L2estimates for the∂-poretaro on complex manifolds
Jean-Pierre Demailly
Universit´edeGrenobleI LaboratoiredeMath´ematiques,UMR5582duCNRS InstitutFourier,BP74,38402Saint-Martind’He`res,France
Abstract.these notes is to describe a powerfulThe main goal of geometric method erential which yields precise existence theorems for solutions of equations∂u=von (pseudoconvex) complex manifolds. The main idea is to combine Hilbert space techniques with a geometric identity known as the Bochner-Kodaira-Nakano identity. The BKN identity relates the complex Laplace operatorsΔ′andΔ′′associated to∂and∂with a suitable curvature tensor. The curvature tensor reﬂects the convexity properties of the manifold, from the viewpoint of complex geometry. In this way, under suitable convexity assumptions, one is able to derive existence theorems for holomorphic functions subject to certain constraints (in the form ofL2estimates). The central ideas go back to Kodaira and Nakano (1954) in the case of compact manifolds, and to Androtti-Vesentini and Ho¨rmander(1965)inthecaseofopenmanifoldswithplurisubharmonicweights.H¨ormander’s estimates can be used for instance to give a quick solution of the Levi problem. They have many other important applications to complex analysis, complex geometry, local algebra and algebraic geometry the last two decades.variants of these estimates have been developped inImportant The ﬁrst ones are theL2estimates of Skoda (1972, 1978), which deal with the problem of solving “Bezout identities”Pfjgj=hwhengjandhare given holomorphic functions and thefj’s are the unknowns. The last ones are theL2estimates of Ohsawa-Takegoshi (1987), which concern the problem of extending a holomorphic function given on a submanifoldY⊂Xto the whole manifoldXtask will be to explain the main techniques leading to a. Our lthree types ofL2-estimates(Ho¨rmander,Skoda,Ohsawa-Takegoshi),andtopresentafewapplications.
Contents
1. Non bounded operators on Hilbert spaces 1 2. Basic concepts of complex analysis in several variables 3 3.Ka¨hlermetricsandK¨ahlermanifolds 12 4. Diﬀerential operators on vector bundles 15 5.OperatorsofKa¨hlergeometryandcommutationidentities 19 6. Connections and curvature 24 7. Bochner-Kodaira-Nakano identity and inequality 27 8.L2estimates for solutions ofd′′qeauitnos- 31 9.SomeapplicationsofHo¨rmander’sL2sematiste 37 10. Further preliminary results of hermitian diﬀerential geometry 45 11. Skoda’sL2estimates for surjective bundle morphisms 53 12. Application of Skoda’sL2estimates to local algebra 59 13. The Ohsawa-TakegoshiL2extension theorem 63 14. Approximation of psh functions by logarithms of holomorphic functions 76 15. Nadel vanishing theorem 78 References 83
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L2estimates for the∂-operator on complex manifolds
1. Non bounded operators on Hilbert spaces
A few preliminary results of functional analysis are needed. LetH1,H2be complex Hilbert spaces. We consider a linear operatorTdeﬁned on a subspace DomT⊂ H1 (called the domain ofT) intoH2. The operatorTis said to bedensely deﬁnedif DomTis dense inH1, andclosedif its graph GrT=(x T x) ;x∈DomT
is closed inH1× H2. Assume now thatTis closed and densely deﬁned. The adjointT⋆ofT(in Von Neumann’s sense) is constructed as follows: DomT⋆is the set ofy∈ H2such that the linear form DomT∋x→−h7T x yi2 is bounded inH1-norm. Since DomTis dense, there exists for everyyin DomT⋆ a unique elementT⋆y∈ H1such thathT x yi2=hx T⋆yi1for allx∈DomT⋆. It is immediate to verify that GrT⋆=Gr(−T)⊥inH1× H2. It follows thatT⋆is closed and that every pair (u v)∈ H1× H2can be written (u v) = (x−T x) + (T⋆y y) x∈DomT y∈DomT⋆
Take in particularu= 0. Then x+T⋆y= 0 v=y−T x=y+T T⋆yhv yi2=kyk22+kT⋆yk12 Ifv∈(DomT⋆)⊥we gethv yi2= 0, thusy and= 0v This implies= 0. (DomT⋆)⊥= 0, henceT⋆is densely deﬁned and our discussion yields
(1.1) Theorem(Von Neumann 1933).IfT:H1H→−2is a closed and densely deﬁned operator, its adjointT⋆is also closed and densely deﬁned and(T⋆)⋆=T. Furthermore, we have the relationKerT⋆= (ImT)⊥and its dual(KerT)⊥= ImT⋆.
