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
![{\displaystyle \forall z\in B_{\varepsilon }(0):\quad f(z)={\frac {1}{2\pi i))\int _{\partial B_{\varepsilon }(0)}{\frac {f(\xi )}{\xi -z))d\xi +{\frac {1}{2\pi i))\iint _{B_{\varepsilon }(0)}{\frac {\partial f}{\partial {\bar {\xi )))){\frac {d\xi \wedge d{\bar {\xi ))}{\xi -z)).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/536e9946cdffe0991dc7189f8e452057efa857b5)
Lemma (
-Poincaré lemma on the complex plane): Let
be as before and
a smooth form, then
![{\displaystyle {\mathcal {C))^{\infty }(U)\ni g(z):={\frac {1}{2\pi i))\int _{B_{\varepsilon }(0)}{\frac {f(\xi )}{\xi -z))d\xi \wedge d{\bar {\xi ))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f72da2bc12f7aa5ad09aa83654b0be418908139)
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
![{\displaystyle f=f_{1}+f_{2}:=\rho f+(1-\rho )f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e34b2ad0c964c2b7a92f19bfedd7e7fb20517e2f)
and define
![{\displaystyle g_{i}:={\frac {1}{2\pi i))\int _{B_{\varepsilon }(0)}{\frac {f_{i}(\xi )}{\xi -z))d\xi \wedge d{\bar {\xi )).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53a21f3e92144e0f2f18f7f074662c7aba3d6726)
Since
in
then
is clearly well-defined and smooth; we note that
![{\displaystyle {\begin{aligned}g_{1}&={\frac {1}{2\pi i))\int _{B_{\varepsilon }(0)}{\frac {f_{1}(\xi )}{\xi -z))d\xi \wedge d{\bar {\xi ))\\&={\frac {1}{2\pi i))\int _{\mathbb {C} }{\frac {f_{1}(\xi )}{\xi -z))d\xi \wedge d{\bar {\xi ))\\&=\pi ^{-1}\int _{0}^{\infty }\int _{0}^{2\pi }f_{1}(z+re^{i\theta })e^{-i\theta }d\theta dr,\end{aligned))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac4b5ec3d1b592c2bc54ce62871e9e1b8d61ab32)
which is indeed well-defined and smooth, therefore the same is true for
. Now we show that
on
.
![{\displaystyle {\frac {\partial g_{2)){\partial {\bar {z))))={\frac {1}{2\pi i))\int _{B_{\varepsilon }(0)}f_{2}(\xi ){\frac {\partial }{\partial {\bar {z)))){\Big (}{\frac {1}{\xi -z)){\Big )}d\xi \wedge d{\bar {\xi ))=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/439f1d46d28f50048f586b46c02b7294ce2b8107)
since
is holomorphic in
.
![{\displaystyle {\begin{aligned}{\frac {\partial g_{1)){\partial {\bar {z))))=&\pi ^{-1}\int _{\mathbb {C} }{\frac {\partial f_{1}(z+re^{i\theta })}{\partial {\bar {z))))e^{-i\theta }d\theta \wedge dr\\=&\pi ^{-1}\int _{\mathbb {C} }{\Big (}{\frac {\partial f_{1)){\partial {\bar {z)))){\Big )}(z+re^{i\theta })e^{-i\theta }d\theta \wedge dr\\=&{\frac {1}{2\pi i))\iint _{B_{\varepsilon }(0)}{\frac {\partial f_{1)){\partial {\bar {\xi )))){\frac {d\xi \wedge d{\bar {\xi ))}{\xi -z))\end{aligned))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1ca7adc853f5e1b0a8f6e27e5f6caf8681764bf)
applying the generalised Cauchy formula to
we find
![{\displaystyle f_{1}(z)={\frac {1}{2\pi i))\int _{\partial B_{\varepsilon }(0)}{\frac {f_{1}(\xi )}{\xi -z))d\xi +{\frac {1}{2\pi i))\iint _{B_{\varepsilon }(0)}{\frac {\partial f_{1)){\partial {\bar {\xi )))){\frac {d\xi \wedge d{\bar {\xi ))}{\xi -z))={\frac {1}{2\pi i))\iint _{B_{\varepsilon }(0)}{\frac {\partial f_{1)){\partial {\bar {\xi )))){\frac {d\xi \wedge d{\bar {\xi ))}{\xi -z))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2411a545aff3322c9736a0f163825d5ec71430ac)
