aa r X i v : . [ m a t h . A P ] F e b COMPLEX b -MANIFOLDS GERARDO A. MENDOZA
Abstract.
A complex b -structure on a manifold M with boundary is an in-volutive subbundle b T , M of the complexification of b T M with the propertythat C b T M = b T , M + b T , M as a direct sum; the interior of M is acomplex manifold. The complex b -structure determines an elliptic complex of b -operators and induces a rich structure on the boundary of M . We study thecohomology of the indicial complex of the b -Dolbeault complex. Introduction
A complex b -manifold is a smooth manifold with boundary together with a com-plex b -structure. The latter is a smooth involutive subbundle b T , M of the com-plexification C b T M of Melrose’s b -tangent bundle [5, 6] with the property that C b T M = b T , M + b T , M as a direct sum. Manifolds with complex b -structures generalize the situation thatarises as a result of spherical and certain anisotropic (not complex) blowups ofcomplex manifolds at a discrete set of points or along a complex submanifold, cf.[7, Section 2], [9], as well as (real) blow-ups of complex analytic varieties with onlypoint singularities.The interior of M is a complex manifold. Its ∂ -complex determines a b -ellipticcomplex, the b ∂ -complex, on sections of the exterior powers of the dual of b T , M ,see Section 2. The indicial families D ( σ ) of the b ∂ -operators at a connected compo-nent N of ∂ M give, for each σ , an elliptic complex, see Section 6. Their cohomologyat the various values of σ determine the asymptotics at N of tempered representa-tives of cohomology classes of the b ∂ -complex, in particular of tempered holomorphicfunctions.Each boundary component N of M inherits from b T , M the following objectsin the C ∞ category:(1) an involutive vector subbundle V ⊂ C T N such that V + V = C T N ;(2) a real nowhere vanishing vector field T such that V ∩ V = span C T ;(3) a class βββ of sections of V ∗ ,where the elements of βββ have additional properties, described in (4) below. Thevector bundle V , being involutive, determines a complex of first order differentialoperators D on sections of the exterior powers of V ∗ , elliptic because of the secondproperty in (1) above. To that list add(4) If β ∈ βββ then D β = 0 and ℑh β, T i = −
1, and if β , β ′ ∈ βββ , then β ′ − β = D u with u real-valued. Mathematics Subject Classification.
Primary 32Q99; Secondary 58J10, 32V05.
Key words and phrases.
Complex manifolds, b -tangent bundle, cohomology. These properties, together with the existence of a Hermitian metric on V invariantunder T make N behave in many ways as the circle bundle of a holomorphicline bundle over a compact complex manifold. These analogies are investigated in[10, 11, 12, 13]. The last of these papers contains a detailed account of circle bundlesfrom the perspective of these boundary structures. The paper [8], a predecessor ofthe present one, contains some facts studied here in more detail.The paper is organized as follows. Section 2 deals with the definition of complex b -structure and Section 3 with holomorphic vector bundles over complex b -manifolds(the latter term just means that the b -tangent bundle takes on a primary role overthat of the usual tangent bundle). The associated Dolbeault complexes are definedin these sections accordingly.Section 4 is a careful account of the structure inherited by the boundary.In Section 5 we show that complex b -structures have no formal local invariantsat boundary points. The issue here is that we do not have a Newlander-Nirenbergtheorem that is valid in a neighborhoods of a point of the boundary, so no explicitlocal model for b -manifolds.Section 6 is devoted to general aspects of b -elliptic first order complexes A . Weintroduce here the set spec qb, N ( A ), the boundary spectrum of the complex in degree q at the component N of M , and prove basic properties of the boundary spectrum(assuming that the boundary component N is compact), including some aspectsconcerning Mellin transforms of A -closed forms. Some of these ideas are illustratedusing the b -de Rham complex.Section 7 is a systematic study of the ∂ b -complex of CR structures on N associ-ated with elements of the class βββ . Each β ∈ βββ defines a CR structure, K β = ker β .Assuming that V admits a T -invariant Hermitian metric, we show that there is β ∈ βββ such that the CR structure K β is T -invariant.In Section 8 we assume that V is T -invariant and show that for T -invariantCR structures, a theorem proved in [13] gives that the cohomology spaces of theassociated ∂ b -complex, viewed as the kernel of the Kohn Laplacian at the variousdegrees, split into eigenspaces of − i L T . The eigenvalues of the latter operator arerelated to the indicial spectrum of the b ∂ -complex.In Section 9 we prove a precise theorem on the indicial cohomology and spectrumfor the b ∂ -complex under the assumption that V admits a T -invariant Hermitianmetric.Finally, we have included a very short appendix listing a number of basic defini-tions in connection with b -operators.2. Complex b -structures Let M be a smooth manifold with smooth boundary. An almost CR b -structureon M is a subbundle W of the complexification, C b T M → M of the b -tangentbundle of M (Melrose [5, 6]) such that(2.1) W ∩ W = 0with W = W . If in addition(2.2) W + W = C b T M then we say that W is an almost complex b -structure and write b T , M instead of W and b T , M for its conjugate. As is customary, the adverb “almost” is dropped OMPLEX b -MANIFOLDS 3 if W is involutive. Note that since C ∞ ( M ; b T M ) is a Lie algebra, it makes senseto speak of involutive subbundles of b T M (or its complexification). Definition 2.3.
A complex b -manifold is a manifold together with a complex b -structure.By the Newlander-Nirenberg Theorem [14], the interior of complex b -manifoldis a complex manifold. However, its boundary is not a CR manifold; rather, as weshall see, it naturally carries a family of CR structures parametrized by the definingfunctions of ∂ M in M which are positive in ◦ M .That C ∞ ( M ; b T M ) is a Lie algebra is an immediate consequence of the definitionof the b -tangent bundle, which indeed can be characterized as being a vector bundle b T M → M together with a vector bundle homomorphismev : b T M → T M covering the identity such that the induced mapev ∗ : C ∞ ( M ; b T M ) → C ∞ ( M ; T M )is a C ∞ ( M ; R )-module isomorphism onto the submodule C ∞ tan ( M ; T M ) of smoothvector fields on M which are tangential to the boundary of M . Since C ∞ tan ( M , T M )is closed under Lie brackets, there is an induced Lie bracket on C ∞ ( M ; b T M ) Thehomomorphism ev is an isomorphism over the interior of M , and its restriction tothe boundary,(2.4) ev ∂ M : b T ∂ M M →
T ∂ M is surjective. Its kernel, a fortiori a rank-one bundle, is spanned by a canonicalsection denoted r ∂ r . Here and elsewhere, r refers to any smooth defining functionfor ∂ M in M , by convention positive in the interior of M .Associated with a complex b -structure on M there is a Dolbeault complex. Let b V ,q M denote the q -th exterior power of the dual of b T , M . Then the operator · · · → C ∞ ( M ; b V ,q M ) b ∂ −→ C ∞ ( M ; b V ,q +1 M ) → · · · is define by(2.5) ( q + 1) b ∂φ ( V , . . . , V q ) = q X j =0 V j φ ( V , . . . , ˆ V j , . . . , V q )+ X j
0, we define b ∂ on forms of type ( p, q ) with p > b -de Rham complex, exactly as in Foland and Kohn [2] for GERARDO A. MENDOZA standard complex structures and de Rham complex. The b -de Rham complex, werecall from Melrose [6], is the complex associated with the dual, C b T ∗ M , of C b T M , · · · → C ∞ ( M ; b V r M ) b d −→ C ∞ ( M ; b V r +1 M ) → · · · where b V q M denotes the r -th exterior power of C b T ∗ M . The operators b d aredefined by the same formula as (2.5), now however with the V j ∈ C ∞ ( M ; C b T M ).On functions f we have b df = ev ∗ df. More generally, ev ∗ ◦ d = b d ◦ ev ∗ in any degree. Also,(2.7) b d ( f φ ) = f b dφ + b df ∧ φ for φ ∈ C ∞ ( M ; b V r M ) and f ∈ C ∞ ( M ) . It is convenient to note here that for f ∈ C ∞ ( M ),(2.8) b df vanishes on ∂ M if f does.Now, with the obvious definition,(2.9) b V r M = M p + q = r b V p,q M . Using the special cases b d : C ∞ ( M ; b V , ) → C ∞ ( M ; b V , ) + C ∞ ( M ; b V , ) , b d : C ∞ ( M ; b V , ) → C ∞ ( M ; b V , ) + C ∞ ( M ; b V , ) , consequences of the involutivity of b T , M and its conjugate, one gets b dφ ∈ C ∞ ( M ; b V p +1 ,q M ) ⊕ C ∞ ( M ; b V p,q +1 M ) if φ ∈ C ∞ ( M ; b V p,q M )for general ( p, q ). Let π p,q : b V k M → b V p,q M be the projection according to thedecomposition (2.9), and define b ∂ = π p +1 ,qb d, b ∂ = π q,p +1 b d, so b d = b ∂ + b ∂ . The operators b ∂ are identical to the ∂ -operators over the interior of M and with the previously defined b ∂ operators on (0 , q )-forms, and give a complex(2.10) · · · → C ∞ ( M ; b V p,q M ) b ∂ −→ C ∞ ( M ; b V p,q +1 M ) → · · · for each p . On functions f : M → C ,(2.11) b ∂f = π , b df. The formula(2.7 ′ ) b ∂f φ = b ∂f ∧ φ + f b ∂φ, f ∈ C ∞ ( M ) , φ ∈ C ∞ ( M ; b V p,q M ) , a consequence of (2.7), implies that b ∂ is a first order operator. As a consequenceof (2.8),(2.8 ′ ) b ∂f vanishes on ∂ M if f does.The operators of the b -de Rham complex are first order operators because of(2.7), and (2.8) implies that these are b -operators, see (A.1). Likewise, (2.7 ′ ) and OMPLEX b -MANIFOLDS 5 (2.8 ′ ) imply that in any bidegree, the operator φ r − b ∂ r φ has coefficients smoothup to the boundary, so(2.12) b ∂ ∈ Diff b ( M ; b V p,q M , b V p,q +1 M ) , see (A.1). We also get from these formulas that the b -symbol of b ∂ is(2.13) b σσ ( b ∂ )( ξ )( φ ) = iπ , ( ξ ) ∧ φ, x ∈ M , ξ ∈ b T ∗ x M , φ ∈ b V p,qx M , see (A.2). Since π , is injective on the real b -cotangent bundle (this follows from(2.2)), the complex (2.10) is b -elliptic.3. Holomorphic vector bundles
