Effective Termination of Kohn's Algorithm for Subelliptic Multipliers
aa r X i v : . [ m a t h . C V ] A ug Effective Termination of Kohn’s Algorithmfor Subelliptic Multipliers
Yum-Tong Siu Dedicated to Professor Joseph J. Kohn on his 75th Birthday
Introduction.
In this note we will discuss the problem of the effective ter-mination of Kohn’s algorithm for subelliptic multipliers for bounded smoothweakly pseudoconvex domains of finite type [Ko79]. We will give a completeproof for the case of special domains and will only indicate briefly how thismethod is to be extended to the case of general bounded smooth weakly pseu-doconvex domains of finite type. The method is rather simple and uses somelocal theory of algebraic geometry. People with some minimal background inalgebraic geometry may find the algebraic-geometric techniques involved inthis note very simple or even completely trivial. Since this topic is of interestmainly to the analysts I will use as much as possible the language of functiontheory to describe our method.In a number of conferences in recent years I gave talks on this topic,but because of time limitation never had the opportunity to present all thedetails. This note is written to make the details available. This note willappear in the special issue of
Pure and Applied Mathematics Quarterly forProfessor Joseph J. Kohn.The termination of Kohn’s algorithm in the real-analytic case was verifiedby Diederich-Fornaess [DF78] without effectiveness. In this note we are alsogoing to formulate Kohn’s algorithm geometrically in terms of the theorem ofFrobenius on integral submanifolds and present a proof from this geometricviewpoint so that one can see clearly how the procedures of Kohn’s algorithmarise naturally in the geometric context and why the real-analyticity facili-tates the proof of the termination of Kohn’s algorithm. We present this moregeometric proof here to provide an alternative to the proof of Proposition 3on pp.380-388 of [DF78] which is the key step of [DF78] and which is stillquite a bit of a challenge to follow. Moreover, the proof of the real-analyticcase of the ineffective termination of Kohn’s algorithm from the geometricviewpoint gives a better understanding of the rˆole played by the real-analytic Partially supported by Grant DMS-0500964 from the National Science Foundation. art I. Global Regularity, Subellipticity, Kohn’s Algorithm, andSpecial Domains (I.1)
The Setting.
We start out with the setting of a bounded domain Ωin C n with smooth boundary defined by r < r on an open neighborhood of the topological closure of Ω. We assume thatΩ is weakly pseudoconvex at all its boundary points in the sense that atany boundary point P of Ω the (1 , ∂ ¯ ∂r is weakly positive definitewhen restricted to the complex tangent space { ∂r = 0 } of the boundary ∂ Ω of Ω at P . (To be more precise, we should have said that √− ∂ ¯ ∂r is weakly positive definite instead of ∂ ¯ ∂r being weakly positive definite, butfor notational simplicity we will drop the factor √− type m at a point P of the boundary ∂ Ω of Ω is the supremum ofthe normalized touching order ord ( r ◦ ϕ )ord ϕ , to ∂ Ω, of all local holomorphic curves ϕ : ∆ → C n with ϕ (0) = P , where∆ is the open unit 1-disk and ord is the vanishing order at the origin 0. Apoint P of the boundary ∂ Ω of Ω is said to be of finite type if the type m at P is finite. This notion of finite type was introduced by D’Angelo [DA79]. Fornotational convenience we also call m the order of finite type instead of justthe “type” to indicate that it is in the sense of D’Angelo’s notion of finitetype.Our goal is to obtain global regularity for any smooth weakly pseudo-convex domain Ω of finite type in the sense that, for any ¯ ∂ -closed smooth(0 , f on Ω which is smooth up to the boundary of Ω, the solution u of ¯ ∂u = f on Ω with u orthogonal to all holomorphic functions on Ω mustalso be smooth up to the boundary of Ω. In this note by “smoothness” wemean infinite differentiability. For notational simplicity we formulate globalregularity only for (0 , , q )-formsfor a general q .Global regularity is a consequence of the subelliptic estimate , which isdefined as follows. For any P ∈ ∂ Ω there exist an open neighborhood U of4 in C n and positive numbers ǫ and C satisfying k| g |k ǫ ≤ C (cid:0) k ¯ ∂g k + k ¯ ∂ ∗ g k + k g k (cid:1) for any (0 , g supported on U ∩ ¯Ω which is in the domain of ¯ ∂ and¯ ∂ ∗ , where k| · |k ǫ is the