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
Hp,q(M,\Complex)
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
\bar{\partial}:\Omegap,q\to\Omegap,q+1
Since
\bar{\partial}2=0
this operator has some associated cohomology. Specifically, define the cohomology to be the quotient space
Hp,q(M,\Complex)=
\ker(\bar\partial:\Omegap,q\to\Omegap,q+1) | |
im(\bar\partial:\Omegap,q-1\to\Omegap,q) |
.
If E is a holomorphic vector bundle on a complex manifold X, then one can define likewise a fine resolution of the sheaf
lO(E)
lO(E)
In particular associated to the holomorphic structure of
E
\bar\partialE:\Gamma(E)\to\Omega0,1(E)
E
(0,1)
E
\bar\partial
(0,1)
E
E
which acts on a section
\alpha ⊗ s\in\Omegap,q(E)
and is extended linearly to any section in
\Omegap,q(E)
\bar
2 | |
\partial | |
E |
=0
E
The Dolbeault cohomology groups do not depend on the choice of Dolbeault operator
\bar\partialE
E
Hp,q(X,E)
\bar\partialE
In order to establish the Dolbeault isomorphism we need to prove the Dolbeault–Grothendieck lemma (or
\bar{\partial}
\bar{\partial}
Proposition: Let
B\varepsilon(0):=\lbracez\in\Complex\mid|z|<\varepsilon\rbrace
0
\varepsilon\in\R>0,
\overline{B\varepsilon(0)}\subseteqU
f\inl{C}infty(U)
\forallz\inB\varepsilon(0): f(z)=
1 | |
2\pii |
\int | |
\partialB\varepsilon(0) |
f(\xi) | d\xi+ | |
\xi-z |
1 | |
2\pii |
\iint | |
B\varepsilon(0) |
\partialf | |
\partial\bar{\xi |
Lemma (
\bar{\partial}
B\varepsilon(0),U
\alpha=f
0,1 | |
d\bar{z}\inl{A} | |
\Complex |
(U)
l{C}infty(U)\nig(z):=
1 | |
2\pii |
\int | |
B\varepsilon(0) |
f(\xi) | |
\xi-z |
d\xi\wedged\bar{\xi}
\alpha=\bar{\partial}g
B\varepsilon(0).
Proof. Our claim is that
g
\alpha=fd\bar{z}=\bar{\partial}g
z\inB\varepsilon(0)
z\inV\subseteqB\varepsilon(0)
\rho:B\varepsilon(0)\to\R
B\varepsilon(0)
\rho|V\equiv1.
f=f1+f2:=\rhof+(1-\rho)f
and define
g | ||||
|
\int | |
B\varepsilon(0) |
fi(\xi) | |
\xi-z |
d\xi\wedged\bar{\xi}.
Since
f2\equiv0
V
g2
\begin{align}
g | ||||
|
\int | |
B\varepsilon(0) |
f1(\xi) | |
\xi-z |
d\xi\wedged\bar{\xi}\\ &=
1 | |
2\pii |
\int\Complex
f1(\xi) | |
\xi-z |
d\xi\wedged\bar{\xi}\\ &=\pi-1
2\pi | |
\int | |
0 |
i\theta | |
f | |
1(z+re |
)e-i\thetad\thetadr, \end{align}
which is indeed well-defined and smooth, therefore the same is true for
g
\bar{\partial}g=\alpha
B\varepsilon(0)
\partialg2 | |
\partial\bar{z |
since
(\xi-z)-1
B\varepsilon(0)\setminusV
\begin{align} | \partialg1 |
\partial\bar{z |
applying the generalised Cauchy formula to
f1
f | ||||