The notion of holomorphic vector bundle in the b -category is a translation ofthe standard one using connections. Let ρ : F → M be a complex vector bundle.Recall from [6] that a b -connection on F is a linear operator b ∇ : C ∞ ( M ; F ) → C ∞ ( M ; b V M ⊗ F )such that(3.1) b ∇ f φ = f b ∇ φ + b df ⊗ φ for each φ ∈ C ∞ ( M ; F ) and f ∈ C ∞ ( M ). This property automatically makes b ∇ a b -operator.A standard connection ∇ : C ∞ ( M ; F ) → C ∞ ( M ; V M ⊗ F ) determines a b -connection by composition withev ∗ ⊗ I : V M ⊗ F → b V M ⊗ F, but b -connections are more general than standard connections. Indeed, the differ-ence between the latter and the former can be any smooth section of the bundleHom( F, b V M ⊗ F ). A b -connection b ∇ on F arises from a standard connection ifand only if b ∇ r ∂ r = 0 along ∂ M .As in the standard situation, the b -connection b ∇ determines operators(3.2) b ∇ : C ∞ ( M ; b V k M ⊗ F ) → C ∞ ( M ; b V k +1 M ⊗ F )by way of the usual formula translated to the b setting:(3.3) b ∇ ( α ⊗ φ ) = ( − k α ∧ b ∇ φ + b dα ∧ φ, φ ∈ C ∞ ( M ; F ) , α ∈ b V k M . Since b ∇ r α ⊗ φ = r b ∇ ( α ⊗ φ ) + b d r ∧ α ⊗ φ is smooth and vanishes on ∂ M , also b ∇ ∈ Diff b ( M ; b V k M ⊗ F, b V k +1 M ⊗ F ) . The principal b -symbol of (3.2), easily computed using (3.3) and b σσ ( b ∇ )( b df )( φ ) = lim τ →∞ e − iτf τ b ∇ e iτf φ for f ∈ C ∞ ( M ; R ) and φ ∈ C ∞ ( M ; b V k M ⊗ F ), is b σσ ( b ∇ )( ξ )( φ ) = iξ ∧ φ, ξ ∈ b T ∗ x M , φ ∈ b V kx M ⊗ F x , x ∈ M . GERARDO A. MENDOZA
As expected, the connection is called holomorphic if the component in b V , M ⊗ F of the curvature operatorΩ = b ∇ : C ∞ ( M ; F ) → C ∞ ( M ; b V M ⊗ F ) , vanishes. Such a connection gives F the structure of a complex b -manifold. Itscomplex b -structure can be described locally as in the standard situation, as follows.Fix a frame η µ for F and let the ω νµ be the local sections of b V , M such that b ∂η µ = X ν ω νµ ⊗ η ν . Denote by ζ µ the fiber coordinates determined by the frame η µ . Let V , . . . , V n +1 be a frame of b T , M over U , denote by ˜ V j the sections of C b T F over ρ − ( U ) whichproject on the V j and satisfy ˜ V j ζ µ = ˜ V j ζ µ = 0 for all µ , and by ∂ ζ µ the verticalvector fields such that ∂ ζ µ ζ ν = δ νµ and ∂ ζ µ ζ ν = 0. Then the sections(3.4) ˜ V j − X µ,ν ζ µ h ω νµ , V j i ∂ ζ ν , j = 1 , . . . , n + 1 , ∂ ζ ν , ν = 1 , . . . , k of C b T F over ρ − ( U ) form a frame of b T , F . As in the standard situation, theinvolutivity of this subbundle of C b T F is equivalent to the condition on the vanishingof the (0 ,
2) component of the curvature of b ∇ . A vector bundle F → M togetherwith the complex b -structure determined by a choice of holomorphic b -connection(if one exists at all) is a holomorphic vector bundle.The ∂ operator of a holomorphic vector bundle is b ∂ = ( π ,q +1 ⊗ I ) ◦ b ∇ : C ∞ ( M ; b V ,q M ⊗ F ) → C ∞ ( M ; b V ,q +1 M ⊗ F ) . As is the case for standard complex structures, the condition on the curvature of b ∇ implies that these operators form a complex, b -elliptic since b σσ ( b ∂ )( ξ )( φ ) = iπ , ( ξ ) ∧ φ, ξ ∈ b T ∗ x M , φ ∈ b V kx M ⊗ F x , x ∈ M and π , ( ξ ) = 0 for ξ ∈ b T ∗ M if and only if ξ = 0.Also as usual, a b -connection b ∇ on a Hermitian vector bundle F → M withHermitian form h is Hermitian if b dh ( φ, ψ ) = h ( b ∇ φ, ψ ) + h ( φ, b ∇ ψ )for every pair of smooth sections φ , ψ of F . In view of the definition of b d thismeans that for every v ∈ C b T M and sections as above,ev( v ) h ( φ, ψ ) = h ( b ∇ v φ, ψ ) + h ( φ, b ∇ v ψ )On a complex b -manifold M , if an arbitrary connection b ∇ ′ and the Hermitianform h are given for a vector bundle F , holomorphic or not, then there is a unique Hermitian b -connection b ∇ such that π , b ∇ = π , b ∇ ′ . Namely, let η µ be a localorthonormal frame of F , let( π , ⊗ I ) ◦ b ∇ ′ η µ = X ν ω νµ ⊗ η ν , and let b ∇ be the connection defined in the domain of the frame by(3.5) b ∇ η µ = ( ω νµ − ω µν ) ⊗ η ν . OMPLEX b -MANIFOLDS 7 If the matrix of functions Q = [ q µλ ] is unitary and ˜ η λ = P µ q µλ η µ , then( π , ⊗ I ) ◦ b ∇ ′ ˜ η λ = X ν ˜ ω σλ ⊗ ˜ η σ with ˜ ω σλ = X µ q µσ b ∂q µλ + X µ,ν q µσ q νλ ω µν , using (3.1), that Q − = [ q µλ ], and that π , b df = b ∂f . Thus˜ ω σλ − ˜ ω λσ = X µ ( q µσ b ∂q µλ − q µλ b ∂q µσ ) + X µ,ν ( q µσ q νλ ω µν − q µλ q νσ ω µν )= X µ ( b ∂q µλ + b ∂q µλ ) q µσ + X µ,ν q νλ ( ω µν − ω νµ ) q µσ = X µ b dq µλ + q µσ + X µ,ν q νλ ( ω µν − ω νµ ) q µσ using that b ∂f = b ∂f and that P µ q µλ b ∂q µσ = − P µ b ∂q µλ q µσ because P µ q µλ q µσ isconstant, and that b ∂q µλ + b ∂q µλ = b dq µλ . Thus there is a globally defined Hermit-ian connection locally given by (3.5). We leave to the reader to verify that thisconnection is Hermitian. Clearly b ∇ is the unique Hermitian connection such that π , b ∇ = π , b ∇ ′ . When b ∇ ′ is a holomorphic connection, b ∇ is the unique Hermitianholomorphic connection. Lemma 3.6.
The vector bundles b V p, M are holomorphic. We prove this by exhibiting a holomorphic b -connection. Fix an auxiliary Her-mitian metric on b V p, M and pick an orthonormal frame ( η µ ) of b V p, M over someopen set U . Let ω νµ be the unique sections of b V , M such that b ∂η µ = X ν ω νµ ∧ η ν , and let b ∇ be the b -connection defined on U by the formula (3.5). As in the previousparagraph, this gives a globally defined b -connection. That it is holomorphic followsfrom b ∂ω νµ + X λ ω νλ ∧ ω λµ = 0 , a consequence of b ∂ = 0. Evidently, with the identifications b V ,q M ⊗ b V p, M = b V p,q M , π p,q +1 b ∇ is the b ∂ operator in (2.12).4. The boundary a complex b -manifold Suppose that M is a complex b -manifold and N is a component of its boundary.We shall assume N compact, although for the most part this is not necessary.The homomorphism ev : C b T M → C T M is an isomorphism over the interior of M , and its restriction to N maps onto C T N with kernel spanned by r ∂ r . Writeev N : C b T N M → C T N GERARDO A. MENDOZA for this restriction and(4.1) Φ : b T , N M → V for of the restriction of ev N to b T , N M . From (2.1) and the fact that the kernel ofev N is spanned by the real section r ∂ r one obtains that Φ is injective, so its image, V = Φ( b T , N M )is a subbundle of C T N .Since b T , M is involutive, so is V , see [7, Proposition 3.12]. From (2.2) and thefact that ev N maps onto C T N , one obtains that(4.2) V + V = C T N , see [7, Lemma 3.13]. Thus Lemma 4.3. V is an elliptic structure. This just means what we just said: V is involutive and (4.2) holds, see Treves [16,17]; the sum need not be direct. All elliptic structures are locally of the same kind,depending only on the dimension of V ∩ V . This is a result of Nirenberg [15] (seealso H¨ormander [4]) extending the Newlander-Nirenberg theorem. In the case athand,
V ∩ V has rank 1 because of the relationrank C ( V ∩ V ) = 2 rank C V − dim N which holds whenever (4.2) holds.(4.4) Every p ∈ N hs a neighborhood in which there coordinates x , . . . , x n , t such that with z j = x j + m x j + n , the vector fields ∂∂z , . . . , ∂∂z n , ∂∂t span V near p . The function ( z , . . . , z n , t ) is called a hypoanalyticchart (Baouendi, Chang, and Treves [1], Treves [17]).The intersection V ∩ V is, in the case we are discussing, spanned by a canonicalglobally defined real vector field. Namely, let r ∂ r be the canonical section of b T M along N . There is a unique section J r ∂ r of b T M along N such that r ∂ r + iJ r ∂ r isa section of b T , M along N . Then T = ev N ( J r ∂ r )is a nonvanishing real vector field in V ∩ V , (see [8, Lemma 2.1]). Using the isomor-phism (4.1) we have T = Φ( J ( r ∂ r ) − i r ∂ r ) . Because V is involutive, there is yet another complex, this time associated withthe exterior powers of the dual of V :(4.5) · · · → C ∞ ( N ; V q V ∗ ) D −→ C ∞ ( N ; V q +1 V ∗ ) → · · · , where D is defined by the formula (2.5) where now the V j are sections of V . Thecomplex (4.5) is elliptic because of (4.2). For a function f we have D f = ι ∗ df ,where ι ∗ : C T ∗ N → V ∗ is the dual of the inclusion homomorphism ι : V → C T N .For later use we show: Lemma 4.6.
Suppose that N is compact and connected. If ζ : N → C solves D ζ = 0 , then ζ is constant. OMPLEX b -MANIFOLDS 9 Proof.
Let p be an extremal point of | ζ | . Fix a hypoanalytic chart ( z, t ) for V centered at p . Since D ζ = 0, ζ ( z, t ) is independent of t and ∂ z ν ζ = 0. So there is aholomorphic function Z defined in a neighborhood of 0 in C n such that ζ = Z ◦ z .Then | Z | has a maximum at 0, so Z is constant near 0. Therefore ζ is constant,say ζ ( p ) = c , near p . Let C = { p : ζ ( p ) = c } , a closed set. Let p ∈ C . Since p isalso an extremal point of ζ , the above argument gives that ζ is constant near p ,therefore equal to c . Thus C is open, and consequently ζ is constant on N . (cid:3) Since the operators b ∂ : C ∞ ( M , b V ,q M ) → C ∞ ( M , b V ,q +1 M ) are totally char-acteristic, they induce operators b ∂ b : C ∞ ( N , b V ,q N M ) → C ∞ ( M , b V ,q +1 N M ) , see (A.3); these boundary operators define a complex because of (A.4). By way ofthe dual(4.7) Φ ∗ : V ∗ → b V , N M of the isomorphism (4.1) the operators b ∂ b become identical to the operators of the D -complex (4.5): The diagram · · · −−−−→ C ∞ ( N ; V q V ∗ ) D −−−−→ C ∞ ( N ; V q +1 V ∗ ) −−−−→ · · · Φ ∗ y y Φ ∗ · · · −−−−→ C ∞ ( N , b V ,q N M ) b ∂ b −−−−→ C ∞ ( M , b V ,q +1 N M ) −−−−→ · · · is commutative and the vertical arrows are isomorphisms. This can be proved bywriting the b ∂ operators using Cartan’s formula (2.5) for b ∂ and D and comparingthe resulting expressions.Let r : M → R be a smooth defining function for ∂ M , r > M . Then b ∂ r is smooth and vanishes on ∂ M , so b ∂ rr is also a smooth b ∂ -closedsection of b V , M . Thus we get a D -closed element(4.8) β r = [Φ ∗ ] − b ∂ rr ∈ C ∞ ( ∂ M ; V ∗ ) . By definition, h β r , T i = h b ∂ rr , J ( r ∂ r ) − i r ∂ r i . Extend the section r ∂ r to a section of b T M over a neighborhood U of N in M withthe property that r ∂ r r = r . In U we have h b ∂ r , J ( r ∂ r ) − i r ∂ r i = ( J ( r ∂ r ) − i r ∂ r ) r = J ( r ∂ r ) r − i r . The function J ( r ∂ r ) r is smooth, real-valued, and vanishes along the boundary. So r − J ( r ∂ r ) r is smooth, real-valued. Thus h β r , T i = a r − i on N for some smooth function a r : N → R , see [8, Lemma 2.5].If r ′ is another defining function for ∂ M , then r ′ = r e u for some smooth function u : M → R . Then b ∂ r ′ = e u b ∂ r + e u r b ∂u and it follows that β r ′ = β r + D u. In particular, a r ′ = a r + T u. Let a t denote the one-parameter group of diffeomorphisms generated by T . Proposition 4.9.