Consider now two closed and densely deﬁned operatorsT,S: H1−T→ H2−S→ H3 such thatS◦T= 0. By this, we mean that the rangeT(DomT) is contained in KerS⊂DomSway that there is no problem for deﬁning the composition, in such a S◦T. The starting point of allL2estimates is the following abstract existence theorem.
(1.2) Theorem.There are orthogonal decompositions H2= (KerS∩KerT⋆)⊕ImT⊕ImS⋆ KerS= (KerS∩KerT⋆)⊕ImT
In order thatImT= KerS, it suﬃces that
1. Non bounded operators on Hilbert spaces 3 (13)kT⋆xk21+kSxk32>Ckxk2∀x∈DomS∩DomT⋆ 2 for some constantC >0. In that case, for everyv∈ H2such thatSv= 0, there existsu∈ H1such thatT u=vand kuk216C1kvk22 In particular
ImT= ImT= KerS
ImS⋆= ImS⋆= KerT⋆
Proof.SinceSis closed, the kernel KerSis closed inH2. The relation (KerS)⊥= ImS⋆implies
(14)H2= KerS⊕ImS⋆ and similarlyH2= KerT⋆⊕ImT. However, the assumptionS◦T= 0 shows that ImT⊂KerS, therefore
(15) KerS= (KerS∩KerT⋆)⊕ImT The ﬁrst two equalities in Th. 1.2 are then equivalent to the conjunction of (1.4) and (1.5). Now, under assumption (1.3), we are going to show that the equationT u=v is always solvable ifSv= 0. Letx∈DomT⋆. One can write ′′⋆ x=x′+xwherex′∈KerSandx′′∈(KerS)⊥⊂(ImT)⊥= KerT Sincex x′′∈DomT⋆, we have alsox′∈DomT⋆. We get hv xi2=hv x′i2+hv x′′i2=hv x′i2 becausev∈KerSandx′′∈(KerS)⊥. AsSx′= 0 andT⋆x′′= 0, the Cauchy-Schwarz inequality combined with (1.3) implies |hv xi2|26kvk22kx′k226C1kvk22kT⋆x′k21=C1kvk22kT⋆xk12 This shows that the linear formT⋆X∋x7→−hx vi2is continuous on ImT⋆⊂ H1 with norm6C−12kvk2. By the Hahn-Banach theorem, this form can be extended to a continuous linear form onH1of norm6C−12kvk2, i.e. we can ﬁndu∈ H1 such thatkuk16C−12kvk2and hx vi2=hT⋆x ui1∀x∈DomT⋆ This means thatu∈Dom(T⋆)⋆= DomTandv=T u. We have thus shown that ImT= KerS, in particular ImTis closed. The dual equality ImS⋆= KerT⋆follows by considering the dual pair (S⋆ T⋆).