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.[1][2] We denote with
the open polydisc 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
![{\displaystyle \alpha =\sum _{IJ}\alpha _{IJ}dz_{I}\wedge d{\bar {z))_{J}=\sum _{J}\left(\sum _{I}\alpha _{IJ}dz_{I}\right)_{J}\wedge d{\bar {z))_{J))](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8b1f71e8f58c7efa3c23f8856c661868910ffe2)
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
![{\displaystyle \omega =d{\bar {z))_{k+1}\wedge \psi +\mu ,\qquad \psi ,\mu \in (d{\bar {z))_{1},\dots ,d{\bar {z))_{k}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9b69c459182e73284f94beeafda1d792ac5ad5f)
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
![{\displaystyle \psi _{J}={\frac {\partial g_{J)){\partial {\bar {z))_{k+1))}\quad {\text{on))\quad B_{\varepsilon _{k+1))(0).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9d0317d2b68058045bc590e65a6a250af122a59)
are also holomorphic in
. Define
![{\displaystyle {\tilde {\psi )):=\sum _{J}g_{J}d{\bar {z))_{J))](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7ded9f3190a80258004bda99109c0b1d9db738e)
then
![{\displaystyle {\begin{aligned}\omega -{\bar {\partial )){\tilde {\psi ))&=d{\bar {z))_{k+1}\wedge \psi +\mu -\sum _{J}{\frac {\partial g_{J)){\partial {\bar {z))_{k+1))}d{\bar {z))_{k+1}\wedge d{\bar {z))_{J}+\sum _{j=1}^{k}\sum _{J}{\frac {\partial g_{J)){\partial {\bar {z))_{j))}d{\bar {z))_{j}\wedge d{\bar {z))_{J\setminus \lbrace j\rbrace }\\&=d{\bar {z))_{k+1}\wedge \psi +\mu -d{\bar {z))_{k+1}\wedge \psi +\sum _{j=1}^{k}\sum _{J}{\frac {\partial g_{J)){\partial {\bar {z))_{j))}d{\bar {z))_{j}\wedge d{\bar {z))_{J\setminus \lbrace j\rbrace }\\&=\mu +\sum _{j=1}^{k}\sum _{J}{\frac {\partial g_{J)){\partial {\bar {z))_{j))}d{\bar {z))_{j}\wedge d{\bar {z))_{J\setminus \lbrace j\rbrace }\in (d{\bar {z))_{1},\dots ,d{\bar {z))_{k}),\end{aligned))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ccee6d4baec848dd16a9b3f91395465f9e07b0f0)
therefore we can apply the induction hypothesis to it, there exists
such that
![{\displaystyle \omega -{\bar {\partial )){\tilde {\psi ))={\bar {\partial ))\eta \quad {\text{on))\quad \Delta _{\varepsilon }^{n}(0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9eec50d656765458acbc361c122ff1326eb7a1eb)
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
![{\displaystyle \beta _{i+1}|_{\Delta _{i))=\beta _{i}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e25c8c45abc60c4cbd7618cc0af3f662e47ef13)
We proceed by induction on
: the case when
holds by the previous lemma. Let the claim be true for
and take
with
![{\displaystyle \Delta _{\varepsilon }^{n}(0)=\bigcup _{i=1}^{k+1}\Delta _{i}\quad {\text{and))\quad {\overline {\Delta _{k))}\subset \Delta _{k+1}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f46a25062c2855c2ed1026f12c3298a4a5ab6176)
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:
![{\displaystyle \operatorname {supp} (\rho _{k})\subset U_{k},\qquad \rho |_{V_{k))=1,\qquad \rho _{k}|_{\Delta _{\varepsilon }^{n}(0)\setminus U_{k))=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/adc0741c6ec5177dfca0b407afdf2e4d7e04d399)
Then
is a well-defined smooth form on
which satisfies