L norm on Ω involving derivatives up to order ǫ in the boundary tangential direction of Ω and k · k is the usual L norm onΩ without involving any derivatives, and ¯ ∂ ∗ is the actual adjoint of ¯ ∂ withrespect to k · k .Kohn [Ko79] introduced the following notion of multipliers to obtain thesubelliptic estimate. At a point P of ∂ Ω a smooth function germ f at P is called a subelliptic multiplier (or simply called a multiplier ) if there existsome open neighborhood U of P in C n and some positive numbers ǫ and C (all three depending on f ) such that(I . . k| f g |k ǫ ≤ C (cid:0) k ¯ ∂g k + k ¯ ∂ ∗ g k + k g k (cid:1) for any (0 , g supported on U ∩ ¯Ω which is in the domain of ¯ ∂ and¯ ∂ ∗ . We call the positive number ε an order of subellipticity for the multiplier f . We also call a subelliptic multiplier a scalar multiplier to emphasize itsdifference from vector-multipliers introduced below. The collection of allmultipliers at P forms a ideal which is called the multiplier ideal and isdenoted by I P .A germ of a smooth (1 , θ at P is called a subelliptic vector-multiplier if there exist some open neighborhood U of P in C n and somepositive numbers ǫ and C (all three depending on θ ) such that(I . . k| ¯ θ · g |k ǫ ≤ C (cid:0) k ¯ ∂g k + k ¯ ∂ ∗ g k + k g k (cid:1) for any (0 , g supported on U ∩ ¯Ω which is in the domain of ¯ ∂ and ¯ ∂ ∗ ,where ¯ θ · g is the function obtained by taking the pointwise inner product ofthe complex-conjugate ¯ θ of θ with g with respect to the Euclidean metric of C n . We call the positive number ε an order of subellipticity for the vector-multiplier θ . The collection of all vector-multipliers at P forms a modulewhich is called the vector-multiplier module and is denoted by A P .The subelliptic estimate holds at a boundary point P of Ω if a nonzeroconstant function belongs to I P . Kohn introduced the following algorithmto generate elements of I P . 5A) Initial Membership .(i) r ∈ I P .(ii) ∂ ¯ ∂ j r belongs to A P for every 1 ≤ j ≤ n − ∂r = dz n at P forsome local holomorphic coordinate system ( z , · · · , z n ) centeredat P , where ∂ j means ∂∂z j .(B) Generation of New Members .(i) If f ∈ I P , then ∂f ∈ A P .(ii) If θ , · · · , θ n − ∈ A P , then the coefficient of θ ∧ · · · ∧ θ n − ∧ ∂r is in I P .(C) Real Radical Property .If g ∈ I P and | f | m ≤ | g | for some positive integer m , then f ∈ I P .The key point of Kohn’s algorithm is to allow certain differential operatorsto lower the vanishing orders of multipliers so that eventually one can geta nonzero constant as a multiplier. However, there are two limitations onthis process of differentiation to lower vanishing orders. One is that only(1 , , assigned order of subellipticity . In addition to keeping track ofthe number and the nature of the steps of the Kohn algorithm used, theeffectiveness of the termination of the Kohn algorithm seeks to keep track ofthe assigned orders of subellipticity for the individual scalar multipliers andvector-multipliers so that the final nowhere zero multiplier has an effective6ositive number as its assigned order of subellipticity. Note that the assignedorder of subellipticity of a scalar multiplier or vector-multiplier constructedin the Kohn algorithm is in general not the maximum ε for the inequality(I.1.1) or (I.1.2).We will adopt the following rule of giving to a scalar multiplier or avector-multiplier in the Kohn algorithm its assigned order of multiplicity. Asits assigned order of subellipticity we give the scalar multiplier r the number1. As its assigned order of subellipticity we give the number to the vector-multiplier ∂ ¯ ∂ j r at P for every 1 ≤ j ≤ n − if ∂r = dz n at P forsome local holomorphic coordinate system ( z , · · · , z n ) centered at P . If theassigned order of subellipticity of the scalar multiplier f is ε , then we giveto the vector-multiplier ∂f as its assigned order of subellipticity the number ε . If the minimum of the assigned orders of subellipticity of the vector-multipliers θ , · · · , θ n − is ε , then we give to the same ε to the coefficientof θ ∧ · · · ∧ θ n − ∧ ∂r as a scalar multiplier as its assigned order of subellipticity. If the assignedorder of subellipticity of the scalar multiplier g is ε and if | f | m ≤ | g | for somepositive integer m , then we give the number εm to the scalar multiplier f asits assigned order of subellipticity.(I.2) Algebraic-Geometric Description of Finite Type for Special Domains.