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L2estimates for the∂-operator on complex manifolds
2. Basic concepts of complex analysis in several variables
For more details on the concepts introduced here, we refer to Thierry Bouche’s lecture notes. LetXbe an-dimensional complex manifold and let (z1 zn) be holomorphic local coordinates on some open set⊂X(we usually think ofas being just an open set inCn). We writezj=xj+ iyjand (21)dzj=dxj+ idyj zj=dxj−idyj (Complex) diﬀerential forms overXcan be deﬁned to be linear combinations Xcα1αℓβ1βmdxα1∧ ∧dxαℓ∧dyβ1∧ ∧dyβm with complex coeﬃcients. Sincedxj=2(dzj+zj) anddyj=2i(dzj−zj), we can rearrange the wedge products as products in the complex linear formsdzj(such that dzj(ξ) =ξj) and the conjugate linear formszj(such thatzj(ξ) =ξj). A (p q)-form is a diﬀerential form of total degreep+qwith complex coeﬃcients, which can be written as (22)u(z) =XuIJ(z)dzI∧zJ |I|=p|J|=q
whereI= (i1 ip) andJ= (j1 jq) are multiindices (arranged in increasing order) and dzI=dzi1∧ ∧dzip zJ=zj1∧ ∧zjq The vector bundle of complex valued (p q)-forms overXwill be denoted byΛpqTX⋆. In this setting, the diﬀerential of aC1functionfcan be expressed as df=16jX∂fyd∂fdj=16jXf∂∂jdzj+z∂f∂jzj 6n∂xjxj+∂yj6nz where ∂∂zfj12=xf∂∂j−if∂y∂j∂∂fzj1=2∂f∂xj+ i∂f∂yj We thus getdf=d′f+d′′f(ordf=∂f+∂fin British-American style), where d′f=16jX6nz∂f∂jdzjresp.d′′f=16jX6nf∂∂zjzj isC-linear (resp. conjugateC-linear). We say thatfis holomorphic ifdfisC-linear, or, in an equivalent way, ifd′′f= 0 (Cauchy-Riemann equation). More generally, the exterior derivativeduof the (p q)-formuis du=|I|p|J|=Xq16k6∂u∂zIJkdzk+∂uIJzdzI∧zJ ∂zkk =n We may therefore writedu=d′u+d′′uwith uniquely deﬁned formsd′uof type (p+ 1 q) andd′′uof type (p q such that+ 1),
2. Basic concepts of complex analysis in several variables
5
′ (23′)d u=X∂zu∂kIJdzk∧dzI∧zJ |I|=p|J|=q16k6n (23′′)d′′u=X∂z∂ukJIzk∧dzI∧zJ |I|=p|J|=q16k6n The operatorsd′′=∂can be viewed as linear diﬀerential operators acting on the bundles of complex (p q)-forms (see§4). As 0 =d2= (d′+d′′)2=d′2+ (d′d′′+d′′d′) +d′′2
where each of the three components are of diﬀerent types, we get the identities (24)d′2= 0 d′′2= 0 d′d′′+d′′d′= 0 Moreover,d′andd′′areconjugate, i.e.,d′u=d′′ufor any (p q)-formuonX. A basic result is the so-called Dolbeault-Grothendieck lemma, which is the complex analogueofthePoincare´lemma.
(2.5) Dolbeault-Grothendieck lemma.Letv=P|J|=qvJzJ,q>1, be a smooth form of bidegree(0 q)on a polydisk=D(0 R) =D(0 R1)× ×D(0 Rn) inCn. Then there is a smooth(0 q−1)-formuonsuch thatd′′u=von.
Proof.We ﬁrst show that a solutionuexists on any smaller polydiskD(0 r)⋐, rj< Rj. Letkbe the smallest integer such that the monomialszJappearing inv only involvez1, ,zk. We prove by induction onkthat the equationd′′u=v can be solved on the polydiskD(0 r). Ifk= 0, thenv= 0 and there is nothing to prove, whilstk=nis the desired result. Now, assume that the result has been settled fork−1, thatvonly involvesz1 zk, and set v=zk∧f+g f=XfJdzJ g=XgJzJ |J|=q−1|J|=q wheref,gonly involvez1, ,zk−1. The assumptiond′′v= 0 implies d′′v=−zk∧d′′f+d′′g= 0 wherezk∧d′′finvolves terms∂fJ∂zℓzk∧zℓ∧zJ,ℓ > k, andd′′gcan only involve one factorzℓwith an indexℓ>k. From this we conclude that∂fJ∂zℓ= 0 forℓ > k. Hence the coeﬃcientsfJare holomorphic inzk+1 zn. Now, let us consider the (0 q−1)-form F=XFJzJ FJ(z) =ψ(zk)fJ(z)⋆kπ1zk |J|=q−1