The functions a supav , a infav : N → R defined by a supav ( p ) = lim sup t →∞ t Z t − t a r ( a s ( p )) ds, a infav ( p ) = lim inf t →∞ t Z t − t a r ( a s ( p )) ds are invariants of the complex b -structure, that is, they are independent of the defin-ing function r . The equality a supav = a infav holds for some r if and only if it holds forall r . Indeed,lim t →∞ (cid:16) t Z t − t a r ′ ( a s ( p )) ds − t Z t − t a r ( a s ( p )) ds (cid:17) = lim t →∞ t Z t − t dds u ( a s ( p )) ds = 0because u is bounded (since N is compact).The functions a supav , a infav are constant on orbits of T , but they may not be smooth. Example 4.10.
Let N be the unit sphere in C n +1 centered at the origin. Write( z , . . . , z n +1 ) for the standard coordinates in C n +1 . Fix τ , . . . , τ n +1 ∈ R \
0, all ofthe same sign, and let T = i n +1 X j =1 τ j ( z j ∂ z j − z j ∂ z j ) . This vector field is real and tangent to N . Let K be the standard CR structure of N as a submanifold of C n +1 (the part of T , C n +1 tangential to N ). The conditionthat the τ j are different from 0 and have the same sign ensures that T is never in K ⊕ K . Indeed, the latter subbundle of C T N is the annihilator of the pullback to N of i∂ P n +1 ℓ =1 | z ℓ | . The pairing of this form with T is h i n +1 X ℓ =1 z ℓ dz ℓ , i n +1 X j =1 τ j ( z j ∂ z j − z j ∂ z j ) i = n +1 X j =1 τ j | z j | , a function that vanishes nowhere if and only if all τ j are different from zero and havethe same sign. Thus V = K ⊕ span C T is a subbundle of C T N of rank n + 1 with theproperty that V + V = C T N . To show that V is involutive we first note that K isthe annihilator of the pullback to N of the span of the differentials dz , . . . , dz n +1 .Let L T denote the Lie derivative with respect to T . Then L T dz j = iτ j dz j , so if L isa CR vector field, then so is [ L, T ]. Since in addition K and span C T are themselvesinvolutive, V is involutive. Thus V is an elliptic structure with V ∩V = span C T . Let β be the section of V ∗ which vanishes on K and satisfies h β, T i = − i . Let D denotethe operators of the associated differential complex. Then D β = 0, since β vanisheson commutators of sections of K (since K is involutive) and on commutators of T with sections of K (since such commutators are in K ).If the τ j are positive (negative), this example may be viewed as the boundary ofa blowup (compactification) of C n +1 , see [9]. OMPLEX b -MANIFOLDS 11 Let now ρ : F → M be a holomorphic vector bundle. Its b ∂ -complex alsodetermines a complex along N ,(4.11) · · · → C ∞ ( N ; V q V ∗ ⊗ F N ) D −→ C ∞ ( N ; V q +1 V ∗ ⊗ F N ) → · · · , where D is defined using the boundary operators b ∂ b and the isomorphism (4.7):(4.12) D ( φ ⊗ η ) = (Φ ∗ ) − b ∂ b [Φ ∗ ( φ ⊗ η )]where Φ ∗ means Φ ∗ ⊗ I . These operators can be expressed locally in terms of theoperators of the complex (4.5). Fix a smooth frame η µ , µ = 1 , . . . , k , of F in aneighborhood U ⊂ M of p ∈ N , and suppose b ∂η µ = X ν ω νµ ⊗ η ν . The ω νµ are local sections of b V , M , and if P µ φ µ ⊗ η µ is a section of b V ,q M ⊗ F over U , then b ∂ X φ µ ⊗ η µ = X ν ( b ∂φ ν + X µ ω νµ ∧ φ µ ) ⊗ η ν . Therefore, using the identification (4.7), the boundary operator b ∂ b is the operatorgiven locally by(4.13) D X φ µ ⊗ η µ = X ν ( D φ ν + X µ ω νµ ∧ φ µ ) ⊗ η ν where now the φ µ are sections of b V q V ∗ , the ω νµ are the sections of V ∗ correspondingto the original ω νµ via Φ ∗ , and D on the right hand side of the formula is the operatorassociated with V .The structure bundle b T , F is locally given as the span of the sections (3.4). Ap-plying the evaluation homomorphism C b T ∂F F → C T ∂F (over N ) to these sectionsgives vector fields on F N forming a frame for the elliptic structure V F inherited by F N . Writing V j = ev V j , this frame is just(4.14) ˜ V j − X µ,ν ζ µ h ω νµ , V j i ∂ ζ ν , j = 1 , . . . , n + 1 , ∂ ζ ν , ν = 1 , . . . , k, where now the ω νµ are the forms associated to the D operator of F N . Alternatively,one may take the D operators of F N and use the formula (4.13) to define a subbundleof C T F locally as the span of the vector fields (4.14), a fortiori an elliptic structureon F N , involutive because D ω ν + X λ ω νλ ∧ ω λµ = 0 . To obtain a formula for the canonical real vector field T F in V F , let J F be thealmost complex b -structure of b T F and consider again the sections (3.4); they aredefined in an open set ρ − ( U ), U a neighborhood in M of a point of N . Since theelements ∂ ζ ν are sections of b T , F ,(4.15) J F ℜ ∂ ζ ν = ℑ ∂ ζ ν . Pick a defining function r for N . Then ˜ r = ρ ∗ r is a defining function for F N .We may take V n +1 = r ∂ r + iJ r ∂ r along U ∩ N . Then ˜ V n +1 = ˜ r ∂ ˜ r + i g J r ∂ r along ρ − ( U ) ∩ F N and so J F ℜ (cid:0) ˜ r ∂ ˜ r + i g J r ∂ r − X µ,ν ζ µ h ω νµ , r ∂ r + iJ r ∂ r i ∂ ζ ν (cid:1) = ℑ (cid:0) ˜ r ∂ ˜ r + i g J r ∂ r − X µ,ν ζ µ h ω νµ , r ∂ r + iJ r ∂ r i ∂ ζ ν (cid:1) along ρ − ( U ) ∩ F N . Using (4.15) this gives J F ˜ r ∂ ˜ r = g J r ∂ r − ℑ X µ,ν ζ µ h ω νµ , r ∂ r + iJ r ∂ r i ∂ ζ ν . Applying the evaluation homomorphism gives(4.16) T F = ˜ T − ℑ X µ,ν ζ µ h ω νµ , r ∂ r + iJ r ∂ r i ∂ ζ ν where ˜ T is the real vector field on ρ − ( U ∩ N ) = ρ − ( U ) ∩ F N which projects on T and satisfies ˜ T ζ µ = 0 for all µ .Let h be a Hermitian metric on F , and suppose that the frame η µ is orthonormal.Applying T E as given in (4.16) to the function | ζ | = P | ζ µ | we get that T F istangent to the unit sphere bundle of F if and only if h ω νµ , r ∂ r + iJ r ∂ r i − h ω µν , r ∂ r + iJ r ∂ r i = 0for all µ, ν . Equivalently, in terms of the isomorphism (4.7),(4.17) h (Φ ∗ ) − ω νµ , T i + h (Φ ∗ ) − ω µν , T i = 0 for all µ, ν.
Definition 4.18.
The Hermitian metric h will be called exact if (4.17) holds.The terminology in Definition 4.18 is taken from the notion of exact Riemannian b -metric of Melrose [6, pg. 31]. For such metrics, the Levi-Civita b -connectionhas the property that b ∇ r ∂ r = 0 [op. cit., pg. 58]. We proceed to show that theHermitian holomorphic connection of an exact Hermitian metric on F also has thisproperty. Namely, suppose that h is an exact Hermitian metric, and let η µ be anorthonormal frame of F . Then for the Hermitian holomorphic connection we have h ω νµ − ω µν , r ∂ r i = h ω νµ , r ∂ r i − h ω µν , r ∂ r i = 12 (cid:0) h ω νµ , r ∂ r + iJ r ∂ r i − h ω µν , r ∂ r + iJ r ∂ r i (cid:1) using that the ω νµ are of type (0 , b ∇ r ∂ r = 0.5. Local invariants
Complex structures have no local invariants: every point of a complex n -manifoldhas a neighborhood biholomorphic to a ball in C n It is natural to ask the samequestion about complex b -structures, namely,is there a local model depending only on dimension for every complex b -stucture?In lieu of a Newlander-Nirenberg theorem, we show that complex b -structures haveno local formal invariants at the boundary. More precisely: OMPLEX b -MANIFOLDS 13 Proposition 5.1.
Every p ∈ N has a neighborhood V in M on which there aresmooth coordinates x j , j = 1 , . . . , n + 2 centered at p with x n +1 vanishing on V ∩ N such that with (5.2) L j = 12 ( ∂ x j + i∂ x j + n +1 ) , j ≤ n, L n +1 = 12 ( x n +1 ∂ x n +1 + i∂ x n +2 ) there are smooth functions γ jk vanishing to infinite order on V ∩ N such that L j = L j + n +1 X k =1 γ kj L k is a frame for b T , M over V . The proof will require some preparation. Let r : M → R be a defining functionfor ∂ M . Let p ∈ N , pick a hypoanalytic chart ( z, t ) (cf. (4.4)) centered at p with T t = 1. Let U ⊂ N be a neighborhood of p contained in the domain of the chart,mapped by it to B × ( − δ, δ ) ⊂ C n × R , where B is a ball with center 0 and δ issome small positive number. For reference purposes we state Lemma 5.3.
On such U , the problem D φ = ψ, ψ ∈ C ∞ ( U ; V q +1 V ∗ | U ) and D ψ = 0 has a solution in C ∞ ( U ; V q V ∗ | U ) . Extend the functions z j and t to a neighborhood of p in M . Shrinking U ifnecessary, we may assume that in some neighborhood V of p in M with V ∩ ∂ M = U , ( z, t, r ) maps V diffeomorphically onto B × ( − δ, δ ) × [0 , ε ) for some δ , ε > β r defined in (4.8) is D -closed, there is α ∈ C ∞ ( U ) such that − i D α = β r . Extend α to V as a smooth function. The section(5.4) b ∂ (log r + iα ) = b ∂ rr + i b ∂α of b V , M over V vanishes on U , since β r + i D α = 0. So there is a smooth section φ of b V , M over V such that b ∂ (log r + iα ) = r e iα φ. Suppose ζ : U → C is a solution of D ζ = 0 on U , and extend it to V . Then b ∂ζ vanishes on U , so again we have b ∂ζ = r e iα ψ. for some smooth section ψ of b V , M over V . The following lemma will be appliedfor f equal to log r + iα or each of the functions z j . Lemma 5.5.
