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
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, 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 lemmaedit
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 lemmaedit
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
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 theoremedit
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,
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
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.
Furthermore we know that is Kähler, and where is the fundamental form associated to the Fubini–Study metric (which is indeed Kähler), therefore and whenever which yields the result.
-lemma, which describes the potential of a -exact differential form in the setting of compact Kähler manifolds.
Footnotesedit
^Serre, Jean-Pierre (1953–1954), "Faisceaux analytiques sur l'espace projectif", Séminaire Henri Cartan, 6 (Talk no. 18): 1–10
^"Calculus on Complex Manifolds". Several Complex Variables and Complex Manifolds II. 1982. pp. 1–64. doi:10.1017/CBO9780511629327.002. ISBN9780521288880.
^In contrast to de Rham cohomology, Dolbeault cohomology is no longer a topological invariant because it depends closely on complex structure.
^Navarro Aznar, Vicente (1987), "Sur la théorie de Hodge–Deligne", Inventiones Mathematicae, 90 (1): 11–76, Bibcode:1987InMat..90...11A, doi:10.1007/bf01389031, S2CID 122772976, Section 8
dolbeault, cohomology, mathematics, particular, algebraic, geometry, differential, geometry, named, after, pierre, dolbeault, analog, rham, cohomology, complex, manifolds, complex, manifold, then, groups, displaystyle, mathbb, depend, pair, integers, realized,. 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 C displaystyle H p q M mathbb C 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 1 Construction of the cohomology groups 2 Dolbeault cohomology of vector bundles 3 Dolbeault Grothendieck lemma 3 1 Proof of Dolbeault Grothendieck lemma 4 Dolbeault s theorem 4 1 Proof 5 Explicit example of calculation 6 See also 7 Footnotes 8 ReferencesConstruction of the cohomology groups editLet Wp 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 Wp q Wp q 1 displaystyle bar partial Omega p q to Omega p q 1 nbsp Since 2 0 displaystyle bar partial 2 0 nbsp this operator has some associated cohomology Specifically define the cohomology to be the quotient space Hp q M C ker Wp q Wp q 1 im Wp q 1 Wp q displaystyle H p q M mathbb C frac ker bar partial Omega p q to Omega p q 1 mathrm im bar partial Omega p q 1 to Omega p q nbsp Dolbeault cohomology of vector bundles editIf E is a holomorphic vector bundle on a complex manifold X then one can define likewise a fine resolution of the sheaf O E displaystyle mathcal O E nbsp of holomorphic sections of E using the Dolbeault operator of E This is therefore a resolution of the sheaf cohomology of O E displaystyle mathcal O E nbsp In particular associated to the holomorphic structure of E displaystyle E nbsp is a Dolbeault operator E G E W0 1 E displaystyle bar partial E Gamma E to Omega 0 1 E nbsp taking sections of E displaystyle E nbsp to 0 1 displaystyle 0 1 nbsp forms with values in E displaystyle E nbsp This satisfies the characteristic Leibniz rule with respect to the Dolbeault operator displaystyle bar partial nbsp on differential forms and is therefore sometimes known as a 0 1 displaystyle 0 1 nbsp connection on E displaystyle E nbsp Therefore in the same way that a connection on a vector bundle can be extended to the exterior covariant derivative the Dolbeault operator of E displaystyle E nbsp can be extended to an operator E Wp q E Wp q 1 E displaystyle bar partial E Omega p q E to Omega p q 1 E nbsp which acts on a section a s Wp q E displaystyle alpha otimes s in Omega p q E nbsp by E a s a s 1 p qa Es displaystyle bar partial E alpha otimes s bar partial alpha otimes s 1 p q alpha wedge bar partial E s nbsp and is extended linearly to any section in Wp q E displaystyle Omega p q E nbsp The Dolbeault operator satisfies the integrability