A special domain Ω in C n +1 (with coordinates w, z , · · · , z n ) is a boundeddomain given by(I . .
1) Re w + N X j =1 | F j ( z , · · · , z n ) | < , where F j ( z , · · · , z n ) defined on some open neighborhood of ¯Ω in C n +1 de-pends only on the variables z , · · · , z n and is holomorphic in z , · · · , z n foreach 1 ≤ j ≤ N . In what follows, when we consider the subelliptic estimateat a boundary point P of Ω and its type of finite order, if there is no confu-sion we will assume without loss of generality and without explicit mentionthat the point P is the origin of the coordinates w, z , · · · , z n and that F j vanishes at P for each 1 ≤ j ≤ N . Such special domains were introduced byKohn [Ko79, p.115, § p which is the smallestpositive integer such that(I . . | z | p ≤ A N X j =1 | F j ( z ) | on some open neighborhood of the origin in C n for some positive constant A , where z = ( z , · · · , z n ) and | z | = P nℓ =1 | z ℓ | . We will verify below in(I.3) that the order of finite type at the origin P is equal to 2 p .We are going to introduce also two other effectively comparable ways todescribe finite type which are both algebraic-geometrical. The first one isthe following. Let m be the maximum ideal m C n , of C n at the origin. Let I be the ideal on C n generated by holomorphic function germs F , · · · , F N on C n at the origin. Let q be the smallest positive integer such that(I . . . m q ⊂ I . We will verify below in (I.4) that the number p is related to the number q by the inequality p ≤ q ≤ ( n + 2) p . This inequality is far from being sharp.The second is the following. Let s be the dimension over C of the quotientof O C n , by the ideal generated in it by the holomorphic function germs F , · · · , F N on C n at the origin. We will verify below in (I.5) that the number q is related to the number s by the two inequalities q ≤ s and s ≤ (cid:0) n + q − q − (cid:1) .Again this pair of inequalities is far from being sharp.When we prove the effective termination of Kohn’s algorithm for specialdomains, we will in different contexts choose to use one of the three effectivelycomparable descriptions p , q , or s of the order of finite type.Let U be an open neighborhood of 0 in C and ψ : U → C n +1 be aholomorphic map with ψ (0) = 0. Write ψ = ( ψ , ψ ) such that ψ : U → C and ψ : U → C n . For j = 0 , ψ j at 0 of ψ j is thepositive integer s such that ψ j (0) = ( dψ j ) (0) = · · · = (cid:0) d s − ψ j (cid:1) (0) = 0and ( d s ψ j ) (0) = 0. This positive integer s can also be described as thelargest integer such that | ψ j ( ζ ) | ≤ A ,j | ζ | s A ,j when the coordinate ζ of C is small. Thevanishing order ord ( ψ ) at the origin of ψ is equal to the minimum of thevanishing orders ord ( ψ ∗ w ) and ord ( ψ ∗ z j ) of the holomorphic functions ψ ∗ w and ψ ∗ z j of ζ at ζ = 0 for 1 ≤ j ≤ n .In the expansion of the function ψ ∗ r = Re (( ψ ∗ w ) ( ζ )) + N X j =1 | F j ( ψ ( ζ )) | in ζ , ¯ ζ , only terms of the form a ν ζ ν and b ν ¯ ζ ν with ν ≥ i.e., purely holomor-phic or purely antiholomorphic terms in ζ ) can occur in Re (( ψ ∗ w ) ( ζ )) andonly terms of the form c µ,ν ζ µ ¯ ζ ν with µ ≥ ν ≥ i.e., never purely holo-morphic or purely antiholomorphic terms in ζ ) occur in P Nj =1 | F j ( ψ ( ζ )) | (where a ν , b ν , c µ,ν are complex constants). Since there is no possibility at allof any term from Re (( ψ ∗ w ) ( ζ )) canceling a term from P Nj =1 | F j ( ψ ( ζ )) | ,it follows that the vanishing order at 0 of ψ ∗ r must be equal always tothe minimum of the order at 0 of Re (( ψ ∗ w ) ( ζ )) and the order at 0 of P Nj =1 | F j ( ψ ( ζ )) | . Thusord ψ ∗ r ord ψ = min (cid:16) ord ψ ∗ w, ord ψ ∗ P Nj =1 | F j | (cid:17) min (cid:16) ord ψ ∗ w, ord ψ ∗ P nj =1 | z j | (cid:17) . (I.3) Lemma.