![{\displaystyle \beta _{k}=\beta '_{k+1}+{\bar {\partial ))(\gamma _{k}\rho _{k})\quad {\text{on))\quad \Delta _{k},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64f4a72f87893db6ef6853a07f2dd917846f5260)
hence the form
![{\displaystyle \beta _{k+1}:=\beta '_{k+1}+{\bar {\partial ))(\gamma _{k}\rho _{k})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7da81db9539407d517c681dbc6e353e60af6db4)
satisfies
![{\displaystyle {\begin{aligned}\beta _{k+1}|_{\Delta _{k))&=\beta '_{k+1}+{\bar {\partial ))\gamma _{k}=\beta _{k}\\{\bar {\partial ))\beta _{k+1}&={\bar {\partial ))\beta '_{k+1}=\alpha |_{\Delta _{k+1))\end{aligned))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/140fe2ca8f3735adf784185485a357caf74b688e)
Case 2. If instead
we cannot apply the Dolbeault-Grothendieck lemma twice; we take
and
as before, we want to show that
![{\displaystyle \left\|\left.\left({\beta _{i))_{I}-{\beta _{i+1))_{I}\right)\right|_{\Delta _{k-1))\right\|_{\infty }<2^{-i}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0036f0436eff010b33de5d30531f696021ee82a7)
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
![{\displaystyle \alpha |_{\Delta _{k+1))={\bar {\partial ))\beta '_{k+1},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17f91d03615fc5d81dfde50ca87d541c8716be53)
which also satisfies
on
, i.e.
is a holomorphic
-form wherever defined, hence by the Stone–Weierstrass theorem we can write it as
![{\displaystyle \beta _{k}-\beta '_{k+1}=\sum _{|I|=p}(P_{I}+r_{I})dz_{I))](https://wikimedia.org/api/rest_v1/media/math/render/svg/37f3e1db562196f0ca70da7d169039972bfda78c)
where
are polynomials and
![{\displaystyle \left\|r_{I}|_{\Delta _{k-1))\right\|_{\infty }<2^{-k},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94cbef9796e598d5c56ca55157d6b3d5f6ede5d6)
but then the form
![{\displaystyle \beta _{k+1}:=\beta '_{k+1}+\sum _{|I|=p}P_{I}dz_{I))](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5631631f3acccd4f6b53c6670405624e0b3ff58)
satisfies
![{\displaystyle {\begin{aligned}{\bar {\partial ))\beta _{k+1}&={\bar {\partial ))\beta '_{k+1}=\alpha |_{\Delta _{k+1))\\\left\|({\beta _{k))_{I}-{\beta _{k+1))_{I})|_{\Delta _{k-1))\right\|_{\infty }&=\|r_{I}\|_{\infty }<2^{-k}\end{aligned))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b076563cf5c40e65f6af34b8fdd4eccaf5eb5a5c)
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[3] 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,
![{\displaystyle H^{p,q}(M)\cong H^{q}(M,\Omega ^{p})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/837f53d3d5115b4a27107ae3faaf88489f1e38ea)
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
![{\displaystyle H^{p,q}(M,E)\cong H^{q}(M,\Omega ^{p}\otimes E).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0fbf4957225011ed6bb99beffd7b300036d047b5)
A version for logarithmic forms has also been established.[4]
Proof
Let
be the fine sheaf of
forms of type
. Then the
-Poincaré lemma says that the sequence
![{\displaystyle \Omega ^{p,q}{\xrightarrow {\overline {\partial ))}{\mathcal {F))^{p,q+1}{\xrightarrow {\overline {\partial ))}{\mathcal {F))^{p,q+2}{\xrightarrow {\overline {\partial ))}\cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/51f9e81341c25fb396c4b3505c64ddbac9b21a24)
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.