condition E2 0 displaystyle bar partial E 2 0 nbsp and so Dolbeault cohomology with coefficients in E displaystyle E nbsp can be defined as above Hp q X E E ker E Wp q E Wp q 1 E im E Wp q 1 E Wp q E displaystyle H p q X E bar partial E frac ker bar partial E Omega p q E to Omega p q 1 E mathrm im bar partial E Omega p q 1 E to Omega p q E nbsp The Dolbeault cohomology groups do not depend on the choice of Dolbeault operator E displaystyle bar partial E nbsp compatible with the holomorphic structure of E displaystyle E nbsp so are typically denoted by Hp q X E displaystyle H p q X E nbsp dropping the dependence on E displaystyle bar partial E nbsp Dolbeault Grothendieck lemma editIn order to establish the Dolbeault isomorphism we need to prove the Dolbeault Grothendieck lemma or displaystyle bar partial nbsp Poincare lemma First we prove a one dimensional version of the displaystyle bar partial nbsp Poincare lemma we shall use the following generalised form of the Cauchy integral representation for smooth functions Proposition Let Be 0 z C z lt e displaystyle B varepsilon 0 lbrace z in mathbb C mid z lt varepsilon rbrace nbsp the open ball centered in 0 displaystyle 0 nbsp of radius e R gt 0 displaystyle varepsilon in mathbb R gt 0 nbsp Be 0 U displaystyle overline B varepsilon 0 subseteq U nbsp open and f C U displaystyle f in mathcal C infty U nbsp then z Be 0 f z 12pi Be 0 f 3 3 zd3 12pi Be 0 f 3 d3 d3 3 z 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 nbsp Lemma displaystyle bar partial nbsp Poincare lemma on the complex plane Let Be 0 U displaystyle B varepsilon 0 U nbsp be as before and a fdz AC0 1 U displaystyle alpha fd bar z in mathcal A mathbb C 0 1 U nbsp a smooth form then C U g z 12pi Be 0 f 3 3 zd3 d3 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 nbsp satisfies a g displaystyle alpha bar partial g nbsp on Be 0 displaystyle B varepsilon 0 nbsp Proof Our claim is that g displaystyle g nbsp defined above is a well defined smooth function and a fdz g displaystyle alpha f d bar z bar partial g nbsp To show this we choose a point z Be 0 displaystyle z in B varepsilon 0 nbsp and an open neighbourhood z V Be 0 displaystyle z in V subseteq B varepsilon 0 nbsp then we can find a smooth function r Be 0 R displaystyle rho B varepsilon 0 to mathbb R nbsp whose support is compact and lies in Be 0 displaystyle B varepsilon 0 nbsp and r V 1 displaystyle rho V equiv 1 nbsp Then we can write f f1 f2 rf 1 r f displaystyle f f 1 f 2 rho f 1 rho f nbsp and define gi 12pi Be 0 fi 3 3 zd3 d3 displaystyle g i frac 1 2 pi i int B varepsilon 0 frac f i xi xi z d xi wedge d bar xi nbsp Since f2 0 displaystyle f 2 equiv 0 nbsp in V displaystyle V nbsp then g2 displaystyle g 2 nbsp is clearly well defined and smooth we note that g1 12pi Be 0 f1 3 3 zd3 d3 12pi Cf1 3 3 zd3 d3 p 1 0 02pf1 z rei8 e i8d8dr displaystyle begin aligned g 1 amp frac 1 2 pi i int B varepsilon 0 frac f 1 xi xi z d xi wedge d bar xi amp frac 1 2 pi i int mathbb C frac f 1 xi xi z d xi wedge d bar xi amp pi 1 int 0 infty int 0 2 pi f 1 z re i theta e i theta d theta dr end aligned nbsp which is indeed well defined and smooth therefore the same is true for g displaystyle g nbsp Now we show that g a displaystyle bar partial g alpha nbsp on Be 0 displaystyle B varepsilon 0 nbsp g2 z 12pi Be 0 f2 3 z 13 z d3 d3 0 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 nbsp since 3 z 1 displaystyle xi z 1 nbsp is holomorphic in Be 0 V displaystyle B varepsilon 0 setminus V nbsp g1 z p 1 C f1 z rei8 z e i8d8 dr p 1 C f1 z z rei8 e i8d8 dr 12pi Be 0 f1 3 d3 d3 3 z displaystyle begin aligned frac partial g 1 partial bar z amp pi 1 int mathbb C frac partial f 1 z re i theta partial bar z e i theta d theta wedge dr amp 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 amp 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 nbsp applying the generalised Cauchy formula to f1 displaystyle f 1 nbsp we find f1 z 12pi Be 0 f1 3 3 zd3 12pi Be 0 f1 3 d3 d3 3 z 12pi Be 0 f1 3 d3 d3 3 z 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 