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L2 estimates for the operator on complex manifolds

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L2 estimates for the ∂-operator on complex manifolds Jean-Pierre Demailly Universite de Grenoble I Laboratoire de Mathematiques, UMR 5582 du CNRS Institut Fourier, BP74, 38402 Saint-Martin d'Heres, France Abstract. The main goal of these notes is to describe a powerful differential geometric method which yields precise existence theorems for solutions of equations ∂u = v on (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 operators ∆? and ∆?? 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 of L2 estimates). The central ideas go back to Kodaira and Nakano (1954) in the case of compact manifolds, and to Androtti-Vesentini and Hormander (1965) in the case of open manifolds with plurisubharmonic weights. Hormander'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 . . . Important variants of these estimates have been developped in the last two decades.

  • let

  • linear forms

  • differential operators

  • imt ? ?

  • h1 ?h2

  • dzk ?

  • lemma

  • since domt

  • cauchy-riemann equation


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L2estimates for the-poretaro on complex manifolds
Jean-Pierre Demailly
Universit´edeGrenobleI LaboratoiredeMath´ematiques,UMR5582duCNRS InstitutFourier,BP74,38402Saint-MartindHe`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 operatorsΔandΔ′′associated toandwith 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¨ormanders 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 submanifoldYXto 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¨rmandersL2sematiste                                          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
2
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) ;xDomT
is closed inH1× H2. Assume now thatTis closed and densely defined. The adjointTofT(in Von Neumann’s sense) is constructed as follows: DomTis the set ofy∈ H2such that the linear form DomTxh7T x yi2 is bounded inH1-norm. Since DomTis dense, there exists for everyyin DomTa unique elementTy∈ H1such thathT x yi2=hx Tyi1for allxDomT. It is immediate to verify that GrT=Gr(T)inH1× H2. It follows thatTis closed and that every pair (u v)∈ H1× H2can be written (u v) = (xT x) + (Ty y) xDomT yDomT
Take in particularu= 0. Then x+Ty= 0 v=yT x=y+T Tyhv yi2=kyk22+kTyk12Ifv(DomT)we gethv yi2= 0, thusy and= 0v This implies= 0. (DomT)= 0, henceTis densely defined and our discussion yields
(1.1) Theorem(Von Neumann 1933).IfT:H1H2is a closed and densely defined operator, its adjointTis 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: H1T→ H2S→ H3 such thatST= 0. By this, we mean that the rangeT(DomT) is contained in KerSDomSway that there is no problem for defining the composition, in such a ST. The starting point of allL2estimates is the following abstract existence theorem.
(1.2) Theorem.There are orthogonal decompositions H2= (KerSKerT)ImTImSKerS= (KerSKerT)ImT 
In order thatImT= KerS, it suffices that
1. Non bounded operators on Hilbert spaces 3 (13)kTxk21+kSxk32>Ckxk2xDomSDomT2for some constantC >0. In that case, for everyv∈ H2such thatSv= 0, there existsu∈ H1such thatT u=vand kuk216C1kvk22In particular
ImT= ImT= KerS
ImS= ImS= KerT
Proof.SinceSis closed, the kernel KerSis closed inH2. The relation (KerS)= ImSimplies
(14)H2= KerSImSand similarlyH2= KerTImT. However, the assumptionST= 0 shows that ImTKerS, therefore
(15) KerS= (KerSKerT)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. LetxDomT. One can write ′′x=x+xwherexKerSandx′′(KerS)(ImT)= KerT  Sincex x′′DomT, we have alsoxDomT. We get hv xi2=hv xi2+hv x′′i2=hv xi2 becausevKerSandx′′(KerS). AsSx= 0 andTx′′= 0, the Cauchy-Schwarz inequality combined with (1.3) implies |hv xi2|26kvk22kxk226C1kvk22kTxk21=C1kvk22kTxk12This shows that the linear formTXx7hx vi2is continuous on ImT⊂ H1 with norm6C12kvk2. By the Hahn-Banach theorem, this form can be extended to a continuous linear form onH1of norm6C12kvk2, i.e. we can findu∈ H1 such thatkuk16C12kvk2and hx vi2=hTx ui1xDomTThis means thatuDom(T)= DomTandv=T u. We have thus shown that ImT= KerS, in particular ImTis closed. The dual equality ImS= KerTfollows by considering the dual pair (S T).
4
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 setX(we usually think ofas being just an open set inCn). We writezj=xj+ iyjand (21)dzj=dxj+ idyj zj=dxjidyj(Complex) differential forms overXcan be defined to be linear combinations Xcα1αβ1βmdxα1∧    ∧dxαdyβ1∧    ∧dyβm with complex coefficients. Sincedxj=2(dzj+zj) anddyj=2i(dzjzj), 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)dzIzJ|I|=p|J|=q
whereI= (i1     ip) andJ= (j1     jq) are multiindices (arranged in increasing order) and dzI=dzi1∧    ∧dzip zJ=zj1∧    ∧zjqThe vector bundle of complex valued (p q)-forms overXwill be denoted byΛpqTX. In this setting, the differential of aC1functionfcan be expressed as df=16jXfydfdj=16jXfjdzj+zfjzj 6n∂xjxj+∂yj6nz where zfj12=xfjifyjfzj1=2fxj+ ifyjWe thus getdf=df+d′′f(ordf=∂f+∂fin British-American style), where df=16jX6nzfjdzjresp.d′′f=16jX6nfzjzj 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|=Xq16k6uzIJkdzk+∂uIJzdzIzJ∂zkk =n We may therefore writedu=du+d′′uwith uniquely defined formsduof 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=XzukIJdzkdzIzJ|I|=p|J|=q16k6n (23′′)d′′u=XzukJIzkdzIzJ|I|=p|J|=q16k6n 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=d2+ (dd′′+d′′d) +d′′2