|
\int | |
\partialB\varepsilon(0) |
f1(\xi) | d\xi+ | |
\xi-z |
1 | |
2\pii |
\iint | |
B\varepsilon(0) |
\partialf1 | |
\partial\bar{\xi |
since
f1|
\partialB\varepsilon(0) |
=0
f=f | ||||
|
V
z
Now are ready to prove the Dolbeault–Grothendieck lemma; the proof presented here is due to Grothendieck.[1] We denote with
n(0) | |
\Delta | |
\varepsilon |
0\in\Complexn
\varepsilon\in\R>0
Lemma (Dolbeault–Grothendieck): Let
p,q | |
\alpha\inl{A} | |
\Complexn |
(U)
n(0)} | |
\overline{\Delta | |
\varepsilon |
\subseteqU
q>0
\bar{\partial}\alpha=0
p,q-1 | |
\beta\inl{A} | |
\Complexn |
(U)
\alpha=\bar{\partial}\beta
n(0). | |
\Delta | |
\varepsilon |
Before starting the proof we note that any
(p,q)
\alpha=\sumIJ\alphaIJdzI\wedged\bar{z}J=\sumJ\left(\sumI\alphaIJdzI\right)J\wedged\bar{z}J
for multi-indices
I,J,|I|=p,|J|=q
0,q | |
\alpha\inl{A} | |
\Complexn |
(U)
Proof. Let
k>0
\alpha\in(d\bar{z}1,...,d\bar{z}k)
l{C}infty
k
k=0
\alpha\equiv0
q>0
\alpha\in(d\bar{z}1,...,d\bar{z}k)
0,q-1 | |
\beta\inl{A} | |
\Complexn |
(U)
\alpha=\bar{\partial}\beta
n(0) | |
\Delta | |
\varepsilon |
\omega\in(d\bar{z}1,...,d\bar{z}k+1)
\omega=d\bar{z}k+1\wedge\psi+\mu, \psi,\mu\in(d\bar{z}1,...,d\bar{z}k).
Since
\omega
\bar{\partial}
\psi,\mu
zk+2,...,zn
n(0) | |
\Delta | |
\varepsilon |
\bar{\partial}
zk+1\mapsto\psiJ(z1,...,zk+1,...,zn)
B | |
\varepsilonk+1 |
(0)
gJ
\psi | ||||
|
k+1
gJ
zk+2,...,zn
\tilde{\psi}:=\sumJgJd\bar{z}J
then
\begin{align} \omega-\bar{\partial}\tilde{\psi}&=d\bar{z}k+1\wedge\psi
+\mu-\sum | ||||
|
k+1
therefore we can apply the induction hypothesis to it, there exists
0,q-1 | |
η\inl{A} | |
\Complexn |
(U)
\omega-\bar{\partial}\tilde{\psi}=\bar{\partial}η on
n(0) | |
\Delta | |
\varepsilon |
and
\zeta:=η+\tilde{\psi}
The previous lemma can be generalised by admitting polydiscs with
\varepsilonk=+infty
Lemma (extended Dolbeault-Grothendieck). If
n(0) | |
\Delta | |
\varepsilon |
\varepsilonk\in\R\cup\lbrace+infty\rbrace
q>0
p,q | |
H | |
\bar{\partial |
Proof. We consider two cases:
p,q+1 | |
\alpha\inl{A} | |
\Complexn |
(U),q>0
p,1 | |
\alpha\inl{A} | |
\Complexn |
(U)
Case 1. Let
p,q+1 | |
\alpha\inl{A} | |
\Complexn |
(U),q>0
n(0) | |
\Delta | |
\varepsilon |
\overline{\Deltai}\subset\Deltai+1
\betai
(p,q-1)
\overline{\Deltai}\subseteqUi
\alpha
| | |
\Deltai |
=\bar{\partial}\betai
\betai+1
| | |
\Deltai |
=\betai.
We proceed by induction on
i
i=1
k>1
\Deltak+1
k+1 | |
\Delta | |
i=1 |
\Deltai and \overline{\Deltak}\subset\Deltak+1.