whereψ(zk) is a cut-oﬀ function with support inD(0 Rk), equal to 1 on some disk D(0 r′k),r′k∈]rk Rk[, and⋆kdenotes a partial convolution with respect tozk. In other words,
6
L2estimates for the∂-operator on complex manifolds Z
FJ(z) =ψ(w)fJ(z1 zk w∈D(0Rj)−1 w zk+1 zn)π(zk1−w)dλ(w) =Zw∈Cψ(zk−w)fJ(z1 zk−1 zk−w zk+1 zn)π1λdw(w)
It follows from diﬀerentiation under integral sign thatFJis a smooth function on which is holomorphic in all variableszk+1 zn. Moreover, as is a fundamental πz solution of∂zinC(that is,∂z(πz) =δ0), we see that ∂∂zkFJ(z) =ψ(zk)fJ(z1 zk−1 zk zk+1 zn) in particular∂zkFJ=fJon some polydiskD(0 r′),rj′∈]rj Rj[. Therefore d′′F=Xkz∂∂fℓJzℓ∧zJ=zk∧f+g1 |J|=q−116ℓ6
whereg1is a (0 q) form which only involvesz1 z k−1. Hence v1:=v−d′′F= (zk∧f+g)−(zk∧f+g1) =g−g1 only involvesz1 zk−1. Asv1is again ad′′-closed form, the induction hypothe-sis applied onD(0 r′) shows that we can ﬁnd a smooth (0 q−1)-formu1onD(0 r) such thatd′′u1=v1. Thereforev=d′′(F+u1) onD(0 r), and we have thus found a solutionu=F+u1onD(0 r)⋐. To conclude the proof, we now show by induction onqthat one can ﬁnd a solutionudeﬁned on all of=D(0 R). SetRν= (R1−2−ν Rn−2−ν). By what we have proved above, there exists a smooth solutionuν∈D(0 R(ν)) of the equationd′′uν=v. Now, ifq= 1, we getd′′(uν+1−uν) = 0 onD(0 R(ν)), i.e., uν+1−uνis holomorphic onD(0 R(ν)). By looking at its Taylor expansion at 0, we get a polynomialPν(equal to the sum of all terms in the Taylor expansion up to a certain degree) such that|uν+1−uν−Pν|62−νonD(0 R(ν−1))⋐D(0 R(ν)). If we seteuν=uν+P1+ +Pν−1, thenueνis a uniform Cauchy sequence on every compact subset ofD(0 R). Sinceeuν+1−euνis holomorphic onD(0 R(ν)), we conclude that the limituis smooth and satisﬁesd′′u=d′′uν=vonD(0 R(ν)) for everyν, QED. Now, ifq>2, the diﬀerenceuν+1−uνisd′′-closed of degree q−1>1 onD(0 R(ν)Hence, by the induction hypothesis, we can ﬁnd a (0). q−2) formwνonD(0 R(ν)) such thatuν+1−uν=d′′wν. If we replace inductivelyuν+1by uν+1−d′′(ψνwν) whereψνis a cut-oﬀ function with support inD(0 R(ν)) equal to 1 onD(0 R(ν−1)we see that we take arrange the sequence so that), uν+1coincides withuνonD(0 R(ν−1)). Hence we get a stationary sequence converging towards a limitusuch thatd′′u=v.
We now introduce the concept of cohomology group. Adiﬀerential complex is a graded moduleK•=Lq∈ZKqover some (commutative) ringR, together with a diﬀerentiald:K•→K•of degree 1, that is, aR-linear map such that d=dq:Kq→Kq+1onKqandd2= 0 (i.e.,dq+1◦dq= 0 for everyq). One deﬁnes thecocycleandcoboundarymodules to be
2. Basic concepts of complex analysis in several variables 7 (26Z)Zq(K•) = Ker(dq:Kq→Kq+1) (26B)Bq(K•) = Im(dq−1:Kq−1→Kq) The assumptiond2= 0 immediately shows thatBq(K•)⊂Zq(K•), and one deﬁnes theq-th cohomology groupofK•to be (27)Hq(K•) =Zq(K•)Bq(K•) A basic example is theDe Rham complexKq=C∞(X ΛqTX⋆) together with the exterior derivatived, deﬁned wheneverXis a smooth diﬀerentiable manifold. Its cohomology groups are denotedHqDR(XR) (resp.HDqR(XC) in the case of complex valued forms) and are called the De Rham cohomology groups ofX. Here, we will be rather concerned with the complex case. IfXis a complexn-dimensional man-ifold, we consider for each integerpﬁxed theDolbeault complex(Kp• d′′) deﬁned byKpq=C∞(X ΛpqT⋆X) together with thed′′-exterior diﬀerential; its cohomology groupsHpq(X) are called the Dolbeault cohomology groups ofX. More generally, let us consider a holomorphic vector bundleE→X. This means that we have a collection of