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 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).
Contents
Construction of the cohomology groups
Dolbeault cohomology of vector bundles
Dolbeault–Grothendieck lemma
Proof of Dolbeault–Grothendieck lemma
Dolbeault's theorem
Proof
Explicit example of calculation
Footnotes
References
Construction of the cohomology groups
Let Ωp,q be the vector bundle 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. Specifically, define the cohomology to be the quotient space
Dolbeault cohomology of vector bundles
If E is a holomorphic vector bundle on a complex manifold X, then one can define likewise a fine resolution of the
sheaf
of holomorphic sections of E. This is therefore a resolution of the sheaf cohomology of
.
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
satisfies
be as before and
a smooth form, then
on
Proof. Our claim is that
we choose a point
defined above is a well-defined smooth function such that
and an open neighbourhood
whose support is compact and lies in
and
is locally -exact. To show this
, then we can find a smooth function
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
since
is holomorphic in
applying the generalised Cauchy formula to
.
we find
on
.
since
, but then
on
. QED
Proof of Dolbeault–Grothendieck lemma
Now are ready to prove the Dolbeault–Grothendieck lemma; the proof presented here is due to Grothendieck[1]. We
denote with
the open polydisc centered in
with radius
Lemma (Dolbeault–Grothendieck): Let
there exists
which satisfies:
Before starting the proof we note that any
for multi-indices
Proof. Let
we have
then there exists
and observe that we can write
. Moreover we can apply the
. Define
-modules, we proceed by
; next we suppose that if
are holomorphic in variables
satisfy
.
in the sheaf of
. Then suppose
on the open ball
are also holomorphic in
, then
-form can be written as
since
on
-closed it follows that
on the polydisc
then
such that
on
be the smallest index such that
such that
is
open and
, therefore we can reduce the proof to the case
induction on . For
Since
where
.
-Poincaré
and smooth in the remaining ones
lemma
to
the
smooth
functions
, hence there exist a family of smooth functions
which
therefore we can apply the induction hypothesis to it, there exists
and
such that
ends the induction step. QED
The previous lemma can be generalised by admitting polydiscs with
the components of the polyradius.
Lemma (extended Dolbeault-Grothendieck). If
is an open polydisc with
Proof. We consider two cases:
and
Case 1. Let
and
, then
.
, and we cover
Grothendieck lemma we can find forms
for some of
with polydiscs
of bidegree
on
, then by the Dolbeault–
open such that
; we want to
show that
We proceed by induction on : the case when
take
and
with
Then we find a
-form
an open neighbourhood of
lemma to find a
and
Then
holds by the previous lemma. Let the claim be true for
defined in an open neighbourhood of
then
-form
on
such that
. Let
be
and we can apply again the Dolbeault-Grothendieck
such that
on
. Now, let
be an open set with
a smooth function such that:
is a well-defined smooth form on
which satisfies
hence the form
satisfies
Case 2. If instead
before, we want to show that
we cannot apply the Dolbeault-Grothendieck lemma twice; we take
and
as
Again, we proceed by induction on : for
suppose that the claim is true for
find a
-form
the answer is given by the Dolbeault-Grothendieck lemma. Next we
. We take
such that
covers
, then we can
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
-form
such that
which uniformly converges to some
. QED
Dolbeault's theorem
Dolbeault's theorem is a complex analog[2] of de Rham's theorem. It asserts that the Dolbeault cohomology is
isomorphic to the sheaf cohomology of the sheaf of holomorphic differential forms. Specifically,
where
is the sheaf of holomorphic p forms on M.
A version for logarithmic forms has also been established.[3]
Proof
Let
be the fine sheaf 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 is
We apply the following well-known fact from Hodge theory:
because
is a compact Kähler complex manifold. Then
Furthermore we know that
is Kähler, and
the Fubini–Study metric (which is indeed Kähler), therefore
and
where
and
is the fundamental form associated to
whenever
which yields the
result.
Footnotes
1. Serre, Jean-Pierre (1953–1954), "Faisceaux analytiques sur l'espace projectif" (http://www.numdam.org/item?id=
SHC_1953-1954__6__A18_0), Séminaire Henri Cartan, 6 (Talk no. 18): 1–10
2. In contrast to de Rham cohomology, Dolbeault cohomology is no longer a topological invariant because it
depends closely on complex structure.
3. Navarro Aznar, Vicente (1987), "Sur la théorie de Hodge–Deligne", Inventiones Mathematicae, 90 (1): 11–76,
doi:10.1007/bf01389031 (https://doi.org/10.1007%2Fbf01389031), Section 8
References
Dolbeault, Pierre (1953). "Sur la cohomologie des variétés analytiques complexes". Comptes rendus de
l'Académie des Sciences. 236: 175–277.
Wells, Raymond O. (1980). Differential Analysis on Complex Manifolds. Springer-Verlag. ISBN 978-0-387-904191.
Gunning, Robert C. (1990). Introduction to Holomorphic Functions of Several Variables, Volume 1. Chapman and
Hall/CRC. p. 198. ISBN 9780534133085.
Griffiths, Phillip; Harris, Joseph (2014). Principles of Algebraic Geometry. John Wiley & Sons. p. 832.
ISBN 9781118626320.
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