Then we find a
(p,q-1)
\beta'k+1
\overline{\Deltak+1
\alpha| | |
\Deltak+1 |
=\bar{\partial}\betak+1
Uk
\overline{\Deltak}
\bar{\partial}(\betak-\beta'k+1)=0
Uk
(p,q-2)
\gammak
\betak-\beta'k+1=\bar{\partial}\gammak
\Deltak
Vk
\overline{\Deltak}\subsetVk\subsetneqUk
\rhok:
n(0)\to\R | |
\Delta | |
\varepsilon |
\operatorname{supp}(\rhok)\subsetUk,
\rho| | |
Vk |
=1, \rhok|
|
=0.
Then
\rhok\gammak
n(0) | |
\Delta | |
\varepsilon |
\betak=\beta'k+1+\bar{\partial}(\gammak\rhok) on \Deltak,
hence the form
\betak+1:=\beta'k+1+\bar{\partial}(\gammak\rhok)
satisfies
\begin{align} \betak+1
| | |
\Deltak |
&=\beta'k+1+\bar{\partial}\gammak=\betak\\ \bar{\partial}\betak+1&=\bar{\partial}\beta'k+1
=\alpha| | |
\Deltak+1 |
\end{align}
Case 2. If instead
p,1 | |
\alpha\inl{A} | |
\Complexn |
(U),
\betai
\Deltai
\left\|\left.\left({\betai}I-{\betai+1
Again, we proceed by induction on
i
i=1
k>1
\Deltak+1\supset\overline{\Deltak}
\Deltak+1\cup\lbrace\Deltai\rbrace
k | |
i=1 |
n(0) | |
\Delta | |
\varepsilon |
(p,0)
\beta'k+1
\alpha| | |
\Deltak+1 |
=\bar{\partial}\beta'k+1,
which also satisfies
\bar{\partial}(\betak-\beta'k+1)=0
\Deltak
\betak-\beta'k+1
(p,0)
\betak-\beta'k+1=\sum|I|=p(PI+rI)dzI
where
PI
\left\|rI|
\Deltak-1 |
\right
-k | |
\| | |
infty<2 |
,
but then the form
\betak+1:=\beta'k+1+\sum|I|=pPIdzI
satisfies
\begin{align} \bar{\partial}\betak+1&=\bar{\partial}\beta'k+1
=\alpha| | |
\Deltak+1 |
\\ \left\|({\betak}I-{\betak+1
which completes the induction step; therefore we have built a sequence
\lbrace\betai\rbracei\in\N
(p,0)
\beta
\alpha| | |||||||
|
=\bar{\partial}\beta
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,
Hp,q(M)\congHq(M,\Omegap)
where
\Omegap
A version of the Dolbeault theorem also holds for Dolbeault cohomology with coefficients in a holomorphic vector bundle
E
A version for logarithmic forms has also been established.[3]
Let
l{F}p,q
Cinfty
(p,q)
\overline{\partial}
\Omegap,q\xrightarrow{\overline{\partial}}l{F}p,q+1\xrightarrow{\overline{\partial}}l{F}p,q+2\xrightarrow{\overline{\partial}} …
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.
The Dolbeault cohomology of the
n
p,q | |
H | |
\bar{\partial |
We apply the following well-known fact from Hodge theory:
k | |
H | |
\rmdR |
\left
n | |
(P | |
\Complex |
,\Complex\right)=oplusp+q=k
p,q | |
H | |
\bar{\partial |
because
n | |
P | |
\Complex |
b2k+1=0
b2k=hk,k+\sump+q=2k,p\nehp,q=1.
Furthermore we know that
n | |
P | |
\Complex |
0\ne[\omegak]\in
k,k | |
H | |
\bar{\partial |
\omega
hk,k=1
hp,q=0
p\neq,
\partial\bar\partial
\bar\partial
. Raymond O. Wells, Jr.. Differential Analysis on Complex Manifolds . Springer-Verlag . 1980 . 978-0-387-90419-1.
. Gunning. Robert C.. Robert C. Gunning. Introduction to Holomorphic Functions of Several Variables, Volume 1. 1990 . Chapman and Hall/CRC. 9780534133085. 198.