trivializationsE↾Uj≃Uj×Cr,r= rankE, such that the transition ma-tricesgjk(z) are holomorphic. We consider the complexKpq=C∞(X ΛpqTX⋆⊗E) E ofE-valued smooth (p q)-forms with values inE. Again,KpqEpossesses a canon-icald′′-operator. Indeed, ifuis a smooth (p q)-section ofErepresented by forms uj∈C∞(Uj ΛpqT⋆X⊗Cr) over the open setsUj, we have the transition relation uj=gjkuk; this relation impliesd′′uj=gjkd′′uk(sinced′′gjk= 0), hence the collection (d′′uj) deﬁnes a unique global (p q+ 1)-sectiond′′u. By deﬁnition, the Dolbeault cohomology groups ofXwith values inEare (28)Hpq(X E) =Hq(KEp• d′′) An important observation is that the Dolbeault complexKpE•is identical to the Dolbeault complexK0Λp•TX⋆⊗E, thanks to the obvious equality ΛpqT⋆X⊗E=Λ0qT⋆X⊗(ΛpT⋆X⊗E) and the fact thatΛpT⋆Xis itself a holomorphic vector bundle. In particular, we get an equality
(29)Hpq(X E) =H0q(X ΛpTX⋆⊗E) IfX=is an open subset ofCn, the bundleΛpT⋆≃ O⊕(nncthirpmososii)ao direct sum ofncopies of the trivial line bundleO, hence we simply get p Hpq( E) =H0q( E)⊗CΛp(Cn)⋆=H0q( E)⊕(np) In this setting, the Dolbeault-Grothendieck lemma can be restated:
(2.10) Corollary.On every polydiskD(0 R) =D(0 R1)× ×D(0 Rn)⊂Cn , we haveHpq(D(0 R)OD(0R)) = 0for allp>0andq>1. We ﬁnally discuss some basic properties of plurisubharmonic functions. In com-plex geometry, plurisubharmonic functions play exactly the same role as convex
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L2estimates for the∂-operator on complex manifolds
functions do in real (aﬃne) geometry. A functionϕ:→[−∞+∞[ on an open subset⊂Cnis said to becnimoarbhsurilup(usually abbreviated as psh) ifϕis upper semicontinuous and satisﬁes the mean value inequality (211)ϕ(z0)61πϕ(dθ 2πZ02z0+a eiθ) for everya∈Cnsuch that the closed diskz0+Dis contained in(hereDdenotes the unit disk inC).
(2.12) Example.Every convex functionϕonis psh, since convexity implies continuity, and since the convexity inequality ϕ(z0)612ϕ(z0+a eiθ) +ϕ(z0−a eiθ) implies (2.11) by computing the average overθ∈[0 π].
Given a closed (euclidean) ballB(z0 r)⊂, the spherical mean value (σ2n−1r2n−1)−Rz∈S(z0r)ϕ(z)dσ(z) is equal to the average of the mean values com-1 puted on each circlez0+a∂D, whenadescribes the sphereS(0 r). Hence, (2.11) implies the weaker mean value inequality (213)ϕ(z0)61 σ2n−1r2n−1ZS(z0r)ϕ(z)dσ(z) for every ballB(z0 r)⊂, in other words, every psh function is subharmonic (with respect to the Euclidean metric). Notice that (2.13) still implies the apparently weaker inequality (213′)ϕ(z0)v2n1r2nZBϕ(z)dV(z) 6 (z0r) by averaging again over all radii in the range ]0 r[, with respect to the density 2n r2n−1drshow that the mean value properties (2.13) and (2(in fact, one can 13′) are equivalent). As a consequence, we get inclusions
(214)
Conv()⊂Psh()⊂Sh()
where Conv(), Psh(), Sh() are the spaces of convex, psh and subharmonic functions, respectively. Now, ifXis a complex manifold, we say that a function ϕ:X→[−∞+∞[ is psh ifϕis psh on every holomorphic coordinate patch, when viewed as a function of the corresponding coordinates. In fact, Property 2.15 j) below shows that the plurisubharmonicity property does not depend on the choice of complex coordinates; this contrasts with convexity or subharmonicity, which do require an additional linear or riemannian structure, respectively.
(2.15) Basic properties of psh functions. a)For any decreasing sequence of psh functionsϕk∈Psh(X), the limitϕ= limϕk is psh onX.