nbsp since f1 Be 0 0 displaystyle f 1 partial B varepsilon 0 0 nbsp but then f f1 g1 z g z displaystyle f f 1 frac partial g 1 partial bar z frac partial g partial bar z nbsp on V displaystyle V nbsp Since z displaystyle z nbsp was arbitrary the lemma is now proved Proof of Dolbeault Grothendieck lemma edit Now are ready to prove the Dolbeault Grothendieck lemma the proof presented here is due to Grothendieck 1 2 We denote with Den 0 displaystyle Delta varepsilon n 0 nbsp the open polydisc centered in 0 Cn displaystyle 0 in mathbb C n nbsp with radius e R gt 0 displaystyle varepsilon in mathbb R gt 0 nbsp Lemma Dolbeault Grothendieck Let a ACnp q U displaystyle alpha in mathcal A mathbb C n p q U nbsp where Den 0 U displaystyle overline Delta varepsilon n 0 subseteq U nbsp open and q gt 0 displaystyle q gt 0 nbsp such that a 0 displaystyle bar partial alpha 0 nbsp then there exists b ACnp q 1 U displaystyle beta in mathcal A mathbb C n p q 1 U nbsp which satisfies a b displaystyle alpha bar partial beta nbsp on Den 0 displaystyle Delta varepsilon n 0 nbsp Before starting the proof we note that any p q displaystyle p q nbsp form can be written as a IJaIJdzI dz J J IaIJdzI J dz J 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 nbsp for multi indices I J I p J q displaystyle I J I p J q nbsp therefore we can reduce the proof to the case a ACn0 q U displaystyle alpha in mathcal A mathbb C n 0 q U nbsp Proof Let k gt 0 displaystyle k gt 0 nbsp be the smallest index such that a dz 1 dz k displaystyle alpha in d bar z 1 dots d bar z k nbsp in the sheaf of C displaystyle mathcal C infty nbsp modules we proceed by induction on k displaystyle k nbsp For k 0 displaystyle k 0 nbsp we have a 0 displaystyle alpha equiv 0 nbsp since q gt 0 displaystyle q gt 0 nbsp next we suppose that if a dz 1 dz k displaystyle alpha in d bar z 1 dots d bar z k nbsp then there exists b ACn0 q 1 U displaystyle beta in mathcal A mathbb C n 0 q 1 U nbsp such that a b displaystyle alpha bar partial beta nbsp on Den 0 displaystyle Delta varepsilon n 0 nbsp Then suppose w dz 1 dz k 1 displaystyle omega in d bar z 1 dots d bar z k 1 nbsp and observe that we can write w dz k 1 ps m ps m dz 1 dz k displaystyle omega d bar z k 1 wedge psi mu qquad psi mu in d bar z 1 dots d bar z k nbsp Since w displaystyle omega nbsp is displaystyle bar partial nbsp closed it follows that ps m displaystyle psi mu nbsp are holomorphic in variables zk 2 zn displaystyle z k 2 dots z n nbsp and smooth in the remaining ones on the polydisc Den 0 displaystyle Delta varepsilon n 0 nbsp Moreover we can apply the displaystyle bar partial nbsp Poincare lemma to the smooth functions zk 1 psJ z1 zk 1 zn displaystyle z k 1 mapsto psi J z 1 dots z k 1 dots z n nbsp on the open ball Bek 1 0 displaystyle B varepsilon k 1 0 nbsp hence there exist a family of smooth functions gJ displaystyle g J nbsp which satisfy psJ gJ z k 1onBek 1 0 displaystyle psi J frac partial g J partial bar z k 1 quad text on quad B varepsilon k 1 0 nbsp gJ displaystyle g J nbsp are also holomorphic in zk 2 zn displaystyle z k 2 dots z n nbsp Define ps JgJdz J displaystyle tilde psi sum J g J d bar z J nbsp then w ps dz k 1 ps m J gJ z k 1dz k 1 dz J j 1k J gJ z jdz j dz J j dz k 1 ps m dz k 1 ps j 1k J gJ z jdz j dz J j m j 1k J gJ z jdz j dz J j dz 1 dz k displaystyle begin aligned omega bar partial tilde psi amp 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 amp 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 amp 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 nbsp therefore we can apply the induction hypothesis to it there exists h ACn0 q 1 U displaystyle eta in mathcal A mathbb C n 0 q 1 U nbsp such that w ps honDen 0 displaystyle omega bar partial tilde psi bar partial eta quad text on quad Delta varepsilon n 0 nbsp and z h ps displaystyle zeta eta tilde psi nbsp ends the induction step QED The previous lemma can be generalised by admitting polydiscs with ek displaystyle varepsilon k infty nbsp for some of the components of the polyradius Lemma extended Dolbeault