Let p be the smallest positive integer which satisfies (I.2.2) forsome positive constant A . Then the order t of finite type at the origin forthe special domain Ω given by (I.2.1) is equal to 2 p . Proof.
We are going to prove t = 2 p by proving the two inequalities t ≤ p and t ≥ p . We first prove the inequality t ≤ p . From the definition of theorder t of finite type we know that there exist some open neighborhood U of0 in C and some holomorphic map ψ = ( ψ , ψ ) : U → C n +1 = C × C n with ψ (0) = 0 such that t = ord ψ ∗ r ord ψ = min (cid:16) ord ψ ∗ w, ord ψ ∗ P Nj =1 | F j | (cid:17) min (cid:16) ord ψ ∗ w, ord ψ ∗ P nj =1 | z j | (cid:17) . We let α = ord ψ ∗ w, β = 12 ord ψ ∗ n X j =1 | z j | , γ = ord ψ ∗ N X j =1 | F j | . p we have γ ≤ pβ . We differentiate among thefollowing three cases.Case 1. α ≤ β .Case 2. β < α ≤ pβ Case 3. α > pβ .For Case 1, we have t = min( α, γ )min( α, β ) ≤ min( α, pβ )min( α, β ) = αα ≤ p. For Case 2, we have t = min( α, γ )min( α, β ) ≤ min( α, pβ )min( α, β ) = αβ ≤ pββ = 2 p. For Case 3, we have t = min( α, γ )min( α, β ) ≤ min( α, pβ )min( α, β ) = 2 pββ = 2 p. Thus in all three cases t ≤ p .We are now going to prove the other inequality 2 p ≤ t . We use a si-multaneous resolution of embedded singularities π : ˜ W → W for some openneighborhood W of the origin in C n with exceptional hypersurfaces { Y j } Jj =1 in ˜ W in normal crossing so that the pullback π ∗ m of the maximum idealon W at the origin is equal to the ideal sheaf of the divisor P Jj =1 σ j Y j forsome nonnegative integers σ , · · · , σ J and the pullback π ∗ I of the ideal sheaf I on C n generated by F , · · · , F N is equal to the ideal sheaf of the divisor P Jj =1 τ j Y j for some nonnegative integers τ , · · · , τ J .Since p is the smallest integer which satisfies condition (I.2.2) for somepositive constant A , it follows that pσ j ≤ τ j for all 1 ≤ j ≤ J . Take any1 ≤ j ≤ J with σ j > ∈ π ( Y j ) such that there is a regular point Q of some Y j with the property that π ( Q ) = 0 and Q does not belong to anyother Y j with j = j . Take a local regular complex curve ˜ C in ˜ W representedby a holomorphic map ˜ ϕ : U → ˜ W from some open neighborhood U of 0 in C to ˜ W such that ˜ ϕ (0) = Q and the local complex curve ˜ C is transversal to10 j and disjoint from any other Y j with j = j . Now we define a holomorphicmap ψ = ( ψ , ψ ) : U → C n +1 = C × C n by ψ ≡ ψ = π ◦ ϕ . Thenord ψ ∗ r ord ψ = min (cid:16) ord ψ ∗ w, ord ψ ∗ P Nj =1 | F j | (cid:17) min (cid:16) ord ψ ∗ w, ord ψ ∗ P nj =1 | z j | (cid:17) = ord ψ ∗ P Nj =1 | F j | ord ψ ∗ P nj =1 | z j | = 2 τ j σ j ≥ p. By the definition of t we have t ≥ p . Putting the two inequalities t ≤ p and t ≥ p together, we get t = 2 p . Q.E.D.(I.4) Lemma.