Let f be smooth in V \ U and suppose that b ∂f = r e iα ψ with ψ smooth on V . Then there is f : V → C smooth vanishing at U such that b ∂ ( f + f ) vanishes to infinite order on U .Proof. Suppose that f , . . . , f N − are defined on V and that(5.6) b ∂ N − X k =0 ( r e iα ) k f k = ( r e iα ) N ψ N holds with ψ N smooth in V ; by the hypothesis, (5.6) holds when N = 1. Using(5.4) we get that b ∂ ( r e iα ) = ( r e iα ) φ , therefore0 = b ∂ (cid:0) ( r e iα ) N ψ N ) = ( r e iα ) N [ b ∂ψ N + N r e iα φ ∧ ψ N ] , which implies that b ∂ψ N = 0 on U . With arbitrary f N we have b ∂ N X k =0 ( r e iα ) k f k = ( r e iα ) N ( ψ N + b ∂f N + N r e iα f N φ ) . Since D ψ N = 0 and H D ( U ) = 0 by Lemma 5.3, there is a smooth function f N definedin U such that D f N = − ψ N in U . So there is χ N such that ψ N + b ∂f N = r e iα χ N .With such f N , (5.6) holds with N + 1 in place of N and some ψ N +1 . Thus there isa sequence { f j } ∞ j =1 such that (5.6) holds for each N . Borel’s lemma then gives f smooth with f ∼ ∞ X k =1 ( r e iα ) k f k on U such that D ( f + f ) vanishes to infinite order on U . (cid:3) Proof of Proposition 5.1.
Apply the lemma with f = log r + iα to get a function f such that b ∂ ( f + f ) vanishes to infinite order at U . Let x n +1 = r e −ℑ α + ℜ f , x n +2 = ℜ α + ℑ f. These functions are smooth up to U .Applying the lemma to each of the functions f = z j , j = 1 , . . . , n gives smoothfunctions ζ j such that ζ j = z j on U and b ∂ζ j = 0 to infinite order at U . Define x j = ℜ ζ j , x j + n +1 = ℑ ζ j , j = 1 , . . . , n. The functions x j , j = 1 . . . , n + 2 are independent, and the forms η j = b dζ j , j = 1 . . . , n, η n +1 = 1 x n +1 e ix n +2 b d [ x n +1 e ix n +2 ]together with their conjugates form a frame for C b T M near p . Let η j , and η j , be the (1 ,
0) and (0 ,
1) components of η j according to the complex b -structure of M . Then η j , = X k p jk η k + q jk η k . Since η j , = b ∂ζ j vanishes to infinite order at U , the coefficients p jk and q jk vanishto infinite order at U . Replacing this formula for η j , in η j = η j , + η j , get X k ( δ jk − p jk ) η k − X k q jk η k = η j , . The matrix I − [ p jk ] is invertible with inverse of the form I + [ P jk ] with P jk vanishingto infinite order at U . So(5.7) η j − X k γ jk η k = X k ( δ jk + P jk ) η k , b -MANIFOLDS 15 with suitable γ jk vanishing to infinite order on U . Define the vector fields L j as in(5.2). The vector fields L j = L j + X k γ kj L k , j = 1 , . . . , n + 1are independent and since h L j , η k i = 0 and h L j , η k i = δ kj , they annihilate each ofthe forms on the left hand side of (5.7). So they annihilate the forms η k , , whichproves that the L j form a frame of b T , M . (cid:3) Indicial complexes
Throughout this section we assume that N is a connected component of theboundary of a compact manifold M . Let(6.1) · · · → C ∞ ( M ; E q ) A q −−→ C ∞ ( M ; E q +1 ) → · · · be a b -elliptic complex of operators A q ∈ Diff b ( M ; E q , E q +1 ); the E q , q = 0 , . . . , r ,are vector bundles over M .Note that since A q is a first order operator,(6.2) A q ( f φ ) = f A q φ − i b σσ ( A q )( b df )( φ ) . This formula follows from the analogous formula for the standard principal symboland the definition of principal b -symbol. It follows from (6.2) and (2.8) that A q defines an operator A b,q : Diff ( N ; E q N , E q +1 N ) . Fix a smooth defining function r : M → R for ∂ M , r > M , let A q ( σ ) : Diff b ( N ; E q N , E q +1 N ) , σ ∈ C denote the indicial family of A q with respect to r , see (A.5). Using (6.2) and definingΛ r ,q = b σσ ( A q )( b d rr ) , the indicial family of A q with respect to r is(6.3) A q ( σ ) = A b,q + σ Λ r ,q : C ∞ ( N ; E q N ) → C ∞ ( N ; E q +1 N ) . Because of (A.4), these operators form an elliptic complex(6.4) · · · → C ∞ ( N ; E q N ) A q ( σ ) −−−−→ C ∞ ( N ; E q +1 N ) → · · · for each σ and each connected component N of ∂ M . The operators depend on r , but the cohomology groups at a given σ for different defining functions r areisomorphic. Indeed, if r ′ is another defining function for ∂ M , then r ′ = e u r forsome smooth real-valued function u , and a simple calculation gives( A b,q + σ Λ r ,q )( e iσu φ ) = e iσu ( A b,q + σ Λ r ′ ,q ) φ. In analogy with the definition of boundary spectrum of an elliptic operator A ∈ Diff mb ( M ; E, F ), we have
Definition 6.5.
Let N be a connected component of ∂ M . The family of complexes(6.4), σ ∈ C , is the indicial complex of (6.1) at N . For each σ ∈ C let H q A ( σ ) ( N )denote the q -th cohomology group of (6.4) on N . The q -th boundary spectrum ofthe complex (6.1) at N is the setspec qb, N ( A ) = { σ ∈ C : H q A ( σ ) ( N ) = 0 } . The q -th boundary spectrum of A is spec qb ( A ) = S N spec qb, N ( A ).The spaces H q A ( σ ) ( N ) are finite-dimensional because (6.4) is an elliptic complexand N is compact. It is convenient to isolate the behavior of the indicial complexaccording to the components of the boundary, since the sets spec qb, N ( A ) can varydrastically from component to component.Suppose that M is a complex b -manifold. Recall that since b ∂ ∈ Diff b ( M ; b V ,q M , b V ,q +1 M ) , there are induced boundary operators b ∂ b ∈ Diff ( N ; b V ,q N M , b V ,q +1 N M )which via the isomorphism (4.1) become the operators of the D -complex (4.5).Combining (2.11) and (2.13) we get b σσ ( b ∂ )( b d rr )( φ ) = i b ∂ rr ∧ φ and using (4.8) we may identify c b ∂ b ( σ ), given by (6.3), with the operator(6.6) D ( σ ) φ = D φ + iσβ r ∧ φ. If E → M is a holomorphic vector bundle, then the indicial family of b ∂ ∈ Diff b ( M ; b V ,q M ⊗ E, b V ,q +1 M ⊗ E )is again given by (6.6), but using the operator D of the complex (4.11).Returning to the general complex (6.1), fix a smooth positive b -density m on M and a Hermitian metric on each E q . Let A ⋆q ( σ ) be the indicial operator ofthe formal adjoint, A ⋆q , of A q . The Laplacian (cid:3) q of the complex (6.1) in degree q belongs to Diff b ( M ; E q M ), is b -elliptic, and its indicial operator is b (cid:3) q ( σ ) = A ⋆q ( σ ) A q ( σ ) + A q − ( σ ) A ⋆q − ( σ ) . The b -spectrum of (cid:3) q at N , see Melrose [6], is the setspec b, N ( (cid:3) q ) = { σ ∈ C : b (cid:3) q ( σ ) : C ∞ ( N ; E q N ) → C ∞ ( N ; E q N ) is not invertible } . Note that unless σ is real, b (cid:3) q ( σ ) is not the Laplacian of the complex (6.4). Proposition 6.7.
For each q , spec qb, N ( A ) ⊂ spec b, N ( (cid:3) q ) . Note that the set spec b, N ( (cid:3) q ) depends on the choice of Hermitian metrics and b -density used to construct the Laplacian, but that the subset spec qb, N ( A ) is inde-pendent of such choices. For a general b -elliptic complex (6.1) it may occur thatspec qb, N ( A ) = spec b, N ( (cid:3) q ). In Example 6.13 we show that spec qb, N ( b d ) ⊂ { } . Asis well known, spec b, N (∆ q ) is an infinite set if dim M >
1. At the end of this
OMPLEX b -MANIFOLDS 17 section we will give an example where spec b, N ( b ∂ ) is an infinite set. A full discus-sion of spec qb, N ( b ∂ ) for any q and other aspects of the indicial complex of complex b -structures is given in Section 9. Proof of Proposition 6.7.
Since (cid:3) q is b -elliptic, the set spec b, N ( (cid:3) q ) is closed anddiscrete. Let H ( N ; E q N ) be the L -based Sobolev space of order 2. For σ / ∈ spec b, N ( (cid:3) q ) let G q ( σ ) : L ( N ; E q N ) → H ( N ; E q N )be the inverse of b (cid:3) q ( σ ). The map σ
7→ G q ( σ ) is meromorphic with poles inspec b ( (cid:3) q ). Since A ⋆q ( σ ) = [ A q ( σ )] ⋆ the operators b (cid:3) q ( σ ) are the Laplacians of the complex (6.4) when σ is real. Thusfor σ ∈ R \ (spec b, N ( (cid:3) q ) ∪ spec b, N ( (cid:3) q +1 )) we have A q ( σ ) G q ( σ ) = G q +1 ( σ ) A q ( σ ) , A q ( σ ) ⋆ G q +1 ( σ ) = G q ( σ ) A ⋆q ( σ )by standard Hodge theory. Since all operators depend holomorphically on σ , thesame equalities hold for σ ∈ R = C \ (spec b, N ( (cid:3) q ) ∪ spec b, N ( (cid:3) q +1 )). It follows that A ⋆q ( σ ) A q ( σ ) G q ( σ ) = G q ( σ ) A ⋆q ( σ ) A q ( σ )in R . By analytic continuation the equality holds on all of C \ spec b, N ( (cid:3) q ). Thus if σ / ∈ spec b, N ( (cid:3) q ) and φ is a A q ( σ )-closed section, A q ( σ ) φ = 0, then the formula φ = [ A ⋆q ( σ ) A q ( σ ) + A q − ( σ ) A ⋆q − ( σ )] G q ( σ ) φ leads to φ = A q − ( σ )[ A ⋆q − ( σ ) G q ( σ ) φ ] . Therefore σ / ∈ spec qb, N ( A ). (cid:3) Since (cid:3) q is b -elliptic, the set spec b, N ( (cid:3) q ) is discrete and intersects each horizontalstrip a ≤ ℑ σ ≤ b in a finite set (Melrose [6]). Consequently: Corollary 6.8.
The sets spec qb, N ( A ) , q = 0 , . . . , are closed, discrete, and intersecteach horizontal strip a ≤ ℑ σ ≤ b in a finite set. We note in passing that the Euler characteristic of the complex (6.4) vanishesfor each σ . Indeed, let σ ∈ C . The Euler characteristic of the A ( σ )-complex isthe index of A ( σ ) + A ( σ ) ⋆ : M q even C ∞ ( N ; E q ) → M q odd C ∞ ( N ; E q ) . The operator A q ( σ ) is equal to A b,q + σ Λ r ,q , see (6.3). Thus A q ( σ ) ⋆ = A ⋆b,q + σ Λ ⋆ r ,q ,and it follows that for any σ , A ( σ ) + A ( σ ) ⋆ = A ( σ ) + A ( σ ) ⋆ + ( σ − σ )Λ r + ( σ − σ )Λ ⋆ r is a compact perturbation of A ( σ )+ A ( σ ) ⋆ . Therefore, since the index is invariantunder compact perturbations, the index of A ( σ ) + A ( σ ) ⋆ is independent of σ . Thenit vanishes, since it vanishes when σ / ∈ S q spec qb, N ( A ).Let Mero q ( N ) be the sheaf of germs of C ∞ ( N ; E q )-valued meromorphic func-tions on C and let Hol q ( N ) be the subsheaf of germs of holomorphic functions.Let S q ( N ) = Mero q ( N ) / Hol q ( N ). The holomorphic family σ
7→ A q ( σ ) gives a sheaf homomorphism A q : Mero q ( N ) → Mero q +1 ( N ) such that A q ( Hol q ( N )) ⊂ Hol q +1 ( N ) and A q +1 ◦ A q = 0, so we have a complex(6.9) · · · → S q ( N ) A q −−→ S q +1 ( N ) → · · · . The cohomology sheafs H qA ( N ) of this complex contain more refined informationabout the cohomology of the complex A . Proposition 6.10.
