma we can find forms
of bidegree
on
open such that
; we want to show that
:
We proceed by induction on
: the case when
holds by the previous lemma. Let the claim be true for
and take
with
:
Then we find a
-form
defined in an open neighbourhood of
such that
. Let
be an open neighbourhood of
then
on
and we can apply again the Dolbeault-Grothendieck lemma to find a
-form
such that
on
. Now, let
be an open set with
and
a smooth function such that:
:
Then
is a well-defined smooth form on
which satisfies
:
hence the form
:
satisfies
:
Case 2. If instead
we cannot apply the Dolbeault-Grothendieck lemma twice; we take
and
as before, we want to show that
:
Again, we proceed by induction on
: for
the answer is given by the Dolbeault-Grothendieck lemma. Next we suppose that the claim is true for
. We take
such that
covers
, then we can find a
-form
such that
:
which also satisfies
on
, i.e.
is a holomorphic
-form wherever defined, hence by the
Stone–Weierstrass theorem we can write it as
:
where
are polynomials and
:
but then the form
:
satisfies
:
which completes the induction step; therefore we have built a sequence
which uniformly converges to some
-form
such that
. QED
Dolbeault's theorem
Dolbeault's theorem is a complex analog of
de Rham's theorem. It asserts that the Dolbeault cohomology is isomorphic to the
sheaf cohomology of the
sheaf
Sheaf may refer to:
* Sheaf (agriculture), a bundle of harvested cereal stems
* Sheaf (mathematics), a mathematical tool
* Sheaf toss, a Scottish sport
* River Sheaf, a tributary of River Don in England
* ''The Sheaf'', a student-run newspaper se ...
of holomorphic differential forms. Specifically,
:
where
is the sheaf of holomorphic ''p'' forms on ''M''.
A version of the Dolbeault theorem also holds for Dolbeault cohomology with coefficients in a holomorphic vector bundle
. Namely one has an isomorphism
A version for
logarithmic form In contexts including complex manifolds and algebraic geometry, a logarithmic differential form is a meromorphic differential form with poles of a certain kind. The concept was introduced by Deligne.
Let ''X'' be a complex manifold, ''D'' ⊂ '' ...
s has also been established.
[, Section 8]
Proof
Let
be the
fine sheaf
In mathematics, injective sheaves of abelian groups are used to construct the resolutions needed to define sheaf cohomology (and other derived functors, such as sheaf Ext).
There is a further group of related concepts applied to sheaves: flabby ( ...
of
forms of type
. Then the
-Poincaré lemma says that the sequence
:
is exact. Like any long exact sequence, this sequence breaks up into short exact sequences. The long exact sequences of cohomology corresponding to these give the result, once one uses that the higher cohomologies of a fine sheaf vanish.
Explicit example of calculation
The Dolbeault cohomology of the
-dimensional
complex projective space
In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a ...
is
:
We apply the following well-known fact from
Hodge theory:
:
because
is a compact
Kähler complex manifold. Then
and
:
Furthermore we know that
is Kähler, and
where
is the fundamental form associated to the
Fubini–Study metric
In mathematics, the Fubini–Study metric is a Kähler metric on projective Hilbert space, that is, on a complex projective space CP''n'' endowed with a Hermitian form. This metric was originally described in 1904 and 1905 by Guido Fubini and Edu ...
(which is indeed Kähler), therefore
and
whenever
which yields the result.
See also
*
Serre duality In algebraic geometry, a branch of mathematics, Serre duality is a duality for the coherent sheaf cohomology of algebraic varieties, proved by Jean-Pierre Serre. The basic version applies to vector bundles on a smooth projective variety, but Al ...
*
-lemma, which describes the potential of a
-exact differential form in the setting of compact
Kähler manifolds.
Footnotes
References
*
*
*
*{{cite book, last1=Griffiths, first1=Phillip, authorlink1=Phillip Griffiths, last2=Harris, first2=Joseph, authorlink2=Joe Harris (mathematician), title=Principles of Algebraic Geometry, date=2014, publisher=John Wiley & Sons, isbn=9781118626320, pages=832
Cohomology theories
Complex manifolds
Hodge theory