Let p be the smallest positive integer which satisfies (I.2.2) forsome positive constant A . Let q be the smallest positive integer such thatthe q -th power m q of the maximum ideal m of C n at the origin is containedin the ideal I generated by the holomorphic function germs F , · · · , F N on C n at the origin. Then p ≤ q ≤ ( n + 2) p . Proof.
From the definition of q it follows that z qℓ ∈ m q ⊂ I = N X j =1 O C n , F j for every 1 ≤ ℓ ≤ n it follows that | z qℓ | ≤ ˜ A ℓ N X j =1 | F j ( z ) | for some positive constant ˜ A ℓ for 1 ≤ ℓ ≤ n on some open neighborhood ofthe origin in C n . Hence | z | q = n X ℓ =1 | z ℓ | ! q ≤ n q max ≤ ℓ ≤ n | z ℓ | q ≤ n q (cid:18) max ≤ ℓ ≤ n ˜ A ℓ (cid:19) N X j =1 | F j ( z ) | and p ≤ q from the definition of p .For the proof of the inequality p ≤ ( n +2) q , we need the following theoremof Skoda [Sk72, Th.1, pp.555-556]. 11et D be a pseudoconvex domain in C n and χ be a plurisubharmonic functionon D . Let g , . . . , g m be holomorphic functions on D . Let α > ℓ =inf( n, m − F on D such that Z D | F | | g | − αℓ − e − χ < ∞ , there exist holomorphic functions f , . . . , f m on Ω such that F = m X j =1 g j f j and Z D | f | | g | − αℓ e − χ ≤ αα − Z D | F | | g | − αℓ − e − χ , where | g | = m X j =1 | g j | ! , | f | = m X j =1 | f j | ! . For nonnegative integers γ , · · · , γ n with γ + · · · + γ n = ( n + 2) p we applySkoda’s theorem to the case of F = n Y j =1 z γ j j , m = N + n,χ ≡ , { g , · · · , g m } = { F , · · · , F N , , · · · , } ,ℓ = n, α = n + 1 n , with D being some small open ball neighborhood of the origin in C n , toconclude from (I.2.2) that n Y j =1 z γ j j ∈ I = N X j =1 O C n , F j . Hence q ≤ ( n + 2) p . Q.E.D.(I.5) Lemma.
Let q be the smallest positive integer such that the q -th power m q of the maximum ideal m of C n at the origin is contained in the ideal I generated by the holomorphic function germs F , · · · , F N on C n at the origin.Let s be the dimension over C of the quotient of O C n , by the ideal I . Then q ≤ s and s ≤ (cid:0) n + q − q − (cid:1) . 12 roof. Let R = O C n , . Since m q ⊂ I , it follows that s = dim C R / I ≤ dim C R / m q = (cid:18) n + q − q − (cid:19) . On the other hand, we consider the following sequence of nested C -linear sub-spaces of the finite-dimensional C -vector spaces R / I of complex dimension s . R / I ⊃ m ( R / I ) ⊃ m ( R / I ) ⊃ · · · ⊃ m ℓ ( R / I ) ⊃ m ℓ +1 ( R / I ) ⊃ · · · There exists some 1 ≤ ℓ ≤ s such that m ℓ ( R / I ) = m ℓ +1 ( R / I ) . By Nakayama’s lemma we have0 = m ℓ ( R / I ) = (cid:0) m ℓ + I (cid:1) / I and we conclude that m ℓ + I = I and m ℓ ⊂ I , which implies that q ≤ s .Q.E.D.Later we will need the following corollary of Lemma (I.5) which is aversion of the effective Nullstellensatz in terms of multiplicity.(I.6) Lemma.
Let I be an ideal in O C n , such that its multiplicity is no morethan some positive integer m in the sense that the complex dimension of thequotient of O C n , by I is no more than m . Let f be a holomorphic functiongerm on C n at the origin which vanishes at the origin. Then f m belongs to I . Proof.