The sheaf H q A ( N ) is supported on spec qb, N ( A ) .Proof. Let σ ∈ C be such that H q A ( σ ) ( N ) = 0 and let(6.11) φ ( σ ) = µ X k =1 φ k ( σ − σ ) k ,µ > φ k ∈ C ∞ ( N ; V q V ∗ ), represent the A -closed element [ φ ] of the stalk of S q ( N )over σ . The condition that A q [ φ ] = 0 means that A q ( σ ) φ ( σ ) is holomorphic, thatis, A q ( σ ) φ µ ( σ − σ ) µ + µ − X k =1 A q ( σ ) φ k + Λ r ,q φ k +1 ( σ − σ ) k = 0 . In particular A q ( σ ) φ µ = 0. Since H q A ( σ ) ( N ) = 0, there is ψ µ ∈ C ∞ ( N ; E q − )such that A q − ( σ ) ψ µ = φ µ . This shows that if µ = 1, then [ φ ] is exact, and thatif µ >
1, then letting φ ′ ( σ ) = φ ( σ ) − A q − ( σ ) ψ µ / ( σ − σ ) µ , that φ is cohomologousto an element [ φ ′ ] represented by a sum as in (6.11) with µ − µ . Byinduction, [ φ ] is exact. (cid:3) Definition 6.12.
The cohomology sheafs H qA ( N ) of the complex (6.9) will bereferred to as the indicial cohomology sheafs of the complex A . If [ φ ] ∈ h qA ( N ) is anonzero element of the stalk over σ , the smallest µ such that there is a meromorphicfunction (6.11) representing [ φ ] will be called the order of the pole of [ φ ].The relevancy of this notion of pole lies in that it predicts, for any given cohomol-ogy class of the complex A , the existence of a representative with the most regularleading term (the smallest power of log that must appear in the expansion at theboundary). We will see later (Proposition 9.5) that for the b -Dolbeault complex,under a certain geometric assumption, the order of the pole of [ φ ] ∈ H q b ∂ ( N ) \ Example 6.13.
For the b -de Rham complex one has spec qb, N ( b d ) ⊂ { } and H q D (0) ( N ) = H q dR ( N ) ⊕ H q − ( N )for each component N of ∂ M , and that every element of the stalk of H q b d ( N )over 0 has a representative with a simple pole. By way of the residue we get anisomorphism from the stalk over 0 onto H q dR ( N ).Since the map (2.4) is surjective with kernel spanned by r ∂ r , the dual map(6.14) ev ∗N : T ∗ N → b T ∗N M is injective with image the annihilator, H , of r ∂ r . Let i r ∂ r : b V q N M → b V q − N M denote interior multiplication by r ∂ r Then V q H = ker( i r ∂ r : b V q N M → b V q − N M ).The isomorphism (6.14) gives isomorphismsev ∗N : V q N → H q OMPLEX b -MANIFOLDS 19 for each q . Fix a defining function r for N and let Π : b V q N M → b V q N M be theprojection on H q according to the decomposition b V q N M = H q ⊕ b d rr ∧ H q − , that is, Π φ = φ − b d rr ∧ i r ∂ r φ. If φ ∈ C ∞ ( N , H q ) and φ ∈ C ∞ ( N , H q − ), then b d ( φ + b d rr ∧ φ ) = Π b dφ + b d rr ∧ ( − Π b dφ ) . Since r − iσb d r iσ φ = b dφ + iσ b d rr ∧ φ, the indicial operator D ( σ ) of b d is D ( σ )( φ + b d rr ∧ φ ) = Π b dφ + b d rr ∧ ( iσφ − Π b dφ ) . If D ( σ )( φ + b d rr ∧ φ ) = 0, then of course Π b dφ = 0 and iσφ = Π b dφ , and itfollows that if σ = 0, then ( φ + b d rr ∧ φ ) = D ( σ ) 1 iσ φ . Thus all cohomology groups of the complex D ( σ ) vanish if σ = 0, i.e., spec qb, N ( b d ) ⊂{ } .It is not hard to verify that Π b d ev ∗N = ev ∗N d. Since r − iσb d r iσ φ = b dφ + iσ b d rr ∧ φ, the indicial operator of b d at σ = 0 can be viewed as the operator (cid:20) d − d (cid:21) : V q N⊕ V q − N → V q N⊕ V q − N . From this we get the cohomology groups of D (0) in terms of the de Rham coho-mology of N : H q D (0) ( N ) = H q dR ( N ) ⊕ H q − ( N ) . Thus the groups H q D (0) ( N ) do not vanish for q = 0, 1, dim M −
1, dim M but mayvanish for other values of q .We now show that every element of the stalk of H q b d ( N ) over 0 has a representativewith a simple pole at 0. Suppose that(6.15) φ ( σ ) = µ X k =1 σ k (cid:18) φ k + b d rr ∧ φ k (cid:19) is such that D ( σ ) φ ( σ ) is holomorphic. Then µ X k =1 σ k (cid:18) dφ k − b d rr ∧ dφ k (cid:19) + b d rr ∧ µ − X k =1 iσ k φ k +1 ! = 0 , hence dφ = 0, dφ µ = 0 and φ k = − idφ k − , k = 2 , . . . , µ . Let ψ ( σ ) = − i µ +1 X k =2 σ k φ k − . Then D ( σ ) ψ ( σ ) = − i µ +1 X k =2 σ k dφ k − + b d rr ∧ µ +1 X k =2 σ k − φ k − = µ X k =2 σ k φ k + b d rr ∧ µ X k =1 σ k φ k so φ ( σ ) − D ( σ ) ψ ( σ ) = 1 σ φ . The map that sends the class of the D ( σ )-closed element (6.15) to the class of φ in H q dR ( N ) is an isomorphism. Example 6.16.
As we just saw, the boundary spectrum of the b d complex in degree0 is just { } . In contrast, spec b, N ( b ∂ ) may be an infinite set. We illustrate this inthe context of Example 4.10. The functions z α = ( z ) α · · · ( z n +1 ) α n +1 , where the α j are nonnegative integers, are CR functions that satisfy T z α = i ( X τ j α j ) z α . This implies that D z α + i ( − i X τ j α j ) βz α = 0with β as in Example 4.10, so the numbers σ α = ( − i P τ j α j ) belong to spec b, N ( b ∂ ).For the sake of completeness we also show that if σ ∈ spec b, N ( b ∂ ), then σ = σ α for some α as above. To see this, suppose that ζ : S n +1 → C is not identicallyzero and satisfies D ζ + iσζβ = 0for some σ = 0. Then ζ is smooth, because the principal symbol of D on functionsis injective. Since h β, T i = − i , T ζ + σζ = 0 . Thus ζ ( a t ( p )) = e − σt ζ ( p ) for any p . Since | ζ ( a t ( p )) | is bounded as a function of t and ζ is not identically 0, σ must be purely imaginary. Since ζ is a CR function,it extends uniquely to a holomorphic function ˜ ζ on B = { z ∈ C n +1 : k z k < } ,necessarily smooth up to the boundary. Let ζ t = ζ ◦ a t . This is also a smoothCR function, so it has a unique holomorphic extension ˜ ζ t to B . The integral curvethrough z = ( z , . . . , z n +10 ) of the vector field T is t a t ( z ) = ( e iτ t z , . . . , e iτ n +1 t z n +10 ) OMPLEX b -MANIFOLDS 21 Extending the definition of a t to allow arbitrary z ∈ C n +1 as argument we thenhave that ˜ ζ t = ˜ ζ ◦ a t . Then ∂ t ˜ ζ t + σ ˜ ζ t = 0gives ˜ ζ ( z ) = X { α : τττ · α = iσ } c α z α for | z | <
1, where τττ = ( τ , . . . , τ n +1 ). Thus σ = − i P τ j α j as claimed. Note that ℑ σ is negative (positive) if the τ j are positive (negative) and α = 0.7. Underlying CR complexes
Again let a : R × N → N be the flow of T . Let L T denote the Lie derivativewith respect to T on de Rham q -forms or vector fields and let i T denote interiormultiplication by T of de Rham q -forms or of elements of V q V ∗ .The proofs of the following two lemmas are elementary. Lemma 7.1. If α is a smooth section of the annihilator of V in C T ∗ N , then ( L T α ) | V = 0 . Consequently, for each p ∈ N and t ∈ R , d a t : C T p N → C T a t ( p ) N maps V p onto V a t ( p ) . It follows that there is a well defined smooth bundle homomorphism a ∗ t : V q V ∗ → V q V ∗ covering a − t . In particular, one can define the Lie derivative L T φ with respectto T of an element in φ ∈ C ∞ ( N ; V q V ∗ ). The usual formula holds: Lemma 7.2. If φ ∈ C ∞ ( N ; V q V ∗ ) , then L T φ = i T D φ + D i T φ . Consequently, foreach t and φ ∈ C ∞ ( N ; V q V ∗ ) , D a ∗ t φ = a ∗ t D φ . For any defining function r of N in M , K r = ker β r is a CR structure of CRcodimension 1: indeed, K r ∩ K r ⊂ span C T but since h β r , T i vanishes nowhere, wemust have
K ∩ K = 0. Since
K ⊕ K ⊕ span C T = C T N , the CR codimension is 1.Finally, if V, W ∈ C ∞ ( N ; K r ), then h β r , [ V, W ] i = V h β r , W i − W h β r , V i − D β ( V, W ) , Since the right hand side vanishes, [
V, W ] is again a section of K r .Since V = K r ⊕ span C T , the dual of K r is canonically isomorphic to the kernel of i T : V ∗ → C . We will write K ∗ for this kernel. More generally, V q K ∗ r and the kernel, V q K ∗ , of i T : V q V ∗ → V q − V ∗ are canonically isomorphic. The vector bundles V q K ∗ are independent of the defining function r . We regard the ∂ b -operators ofthe CR structure as operators C ∞ ( N ; V q K ∗ ) → C ∞ ( N ; V q +1 K ∗ ) . They do depend on r but we will not indicate this in the notation.To get a formula for ∂ b , let ˜ β r = ii − a r β r (so that h i ˜ β r , T i = 1). The projection Π r : V q V ∗ → V q V ∗ on V q K ∗ according tothe decomposition(7.3) V q V ∗ = V q K ∗ ⊕ i ˜ β r ∧ V q − K ∗ is(7.4) Π r φ = φ − i ˜ β r ∧ i T φ. Lemma 7.5.
With the identification of V q K ∗ r with V q K ∗ described above, the ∂ b -operators of the CR structure K r are given by (7.6) ∂ b φ = Π r D φ if φ ∈ C ∞ ( N , V q K ∗ ) , Proof.