Grothendieck If Den 0 displaystyle Delta varepsilon n 0 nbsp is an open polydisc with ek R displaystyle varepsilon k in mathbb R cup lbrace infty rbrace nbsp and q gt 0 displaystyle q gt 0 nbsp then H p q Den 0 0 displaystyle H bar partial p q Delta varepsilon n 0 0 nbsp Proof We consider two cases a ACnp q 1 U q gt 0 displaystyle alpha in mathcal A mathbb C n p q 1 U q gt 0 nbsp and a ACnp 1 U displaystyle alpha in mathcal A mathbb C n p 1 U nbsp Case 1 Let a ACnp q 1 U q gt 0 displaystyle alpha in mathcal A mathbb C n p q 1 U q gt 0 nbsp and we cover Den 0 displaystyle Delta varepsilon n 0 nbsp with polydiscs Di Di 1 displaystyle overline Delta i subset Delta i 1 nbsp then by the Dolbeault Grothendieck lemma we can find forms bi displaystyle beta i nbsp of bidegree p q 1 displaystyle p q 1 nbsp on Di Ui displaystyle overline Delta i subseteq U i nbsp open such that a Di bi displaystyle alpha Delta i bar partial beta i nbsp we want to show that bi 1 Di bi displaystyle beta i 1 Delta i beta i nbsp We proceed by induction on i displaystyle i nbsp the case when i 1 displaystyle i 1 nbsp holds by the previous lemma Let the claim be true for k gt 1 displaystyle k gt 1 nbsp and take Dk 1 displaystyle Delta k 1 nbsp with Den 0 i 1k 1DiandDk Dk 1 displaystyle Delta varepsilon n 0 bigcup i 1 k 1 Delta i quad text and quad overline Delta k subset Delta k 1 nbsp Then we find a p q 1 displaystyle p q 1 nbsp form bk 1 displaystyle beta k 1 nbsp defined in an open neighbourhood of Dk 1 displaystyle overline Delta k 1 nbsp such that a Dk 1 bk 1 displaystyle alpha Delta k 1 bar partial beta k 1 nbsp Let Uk displaystyle U k nbsp be an open neighbourhood of Dk displaystyle overline Delta k nbsp then bk bk 1 0 displaystyle bar partial beta k beta k 1 0 nbsp on Uk displaystyle U k nbsp and we can apply again the Dolbeault Grothendieck lemma to find a p q 2 displaystyle p q 2 nbsp form gk displaystyle gamma k nbsp such that bk bk 1 gk displaystyle beta k beta k 1 bar partial gamma k nbsp on Dk displaystyle Delta k nbsp Now let Vk displaystyle V k nbsp be an open set with Dk Vk Uk displaystyle overline Delta k subset V k subsetneq U k nbsp and rk Den 0 R displaystyle rho k Delta varepsilon n 0 to mathbb R nbsp a smooth function such that supp rk Uk r Vk 1 rk Den 0 Uk 0 displaystyle operatorname supp rho k subset U k qquad rho V k 1 qquad rho k Delta varepsilon n 0 setminus U k 0 nbsp Then rkgk displaystyle rho k gamma k nbsp is a well defined smooth form on Den 0 displaystyle Delta varepsilon n 0 nbsp which satisfies bk bk 1 gkrk onDk displaystyle beta k beta k 1 bar partial gamma k rho k quad text on quad Delta k nbsp hence the form bk 1 bk 1 gkrk displaystyle beta k 1 beta k 1 bar partial gamma k rho k nbsp satisfies bk 1 Dk bk 1 gk bk bk 1 bk 1 a Dk 1 displaystyle begin aligned beta k 1 Delta k amp beta k 1 bar partial gamma k beta k bar partial beta k 1 amp bar partial beta k 1 alpha Delta k 1 end aligned nbsp Case 2 If instead a ACnp 1 U displaystyle alpha in mathcal A mathbb C n p 1 U nbsp we cannot apply the Dolbeault Grothendieck lemma twice we take bi displaystyle beta i nbsp and Di displaystyle Delta i nbsp as before we want to show that biI bi 1I Dk 1 lt 2 i displaystyle left left left beta i I beta i 1 I right right Delta k 1 right infty lt 2 i nbsp Again we proceed by induction on i displaystyle i nbsp for i 1 displaystyle i 1 nbsp the answer is given by the Dolbeault Grothendieck lemma Next we suppose that the claim is true for k gt 1 displaystyle k gt 1 nbsp We take Dk 1 Dk displaystyle Delta k 1 supset overline Delta k nbsp such that Dk 1 Di i 1k displaystyle Delta k 1 cup lbrace Delta i rbrace i 1 k nbsp covers Den 0 displaystyle Delta varepsilon n 0 nbsp then we can find a p 0 displaystyle p 0 nbsp form bk 1 displaystyle beta k 1 nbsp such that a Dk 1 bk 1 displaystyle alpha Delta k 1 bar partial beta k 1 nbsp which also satisfies bk bk 1 0 displaystyle bar partial beta k beta k 1 0 nbsp on Dk displaystyle Delta k nbsp i e bk bk 1 displaystyle beta k beta k 1 nbsp is a holomorphic p 0 displaystyle p 0 nbsp