where each of the three components are of different types, we get the identities (24)d2= 0 d′′2= 0 dd′′+d′′d= 0Moreover,dandd′′areconjugate, i.e.,du=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 q1)-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 fork1, thatvonly involvesz1     zk, and set v=zkf+g f=XfJdzJ g=XgJzJ |J|=q1|J|=q wheref,gonly involvez1,  ,zk1. The assumptiond′′v= 0 implies d′′v=zkd′′f+d′′g= 0 wherezkd′′finvolves terms∂fJ∂zzkzzJ,ℓ > k, andd′′gcan only involve one factorzwith an index>k. From this we conclude that∂fJ∂z= 0 forℓ > k. Hence the coefficientsfJare holomorphic inzk+1     zn. Now, let us consider the (0 q1)-form F=XFJzJ FJ(z) =ψ(zk)fJ(z)kπ1zk|J|=q1
whereψ(zk) is a cut-off function with support inD(0 Rk), equal to 1 on some disk D(0 rk),rk]rk Rk[, andkdenotes 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 wD(0Rj)1 w zk+1     zn)π(zk1w)(w)=ZwCψ(zkw)fJ(z1     zk1 zkw zk+1     zn)π1λdw(w)
It follows from differentiation under integral sign thatFJis a smooth function onwhich 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     zk1 zk zk+1     zn)in particular∂zkFJ=fJon some polydiskD(0 r),rj]rj Rj[. Therefore d′′F=XkzfJzzJ=zkf+g1 |J|=q1166
whereg1is a (0 q) form which only involvesz1 z    k1. Hence v1:=vd′′F= (zkf+g)(zkf+g1) =gg1 only involvesz1     zk1. Asv1is again ad′′-closed form, the induction hypothe-sis applied onD(0 r) shows that we can find a smooth (0 q1)-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= (R12ν     Rn2ν). By what we have proved above, there exists a smooth solutionuνD(0 R(ν)) of the equationd′′uν=v. Now, ifq= 1, we getd′′(uν+1uν) = 0 onD(0 R(ν)), i.e., uν+1uνis 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+1uνPν|62νonD(0 R(ν1))D(0 R(ν)). If we seteuν=uν+P1+  +Pν1, thenueνis a uniform Cauchy sequence on every compact subset ofD(0 R). Sinceeuν+1euνis 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+1uνisd′′-closed of degree q1>1 onD(0 R(ν)Hence, by the induction hypothesis, we can find a (0).  q2) formwνonD(0 R(ν)) such thatuν+1uν=d′′wν. If we replace inductivelyuν+1by uν+1d′′(ψνwν) whereψνis a cut-off function with support inD(0 R(ν)) equal to 1 onD(0 R(ν1)we see that we take arrange the sequence so that), uν+1coincides withuνonD(0 R(ν1)). 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=LqZKqover some (commutative) ringR, together with a differentiald:KKof degree 1, that is, aR-linear map such that d=dq:KqKq+1onKqandd2= 0 (i.e.,dq+1dq= 0 for everyq). One defines thecocycleandcoboundarymodules to be
2. Basic concepts of complex analysis in several variables 7 (26Z)Zq(K) = Ker(dq:KqKq+1)(26B)Bq(K) = Im(dq1:Kq1Kq)The assumptiond2= 0 immediately shows thatBq(K)Zq(K), and one defines theq-th cohomology groupofKto 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(XR) (resp.HDqR(XC) 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 byKpq=C(X ΛpqTX) together with thed′′-exterior differential; its cohomology groupsHpq(X) are called the Dolbeault cohomology groups ofX. More generally, let us consider a holomorphic vector bundleEX. This means that we have a collection of trivializationsEUjUj×Cr,r= rankE, such that the transition ma-tricesgjk(z) are holomorphic. We consider the complexKpq=C(X ΛpqTXE) 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 ujC(Uj ΛpqTXCr) 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)Hpq(X E) =Hq(KEp d′′)An important observation is that the Dolbeault complexKpEis identical to the Dolbeault complexK0ΛpTX⋆E, thanks to the obvious equality ΛpqTXE=Λ0qTX(ΛpTXE) and the fact thatΛpTXis itself a holomorphic vector bundle. In particular, we get an equality
(29)Hpq(X E) =H0q(X ΛpTXE)IfX=is an open subset ofCn, the bundleΛpT≃ O(nncthirpmososii)ao direct sum ofncopies of the trivial line bundleO, hence we simply get p Hpq( E) =H0q( E)CΛp(Cn)=H0q( 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 haveHpq(D(0 R)OD(0R)) = 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
8
L2estimates for the-operator on complex manifolds
functions do in real (affine) geometry. A functionϕ:[−∞+[ on an open subsetCnis said to becnimoarbhsurilup(usually abbreviated as psh) ifϕis upper semicontinuous and satisfies the mean value inequality (211)ϕ(z0)61πϕ(2πZ02z0+a e) for everyaCnsuch 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 e) +ϕ(z0a e)implies (2.11) by computing the average overθ[0 π].
Given a closed (euclidean) ballB(z0 r), the spherical mean value (σ2n1r2n1)RzS(z0r)ϕ(z)(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 σ2n1r2n1ZS(z0r)ϕ(z)(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 (z0r) by averaging again over all radii in the range ]0 r[, with respect to the density 2n r2n1drshow 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 functionsϕkPsh(X), the limitϕ= limϕk is psh onX.