Suppose that ( z, t ) is a hypoanalytic chart for V on some open set U , with T t = 1. So ∂ z µ , µ = 1 . . . , n , T = ∂ t is a frame for V over U with dual frame D z µ , D t . If β r = n X µ =1 β µ D z µ + β D t. then L µ = ∂ z µ − β µ β ∂ t , µ = 1 , . . . , n is a frame for K r over U . Let η µ denote the dual frame (for K ∗ r ). Since the L µ commute, ∂ b η µ = 0, so if φ = P ′| I | = q φ I η I , then (with the notation as in eg.Folland and Kohn [2]) ∂ b φ = X ′| J | = q +1 X ′| I | = q X µ ǫ µIJ L µ φ I η J . On the other hand, the frame of V ∗ dual to the frame L µ , µ = 1 , . . . , n , T of V is D z µ , i ˜ β r , and the identification of K ∗ r with K ∗ maps the η µ to the D z µ . So, as asection of V q V ∗ , φ = X ′| I | = q φ I D z I and D φ = X ′| J | = q +1 X ′| I | = q ǫ µIJ L µ φ I D z J + i ˜ β r ∧ X ′| I | = q T φ I D z I . Thus Π r D φ is the section of V q +1 K ∗ associated with ∂ b φ by the identifying map. (cid:3) Using (7.4) in (7.6) and the fact that i T D φ = L T φ if φ ∈ C ∞ ( N ; V q K ∗ ) we get(7.7) ∂ b φ = D φ − i ˜ β r ∧ L T φ if φ ∈ C ∞ ( N , V q K ∗ ) . The D operators can be expressed in terms of the ∂ b operators. Suppose φ ∈ C ∞ ( N ; V q V ∗ ). Then φ = φ + i ˜ β r ∧ φ with unique φ ∈ C ∞ ( N ; V q K ∗ ) and φ ∈ C ∞ ( N ; V q − K ∗ ), and D φ = ∂ b φ + i ˜ β r ∧ L T φ , see (7.7). Using D ˜ β r = D a r i − a r ∧ ˜ β r and (7.7) again we get D ( i ˜ β r ∧ φ ) = i ˜ β r ∧ (cid:0) − D a r i − a r ∧ φ − D φ (cid:1) = i ˜ β r ∧ (cid:0) − ∂ b a r i − a r ∧ φ − ∂ b φ (cid:1) . OMPLEX b -MANIFOLDS 23 This gives(7.8) D = ∂ b L T − ∂ b − ∂ b a r i − a r : C ∞ ( N ; V q K ∗ ) ⊕ C ∞ ( N ; V q − K ∗ ) → C ∞ ( N ; V q +1 K ∗ ) ⊕ C ∞ ( N ; V q K ∗ ) . Since T itself is T -invariant, i T a ∗ t = a ∗ t i T : the subbundle K ∗ of V ∗ is invariantunder a ∗ t for each t . This need not be true of K r , i.e., the statement that for all t , d a t ( K r ) ⊂ K r , equivalently, L ∈ C ∞ ( M ; K r ) = ⇒ [ T , L ] ∈ C ∞ ( M ; K r ) , may fail to hold. Since D β r = 0, the formula0 = T h β r , L i − L h β r , T i − h β r , [ T , L ] i with L ∈ C ∞ ( N ; K r ) gives that K r is invariant under d a t if and only if La r = 0for each CR vector field, that is, if and only if a r is a CR function. This provesthe equivalence between the first and last statements in the following lemma. Thethird statement is the most useful. Lemma 7.9.
Let r be a defining function for N in M and let ∂ b denote the oper-ators of the associated CR complex. The following are equivalent:(1) The function a r is CR;(2) L T ˜ β r = 0 ;(3) L T ∂ b − ∂ b L T = 0 ;(4) K r is T -invariant.Proof. From β r = ( a r − i ) i ˜ β r and L T β r = D a r we obtain D a r = ( L T a r ) i ˜ β r + ( a r − i ) i L T ˜ β r , so ∂ b a r = D a r − ( L T a r ) i ˜ β r = ( a r − i ) i L T ˜ β r . Thus a r is CR if and only if L T ˜ β r = 0.Using L T D = D L T and the definition of ∂ b we get L T ∂ b φ = L T ( D φ − i ˜ β r ∧ L T φ ) = ∂ b L T φ − i ( L T ˜ β r ) ∧ L T φ for φ ∈ C ∞ ( N ; V q K ∗ ). Thus L T ∂ b − ∂ b L T = 0 if and only if L T ˜ β r = 0. (cid:3) Lemma 7.10.
Suppose that V admits a T -invariant metric. Then there is a defin-ing function r for N in M such that a r is constant. If r and r ′ are defining functionssuch that a r and a r ′ are constant, then a r = a r ′ . This constant will be denoted a av .Proof. Let h be a metric as stated. Let H , be the subbundle of V orthogonalto T . This is T -invariant, and since the metric is T -invariant, H , has a T -invariant metric. This metric gives canonically a metric on H , = H , . Using thedecomposition C T N = H , ⊕ H , ⊕ span C T we get a T -invariant metric on C T N for which the decomposition is orthogonal. This metric is induced by a Riemannianmetric g . Let m be the corresponding Riemannian density, which is T -invariantbecause g is. Since D , h , and m are T -invariant, so are the formal adjoint D ⋆ of D and the Laplacians of the D -complex, and if G denotes the Green’s operators forthese Laplacians, then G is also T -invariant, as is the orthogonal projection Π on the space of D -harmonic forms. Arbitrarily pick a defining function r for N in M .Then a r − G D ⋆ D a r = Π a r where Π a r is a constant function by Lemma 4.6. Since β r is D -closed, D a r = L T β r .Thus G D ⋆ D a r = T G D ⋆ β r , and since a r is real valued and T is a real vector field, a r − T ℜ G D ⋆ β r = ℜ Π a r . Extend the function u = ℜ G D ⋆ β r to M as a smooth real-valued function. Then r ′ = e − u r has the required property.Suppose that r , r ′ are defining functions for N in M such that a r and a r ′ areconstant. Then these functions are equal by Proposition 4.9. (cid:3) Note that if for some r , the subbundle K r is T -invariant and admits a T invariantHermitian metric, then there is a T -invariant metric on V .Suppose now that ρ : F → M is a holomorphic vector bundle over M . Usingthe operators D : C ∞ ( N ; V q V ∗ ⊗ F N ) → C ∞ ( N ; V q +1 V ∗ ⊗ F N ) , see (4.12), define operators(7.11) · · · → C ∞ ( N ; V q K ∗ ⊗ F N ) ∂ b −→ C ∞ ( N ; V q +1 K ∗ ⊗ F N ) → · · · by ∂ b φ = Π r D φ, φ ∈ C ∞ ( N ; V q K ∗ ⊗ F N )where Π r means Π r ⊗ I with Π r defined by (7.4). The operators (7.11) form acomplex. Define also L T = i T D + D i T where i T stands for i T ⊗ I . Then i T L T = L T i T , L T D = D L T . The first of these identities implies that the image of C ∞ ( N ; V q K ∗ ⊗ F N ) by L T is contained in C ∞ ( N ; V q K ∗ ⊗ F N ). With these definitions, D as an operator D : C ∞ ( N ; V q K ∗ ⊗ F N ) ⊕ C ∞ ( N ; V q − K ∗ ⊗ F N ) → C ∞ ( N ; V q +1 K ∗ ⊗ F N ) ⊕ C ∞ ( N ; V q K ∗ ⊗ F N ) . is given by the matrix in (7.8) with the new meanings for ∂ b and L T .Assume that there is a T -invariant Riemannian metric on N , that r has be chosenso that a r is constant, that K r is orthogonal to T , and that T has unit length. Thenthe term involving ∂ b a r in the matrix (7.8) is absent, and since D = 0, L T ∂ b = ∂ b L T . Write h V ∗ for the metric induced on the bundles V q V ∗ or V q K ∗ .If η µ , µ = 1 , . . . , k is a local frame of F N over an open set U ⊂ N and φ is alocal section of V q V ∗ ⊗ F N over U , then for some smooth sections φ µ of V q V ∗ and ω νµ of V ∗ over U , φ = X µ φ µ ⊗ η µ , D X µ φ µ ⊗ η µ = X ν ( D φ ν + X µ ω νµ ∧ φ µ ) ⊗ η ν . OMPLEX b -MANIFOLDS 25 This gives ∂ b X µ φ µ ⊗ η µ = X ν ( ∂ b φ ν + X µ Π r ω νµ ∧ φ µ ) ⊗ η ν and L T X µ φ µ ⊗ η µ = X ν ( L T φ ν + X µ h ω νµ , T i φ µ ) ⊗ η ν . Suppose now that h F is a Hermitian metric on F . With this metric and themetric h V ∗ we get Hermitian metrics h on each of the bundles V q V ∗ ⊗ F N . If η µ is an orthonormal frame of F N and φ = P φ µ ⊗ η µ , ψ = P ψ µ ⊗ η µ are sections of V q V ∗ ⊗ F N , then h ( φ, ψ ) = X ν h V ∗ ( φ µ , ψ µ ) . Therefore h ( L T φ, ψ ) + h ( φ, L T ψ )= X ν h V ∗ ( L T φ ν + X µ h ω νµ , T i φ µ , ψ ν ) + X µ h V ∗ ( φ µ , L T ψ µ + h ω µν , T i ψ ν )= X ν T h V ∗ ( φ ν , ψ ν ) + X µ,ν ( h ω νµ , T i + h ω µν , T i ) h V ∗ ( φ µ , ψ ν )= T h ( φ, ψ ) + X µ,ν ( h ω νµ , T i + h ω µν , T i ) h V ∗ ( φ µ , ψ ν ) . Thus T h ( φ, ψ ) = h ( L T φ, ψ ) + h ( φ, L T ψ ) if and only if(7.12) h ω νµ , T i + h ω µν , T i = 0 for all µ, ν.
This condition is (4.17); just note that by the definition of D , the forms (Φ ∗ ) − ω νµ in (4.17) are the forms that we are denoting ω νµ here. Thus (7.12) holds if and onlyif h F is an exact Hermitian metric, see Definition (4.18).Consequently, Lemma 7.13.
The statement (7.14) T h ( φ, ψ ) = h ( L T φ, ψ ) + h ( φ, L T ψ ) ∀ φ, ψ ∈ C ∞ ( N ; V q V ∗ ⊗ F N ) holds if and only the Hermitian metric h F is exact. Spectrum
Suppose that V admits an invariant Hermitian metric. Let r be a defining func-tion for N in M such that a r is constant. By Lemma (7.9) K r is T -invariant,so the restriction of the metric to this subbundle gives a T -invariant metric; weuse the induced metric on the bundles V q K ∗ in the following. As in the proof ofLemma 7.10, there is a T -invariant density m on N .Let ρ : F → M be a Hermitian holomorphic vector bundle, assume that theHermitian metric of F is exact, so with the induced metric h on the vector bundles V q V ∗ ⊗ F N , (7.14) holds. We will write F in place of F N .Let ∂ ⋆b be the formal adjoint of the b ∂ operator (7.11) with respect to the inneron the bundles V q K ∗ ⊗ F and the density m , and let (cid:3) b,q = ∂ b ∂ ⋆b + ∂ ⋆b ∂ b be the formal ∂ b -Laplacian. Since − i L T is formally selfadjoint and commutes with ∂ b , L T commutes with (cid:3) b,q . Let H q∂ b ( N ; F ) = ker (cid:3) b,q = { φ ∈ L ( N ; V q K ∗ ⊗ F ) : (cid:3) b,q φ = 0 } and let Dom q ( L T ) = { φ ∈ H q∂ b ( N ; F ) and L T φ ∈ H q∂ b ( N ; F ) } . The spaces H q∂ b ( N ; F ) may be of infinite dimension, but in any case they are closedsubspaces of L ( N ; V q K ∗ ⊗ F ), so they may be regarded as Hilbert spaces on theirown right. If φ ∈ H q∂ b ( N ; F ), the condition L T φ ∈ H q∂ b ( N ; F ) is equivalent to thecondition L T φ ∈ L ( N ; V q K ∗ ⊗ F ) . So we have a closed operator(8.1) − i L T : Dom q ( L T ) ⊂ H q∂ b ( N ; F ) → H q∂ b ( N ; F ) . The fact that (cid:3) b,q − L T is elliptic, symmetric, and commutes with L T implies that(8.1) is a selfadjoint Fredholm operator with discrete spectrum (see [13, Theorem2.5]). Definition 8.2.