form wherever defined hence by the Stone Weierstrass theorem we can write it as bk bk 1 I p PI rI dzI displaystyle beta k beta k 1 sum I p P I r I dz I nbsp where PI displaystyle P I nbsp are polynomials and rI Dk 1 lt 2 k displaystyle left r I Delta k 1 right infty lt 2 k nbsp but then the form bk 1 bk 1 I pPIdzI displaystyle beta k 1 beta k 1 sum I p P I dz I nbsp satisfies bk 1 bk 1 a Dk 1 bkI bk 1I Dk 1 rI lt 2 k displaystyle begin aligned bar partial beta k 1 amp bar partial beta k 1 alpha Delta k 1 left beta k I beta k 1 I Delta k 1 right infty amp r I infty lt 2 k end aligned nbsp which completes the induction step therefore we have built a sequence bi i N displaystyle lbrace beta i rbrace i in mathbb N nbsp which uniformly converges to some p 0 displaystyle p 0 nbsp form b displaystyle beta nbsp such that a Den 0 b displaystyle alpha Delta varepsilon n 0 bar partial beta nbsp QEDDolbeault s theorem editDolbeault 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 Hp q M Hq M Wp displaystyle H p q M cong H q M Omega p nbsp where Wp displaystyle Omega p nbsp 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 E displaystyle E nbsp Namely one has an isomorphismHp q M E Hq M Wp E displaystyle H p q M E cong H q M Omega p otimes E nbsp A version for logarithmic forms has also been established 4 Proof edit Let Fp q displaystyle mathcal F p q nbsp be the fine sheaf of C displaystyle C infty nbsp forms of type p q displaystyle p q nbsp Then the displaystyle overline partial nbsp Poincare lemma says that the sequence Wp q Fp q 1 Fp q 2 displaystyle Omega p q xrightarrow overline partial mathcal F p q 1 xrightarrow overline partial mathcal F p q 2 xrightarrow overline partial cdots nbsp 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 editThe Dolbeault cohomology of the n displaystyle n nbsp dimensional complex projective space is H p q PCn Cp q0otherwise displaystyle H bar partial p q P mathbb C n begin cases mathbb C amp p q 0 amp text otherwise end cases nbsp We apply the following well known fact from Hodge theory HdRk PCn C p q kH p q PCn displaystyle H rm dR k left P mathbb C n mathbb C right bigoplus p q k H bar partial p q P mathbb C n nbsp because PCn displaystyle P mathbb C n nbsp is a compact Kahler complex manifold Then b2k 1 0 displaystyle b 2k 1 0 nbsp and b2k hk k p q 2k p qhp q 1 displaystyle b 2k h k k sum p q 2k p neq q h p q 1 nbsp Furthermore we know that PCn displaystyle P mathbb C n nbsp is Kahler and 0 wk H k k PCn displaystyle 0 neq omega k in H bar partial k k P mathbb C n nbsp where w displaystyle omega nbsp is the fundamental form associated to the Fubini Study metric which is indeed Kahler therefore hk k 1 displaystyle h k k 1 nbsp and hp q 0 displaystyle h p q 0 nbsp whenever p q displaystyle p neq q nbsp which yields the result See also editSerre duality displaystyle partial bar partial nbsp lemma which describes the potential of a displaystyle bar partial nbsp exact differential form in the setting of compact Kahler manifolds Footnotes edit Serre Jean Pierre 1953 1954 Faisceaux analytiques sur l espace projectif Seminaire Henri Cartan 6 Talk no 18 1 10 Calculus on Complex Manifolds Several Complex Variables and Complex Manifolds II 1982 pp 1 64 doi 10 1017 CBO9780511629327 002 ISBN 9780521288880 In contrast to de Rham cohomology Dolbeault cohomology is no longer a topological invariant because it depends closely on complex structure Navarro Aznar Vicente 1987 Sur la theorie de Hodge Deligne Inventiones Mathematicae 90 1 11 76 Bibcode 1987InMat 90 11A doi 10 1007 bf01389031 S2CID 122772976 Section 8References editDolbeault Pierre 1953 Sur la cohomologie des varietes analytiques complexes Comptes rendus de l Academie des Sciences 236 175 277 Wells Raymond O 1980 Differential Analysis on Complex Manifolds Springer Verlag ISBN 978 0 387 90419 1 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 amp Sons p 832 ISBN 9781118626320 Retrieved from https en wikipedia org w index php title Dolbeault cohomology amp oldid 1157977542, wikipedia, wiki, book, books, library,