Let spec q ( − i L T ) be the spectrum of the operator (8.1), and let H q∂ b ,τ ( N ; F ) be the eigenspace of − i L T in H q∂ b ( N ; F ) corresponding to the eigen-value τ .Let τττ denote the principal symbol of − i T . Then the principal symbol of L T acting on sections of V q K ∗ is τττ I . Because (cid:3) b,q − L T is elliptic, Char( (cid:3) b,q ), thecharacteristic variety of (cid:3) b,q , lies in τττ = 0. LetChar ± ( (cid:3) b,q ) = { ν ∈ Char( (cid:3) b,q ) : τττ ( ν ) ≷ } . By [13, Theorem 4.1], if (cid:3) b,q is microlocally hypoelliptic on Char ± ( (cid:3) b,q ), then { τ ∈ spec q ( − i L T ) : τ ≷ } is finite. We should perhaps point out that Char( (cid:3) b,q ) is equal to the characteristicvariety, Char( K r ), of the CR structure.As a special case consider the situation where F is the trivial line bundle. Let θ r be the real 1-form on N which vanishes on K r and satisfies h θ r , T i = 1; thus θ r is smooth, spans Char( K r ), and has values in Char + ( K r ). The Levi form of thestructure is Levi θ r ( v, w ) = − idθ r ( v, w ) , v, w ∈ K r ,p , p ∈ N . Suppose that Levi θ r is nondegenerate, with k positive and n − k negative eigenvalues.It is well known that then (cid:3) b,q is microlocally hypoelliptic at ν ∈ Char K r for all q except if q = k and τττ ( ν ) < q = n − k and τττ ( ν ) > Theorem 8.3 ([13, Theorem 6.1]) . Suppose that V admits a Hermitian metric andthat for some defining function r such that a r is constant, Levi θ r is nondegeneratewith k positive and n − k negative eigenvalues. Then(1) spec q ( − i L T ) is finite if q = k, n − k ;(2) spec k ( − i L T ) contains only finitely many positive elements, and(3) spec n − k ( − i L T ) contains only finitely many negative elements. OMPLEX b -MANIFOLDS 27 Indicial cohomology
Suppose that there is a T -invariant Hermitian metric ˜ h on V . By Lemma 7.10there is a defining function r such that h β r , T i is constant, equal to a av − i . Therefore K r is T -invariant. Let h be the metric on V which coincides with ˜ h on K r , makes thedecomposition V = K r ⊕ span C T orthogonal, and for which T has unit length. Themetric h is T -invariant. We fix r and such a metric, and let m be the Riemannianmeasure associated with h . The decomposition (7.3) of V q V ∗ is an orthogonaldecomposition.Recall that D ( σ ) φ = D φ + iσβ r ∧ φ . Since a r = a av is constant (in particularCR), D ( σ )( φ + i ˜ β r ∧ φ ) = ∂ b φ + i ˜ β r ∧ (cid:2)(cid:0) L T + (1 + ia av ) σ (cid:1) φ − ∂ b φ (cid:3) if φ ∈ C ∞ ( N ; V q K ∗ r ) and φ ∈ C ∞ ( N ; V q − K ∗ r ). So D ( σ ) can be regarded as theoperator(9.1) D ( σ ) = (cid:20) ∂ b L T + (1 + ia av ) σ − ∂ b (cid:21) : C ∞ ( N ; V q K ∗ r ) ⊕ C ∞ ( N ; V q − K ∗ r ) → C ∞ ( N ; V q +1 K ∗ r ) ⊕ C ∞ ( N ; V q K ∗ r ) . Since the subbundles V q K r and ˜ β ∧ V q − K r are orthogonal with respect to themetric induced by h on V q V , the formal adjoint of D ( σ ) with respect to this metricand the density m is D ( σ ) ⋆ = " ∂ ⋆b −L T + (1 − ia av ) σ − ∂ ⋆b : C ∞ ( N ; V q +1 K ∗ r ) ⊕ C ∞ ( N ; V q K ∗ r ) → C ∞ ( N ; V q K ∗ r ) ⊕ C ∞ ( N ; V q − K ∗ r )where ∂ ⋆b is the formal adjoint of ∂ b . So the Laplacian, (cid:3) D ( σ ) ,q , of the D ( σ )-complexis the diagonal operator with diagonal entries P q ( σ ), P q − ( σ ) where P q ( σ ) = (cid:3) b,q + ( L T + (1 + ia av ) σ )( −L T + (1 − ia av ) σ )acting on C ∞ ( N ; V q K ∗ r ) and P q − ( σ ) is the “same” operator, acting on sections of V q − K ∗ r ; recall that L T commutes with ∂ b and since L ⋆ T = −L T , also with ∂ ⋆b , andthat a av is constant. Note that P q ( σ ) is an elliptic operator.Suppose that φ ∈ C ∞ ( N ; V q K ∗ r ) is a nonzero element of ker P q ( σ ); the complexnumber σ is fixed. Since P q ( σ ) is elliptic, ker P q ( σ ) is a finite dimensional space,invariant under − i L T since the latter operator commutes with P q ( σ ). As an oper-ator on ker P q ( σ ), − i L T is selfadjoint, so there is a decomposition of ker P q ( σ ) intoeigenspaces of − i L T . Thus φ = N X j =1 φ j , − i L T φ j = τ j φ j where the τ j are distinct real numbers and φ j ∈ ker P q ( σ ), φ j = 0. In particular, (cid:3) b,q φ j + ( L T + (1 + ia av ) σ )( −L T + (1 − ia av ) σ ) φ j = 0 , for each j , that is, (cid:3) b,q φ j + | iτ j + (1 + ia av ) σ | φ j = 0 . Since (cid:3) b,q is a nonnegative operator and φ j = 0, iτ j + (1 + ia av ) σ = 0 and φ j ∈ ker (cid:3) b,q . Since σ is fixed, all τ j are equal, which means that N = 1. Conversely, if φ ∈ C ∞ ( N ; V q K ∗ r ) belongs to ker (cid:3) b,q and − i L T φ = τ φ , then P q ( σ ) φ = 0 with σ such that τ = ( i − a av ) σ .Let H q D ( σ ) ( N ) be the kernel of (cid:3) D ( σ ) ,q . Theorem 9.2.
Suppose that V admits a T -invariant metric and let r be a definingfunction for N in M such that h β r , T i = a av − i is constant. Then spec qb, N ( b ∂ ) = ( i − a av ) − spec q ( − i L T ) ∪ ( i − a av ) − spec q − ( − i L T ) , and if σ ∈ spec qb, N ( b ∂ ) , then, with the notation in Definition 8.2 H q D ( σ ) ( N ) = H q∂ b ,τ ( σ ) ( N ) ⊕ H q − ∂ b ,τ ( σ ) ( N ) with τ ( σ ) = ( i − a av ) σ . If the CR structure K r is nondegenerate, Proposition 8.3 gives more specificinformation on spec qb, N ( b ∂ ). In particular, Proposition 9.3.
With the hypotheses of Theorem 9.2, suppose that
Levi θ r isnondegenerate with k positive and n − k negative eigenvalues. If k > , then spec b, N ⊂ { σ ∈ C : ℑ σ ≤ } , and if n − k > , then spec b, N ( b ∂ ) ⊂ { σ ∈ C : ℑ σ ≥ } . Remark 9.4.
The b -spectrum of the Laplacian of the b ∂ -complex in any degree canbe described explicitly in terms of the joint spectra spec( − i L T , (cid:3) b,q ). We brieflyindicate how. With the metric h and defining function r as in the first paragraphof this section, suppose that h is extended to a metric on b T , M . This gives aRiemannian b -metric on M that in turn gives a b -density m on M . With these weget formal adjoints b ∂ ⋆ whose indicial families D ⋆ ( σ ) are related to those of b ∂ by D ⋆ ( σ ) = c b ∂ ⋆ ( σ ) = [ b b ∂ ( σ )] ⋆ = D ( σ ) ⋆ . By (9.1), D ⋆ ( σ ) = " ∂ ⋆b −L T + (1 − ia av ) σ − ∂ ⋆b . Using this one obtains that the indicial family of the Laplacian (cid:3) q of the b ∂ -complexin degree q is a diagonal operator with diagonal entries P ′ q ( σ ), P ′ q − ( σ ) with P ′ q ( σ ) = (cid:3) b,q + ( L T + (1 + ia av ) σ )( −L T + (1 − ia av ) σ )and the analogous operator in degree q −
1. The set spec b ( (cid:3) q ) is the set of valuesof σ for which either P ′ q ( σ ) or P ′ q − ( σ ) is not injective. These points can written interms of the points spec( − i L T , (cid:3) b ) as asserted. In particular one getsspec b ( (cid:3) q ) ⊂ { σ : |ℜ σ | ≤ | a av ||ℑ σ |} with spec qb, N ( b ∂ ) being a subset of the boundary of the set on the right.We now discuss the indicial cohomology sheaf of b ∂ , see Definition 6.12. We willshow: Proposition 9.5.
Let σ ∈ spec qb, N ( b ∂ ) . Every element of the stalk of H q b ∂ ( N ) at σ has a representative of the form σ − σ (cid:20) φ (cid:21) where φ ∈ H q∂ b ,τ ( N ) , τ = ( i − a av ) σ . OMPLEX b -MANIFOLDS 29 Proof.
Let(9.6) φ ( σ ) = µ X k =1 σ − σ ) k (cid:20) φ k φ k (cid:21) represent an element in the stalk at σ of the sheaf of germs of C ∞ ( N ; V q V ∗ ⊗ F )-valued meromorphic functions on C modulo the subsheaf of holomorphic elements.Letting α = 1 + ia av we have D ( σ ) φ ( σ ) = µ X k =1 σ − σ ) k (cid:20) ∂ b φ k (cid:0) L T + ασ (cid:1) φ k − ∂ b φ k (cid:21) + µ − X k =0 α ( σ − σ ) k (cid:20) φ k +1 (cid:21) , so the condition that D ( σ ) φ ( σ ) is holomorphic is equivalent to(9.7) ∂ b φ k = 0 , k = 1 , . . . , µ and(9.8) ( L T + ασ ) φ µ − ∂ b φ µ = 0 , ( L T + ασ ) φ k − ∂ b φ k + αφ k +1 = 0 , k = 1 , . . . , µ − . Let P q ′ = (cid:3) b,q ′ − L T in any degree q ′ . For any ( τ, λ ) ∈ R and q ′ let E q ′ τ,λ = { ψ ∈ C ∞ ( N ; V q ′ V ∗ ⊗ F ) : P q ′ ψ = λψ, − i L T ψ = τ ψ } . This space is zero if ( τ, λ ) is not in the joint spectrum Σ q ′ = spec q ′ ( − i L T , P q ′ ). Each φ ik decomposes as a sum of elements in the spaces E q − iτ,λ , ( τ, λ ) ∈ Σ q − i . Supposethat already φ ik ∈ E q − iτ,λ : P q − i φ ik = λφ ik , − i L T φ ik = τ φ ik , i = 0 , , k = 1 , . . . , µ. Then (9.8) becomes(9.9) ( iτ + ασ ) φ µ − ∂ b φ µ = 0 , ( iτ + ασ ) φ k − ∂ b φ k + αφ k +1 = 0 , k = 1 , . . . , µ − . If τ = τ , then iτ + ασ = 0, and we get φ k = ∂ b ψ k for all k with ψ k = µ − k X j =0 ( − α ) j ( iτ + ασ ) j +1 φ k + j . Trivially (cid:0) L T + ασ (cid:1) ψ µ = φ µ and also (cid:0) L T + ασ (cid:1) ψ k + αψ k +1 = φ k , k = 1 , . . . , µ − , so φ ( σ ) − D ( σ ) µ X k =1 σ − σ ) k (cid:20) ψ k (cid:21) = 0modulo an entire element.Suppose now that the φ ik are arbitrary and satisfy (9.7)-(9.8). The sum(9.10) φ ik = X ( τ,λ ) ∈ Σ q − i φ ik,τ,λ , φ ik,τ,λ ∈ E q − iτ,λ converges in C ∞ , indeed for each N there is C i,k,N such that(9.11) sup p ∈N k φ ik,τ,λ ( p ) k ≤ C i,k,N (1 + λ ) − N for all τ, λ. Since D ( σ ) preserves the spaces E qτ,λ ⊕ E q − τ,λ , the relations (9.9) hold for the φ ik,τ,λ for each ( τ, λ ). Therefore, with(9.12) ψ k = X ( τ,λ ) ∈ Σ q − τ = τ µ − k X j =0 ( − α ) j ( iτ + ασ ) j +1 φ k + j,τ,λ we have formally that φ ( σ ) − D ( σ ) µ X k =1 σ − σ ) k (cid:20) ψ k (cid:21) = µ X k =1 σ − σ ) k (cid:20) ˜ φ k ˜ φ k (cid:21) with(9.13) ˜ φ ik = X ( τ,λ ) ∈ Σ q − τ = τ φ ik,τ,λ , φ ik,τ,λ ∈ E q − iτ,λ . However, the convergence in C ∞ of the series (9.12) is questionable since there maybe a sequence { ( τ ℓ , λ ℓ ) } ∞ ℓ =1 ⊂ spec( − i L T , P q − ) of distinct points such that τ ℓ → τ as ℓ → ∞ , so that the denominators iτ ℓ + ασ in the formula for ψ k tend to zero sofast that for some nonnegative N , λ − Nℓ / ( iτ ℓ + ασ ) is unbounded. To resolve thisdifficulty we will first show that φ ( σ ) is D ( σ )-cohomologous (modulo holomorphicterms) to an element of the same form as φ ( σ ) for which in the series (9.10) theterms φ ik,τ,λ vanish if λ − τ > ε ; the number ε > τ , λ ) ∈ Σ q ∪ Σ q − = ⇒ λ = τ or λ ≥ τ + ε. Recall that spec q ′ ( − i L T , P q ′ ) ⊂ { ( τ, λ ) : λ ≥ τ } .For any V ⊂ S q ′ Σ q ′ letΠ q ′ V : L ( N ; V q ′ V ∗ ⊗ F ) → L ( N ; V q ′ V ∗ ⊗ F )be the orthogonal projection on L ( τ,λ ) ∈ V E q ′ τ,λ . If ψ ∈ C ∞ ( N ; V q ′ V ∗ ⊗ F ), then theseries Π q ′ V ψ = X ( τ,λ ) ∈ V ψ τ,λ , ψ τ,λ ∈ E q ′ τ,λ converges in C ∞ . It follows that (cid:3) b,q ′ and L T commute with Π q ′ V and that ∂ b Π q ′ V =Π q ′ +1 V ∂ b . Since the Π q ′ V are selfadjoint, also ∂ ⋆b Π q ′ +1 V = Π q ′ V ∂ ⋆b .Let U = { ( τ, λ ) ∈ Σ q ∪ Σ q − : λ < τ + ε } , U c = Σ q ∪ Σ q − \ U. Then, for any sequence { ( τ ℓ , λ ℓ ) } ⊂ U of distinct points we have | τ ℓ | → ∞ as ℓ → ∞ . Define G q ′ U c ψ = X ( τ,λ ) ∈ U c λ − τ ψ τ,λ In this definition the denominators λ − τ are bounded from below by ε , so G q ′ U c is a bounded operator in L and maps smooth sections to smooth sections because OMPLEX b -MANIFOLDS 31 the components of such sections satisfy estimates as in (9.11). The operators areanalogous to Green operators: we have(9.15) (cid:3) b,q ′ G q ′ U c = G q ′ U c (cid:3) b,q ′ = I − Π q ′ U so if ∂ b ψ = 0, then(9.16) (cid:3) b,q ′ G q ′ U c ψ = ∂ b ∂ ⋆b G q ′ U c ψ since ∂ b G q ′ U c = G q ′ +1 U c ∂ b .Write φ ( σ ) in (9.6) as φ ( σ ) = Π U c φ ( σ ) + Π U φ ( σ )whereΠ U c φ ( σ ) = µ X k =1 σ − σ ) k (cid:20) Π qU c φ k Π q − U c φ k (cid:21) , Π U φ ( σ ) = µ X k =1 σ − σ ) k (cid:20) Π qU φ k Π q − U φ k (cid:21) . Since D ( σ ) φ ( σ ) is holomorphic, so are D ( σ )Π U c φ ( σ ) and D ( σ )Π U φ ( σ ).We show that Π U c φ ( σ ) is exact modulo holomorphic functions. Using (9.7),(9.15), and (9.16), Π qU c φ k = ∂ ⋆b ∂ b Π qU c φ k . ThenΠ U c φ ( σ ) − D ( σ ) µ X k =1 σ − σ ) k (cid:20) ∂ ⋆b G qU c Π qU φ k (cid:21) = µ X k =1 σ − σ ) k (cid:20) φ k (cid:21) modulo a holomorphic term for some ˆ φ k with Π q − U c ˆ φ k = ˆ φ k . The element on theright is D ( σ )-closed modulo a holomorphic function, so its components satisfy (9.7),(9.8), which give that the ˜ φ k are ∂ b -closed. Using again (9.15) and (9.16) we seethat Π U c φ ( σ ) represent an exact element.We may thus assume that Π qU c φ ( σ ) = 0. If this is the case, then the series (9.12)converges in C ∞ , so φ ( σ ) is cohomologous to the element˜ φ ( σ ) = µ X k =1 σ − σ ) k (cid:20) ˜ φ k ˜ φ k (cid:21) where the ˜ φ ik are given by (9.13) and satisfy Π q − iU c ˜ φ ik = 0. By (9.14), ˜ φ ik ∈ E q − iτ ,τ .In particular, (cid:3) b,q − i φ ik = 0.Assuming now that already φ ik ∈ E q − iτ ,τ , the formulas (9.9) give (since τ = τ and iτ + ασ = 0) ∂ b φ µ = 0 , φ k = ∂ b α φ k − , k = 2 , . . . , µ. Then φ ( σ ) − α D ( σ ) µ +1 X k =2 σ − σ ) k (cid:20) φ k − (cid:21) = 1 σ − σ (cid:20) φ (cid:21) with (cid:3) b,q φ = 0. (cid:3) Appendix A. Totally characteristic differential operators
We review here some basic definitions and notation concerning totally charac-teristic differential operators.Let E , F → M be vector bundles and let Diff m ( M ; E, F ) be the space ofdifferential operators C ∞ ( M ; E ) → C ∞ ( M ; F ) of order m . Then(A.1) Diff mb ( M ; E, F ), the space of totally characteristic differential opera-tors of order m , consists of those elements P ∈ Diff m ( M ; E, F ) withthe property r − ν P r ν ∈ Diff m ( M ; E, F ) , ν = 1 , . . . , m i.e., r − ν P r ν has coefficients smooth up to the boundary.Let π : T ∗ M → M and b π : b T ∗ M → M be the canonical projections. Suppose P ∈ Diff mb ( M ; E, F ). Since P is in particular a differential operator, it has aprincipal symbol σσ ( P ) ∈ C ∞ ( T ∗ M ; Hom( π ∗ E, π ∗ F )) . The fact that P is totally characteristic implies that σσ ( P ) lifts to a section b σσ ( P ) ∈ C ∞ ( b T ∗ M ; Hom( b π ∗ E, b π ∗ F )) , the principal b -symbol of P , characterized by(A.2) b σσ ( P )(ev ∗ ξ ) = σσ ( P )( ξ ) . If P ∈ Diff mb ( M ; E, F ), then P induces a differential operator(A.3) P b ∈ Diff mb ( M ; E ∂ M , F ∂ M ) , as follows. If φ ∈ C ∞ ( ∂ M ; E ∂ M ), let ˜ φ ∈ C ∞ ( M ; E ) be an extension of φ and let P b φ = ( P ˜ φ ) | ∂ M . The condition (A.1) ensures that P ˜ φ | ∂ M is independent of the extension of φ used.Clearly if P and Q are totally characteristic differential operators, then so is P Q ,and(A.4) (
P Q ) b = P b Q b . The indicial family of P ∈ Diff mb ( M ; E, F ) is defined as follows. Fix a definingfunction r for ∂ M . Then for any σ ∈ C , P ( σ ) = r − iσ P r iσ ∈ Diff mb ( M ; E, F ) . Let(A.5) b P ( σ ) = P ( σ ) b . References [1] Baouendi, M. S., Chang, C. H., Treves, F.,
Microlocal hypo-analyticity and extension of CRfunctions , J. Differential Geom. (1983), 331–391.[2] Folland, G., Kohn, J., The Neumann problem for the Cauchy-Riemann complex , Annals ofMathematics Studies . Princeton University Press, 1972.[3] Helgason, S., Differential geometry, Lie groups, and symmetric spaces . Pure and AppliedMathematics, 80. Academic Press, Inc, New York-London, 1978.[4] H¨ormander, L.,
The Frobenius-Nirenberg theorem , Ark. Mat. 5 1965 425–432 (1965).[5] Melrose, R. B.,
Transformation of boundary problems
Acta Math. (1981), 149–236.[6] ,
The Atiyah-Patodi-Singer index theorem , Research Notes in Mathematics, A. K.Peters, Ltd., Wellesley, MA, 1993.
OMPLEX b -MANIFOLDS 33 [7] Mendoza, G., Strictly pseudoconvex b-CR manifolds , Comm. Partial Differential Equations (2004) 1437–1503.[8] , Boundary structure and cohomology of b -complex manifolds . In “Partial DifferentialEquations and Inverse Problems”, C. Conca et al., eds., Contemp. Math., vol. 362 (2004),303–320.[9] , Anisotropic blow-up and compactification , In ”Recent Progress on some Problemsin Several Complex Variables and Partial Differential Equations”, S. Berhanu et al., eds.,Contemp. Math., vol. 400 (2006) 173–187.[10] ,
Characteristic classes of the boundary of a complex b -manifold , in Complex analysis ,245–262, Trends Math., Birkh¨auser/Springer Basel AG, Basel, 2010. Dedicated to LindaP. Rothschild.[11] ,
A Gysin sequence for manifolds with R-action , Geometric Analysis of Several Com-plex Variables and Related Topics, Contemporary Mathematics, vol. 550, Amer. Math. Soc.,Providence, RI, 2011, pp. 139-154.[12] ,
Two embedding theorems , in
From Fourier Analysis and Number Theory to RadonTransforms and Geometry , 399-429, H. M. Farkas et al. (eds.), Developments in Mathematics28, Springer Verlag. Dedicated to Leon Ehrenpreis, in memoriam.[13] ,
Hypoellipticity and vanishing theorems , submitted.[14] Newlander, A., Nirenberg, L.,
Complex analytic coordinates in almost complex manifolds ,Ann. of Math. (1957), 391–404.[15] L. Nirenberg, A complex Frobenius theorem , Seminar on analytic functions I, Princeton,(1957) 172–189.[16] Treves, F.,
Approximation and representation of functions and distributions annihilated by asystem of complex vector fields , Centre Math. Ecole Polytechnique, Paliseau, France (1981).[17] Treves, F.,
Hypo-analytic structures. Local theory , Princeton Mathematical Series, , Prince-ton University Press, Princeton, NJ, 1992. E-mail address : [email protected]@math.temple.edu