Let I = T Jj =1 Q j be the primary decomposition of the ideal I inthe Noetherian ring O C n , and let P j be the radical of the primary ideal Q j .Since m is the multiplicity of I , we have J ≤ m . Let Z j be the subvarietygerm of C n at the origin whose ideal at the origin is P j and let Z j be the setof all regular points of Z j , which without loss of generality we can assumeto be connected. Let n j be the complex codimension of Z j at the origin.Take a generic point Q j in Z j and let Π j be a complex C -linear subspace ofcomplex dimension n j C n which is transversal to Z j at the point Q j . Themultiplicity of the ideal at Q j induced by Q j is no more than m (if we assumewithout loss of generality that Q j is sufficiently close to the origin). Let J j
13e the ideal on Π j ≈ C n j at Q j induced by Q j . Then the multiplicity of J j at the point Q j is no more than m . By Lemma (I.5) applied to Π j ≈ C n j andthe ideal J j on Π j ≈ C n j , we conclude that the holomorphic function germ (cid:0) f | Π j (cid:1) m on Π j ≈ C n j at Q j belongs to the ideal J j . Since Q j is a genericpoint in Z j and since Q j is a primary ideal, it follows that the holomorphicfunction germ f m on C n at the origin belongs to Q j . From J ≤ m it followsthat the holomorphic function germ f m on C n at the origin belongs to theproduct of Q j for 1 ≤ j ≤ J . In particular, holomorphic function germ f m on C n at the origin belongs to I . Q.E.D.14 art II. Algebraic Formulation and Sketches of Techniques (II.1) Algebraic Formulation of Kohn’s Algorithm for Special Domains . Theeffective termination of Kohn’s algorithm for a special domain is reduced tothe purely algebraic-geometric description of items (i) through (vii) listedbelow. For the case of a special domain the setting is as follows. We have aspecial domain Ω in C n +1 (with coordinates w, z , · · · , z n ) defined by r := Re w + N X j =1 | F j ( z , · · · , z n ) | < , where for each 1 ≤ j ≤ N , F j ( z , · · · , z n ) is a holomorphic function van-ishing at the origin which is independent of w and is defined on some openneighborhood of ¯Ω in C n +1 . The boundary point of Ω under consideration isthe origin of C n +1 .In this setting, first of all, from dr = dw at the origin and ∂ ¯ ∂r = N X j =1 dF j ∧ dF j we conclude from (I.1)(A)(ii) and standard techniques of estimates in Kohn’stheory of multipliers [Ko79] that dF j is a vector multiplier which can be given as its assigned order of subellipticity, because the vector-multiplier ∂ (cid:18) ∂r∂z j (cid:19) = N X ℓ =1 (cid:18) ∂h ℓ ∂z j (cid:19) dh ℓ at the origin can be given as its assigned order of subellipticity for 1 ≤ j ≤ n .(i) We start out with the N given holomorphic function germs F , · · · , F N on C n at the origin with the origin as their only common zero-point. Themultiplicity q of the ideal generated by F , · · · , F N at the origin is what weuse for effectiveness statements. That is, a number is considered effective ifit can be estimated by an explicit expression in q .(ii) Select n C -linear combinations g , · · · , g n of F , · · · , F N .(iii) Form the Jacobian determinant of g , · · · , g n with respect to z , · · · , z n .15iv) Take the ideal I generated by all such Jacobian determinants.(v) Choose a finite subset ϕ , · · · , ϕ ℓ of the radical J of I and let σ be apositive number such that ( ϕ j ) σ ∈ I for 1 ≤ j ≤ ℓ .(vi) Replace the set F , · · · , F N by F , · · · , F N , ϕ , · · · , ϕ ℓ and repeat theabove procedure.(vi) Repeat until we get to the point that ϕ can be chosen to be nonzero atthe origin.(vii) Effectiveness means that we have an effective number of steps and alsoan effective bound on σ at each step.(II.2) Sketch of Proof of Effectiveness for Special Domains . We now give anoverview of the logical framework for the proof of the effective terminationof the Kohn algorithm for special domains. Details for the derivation of thebounds of the multiplicities of functions constructed from generic C -linearcombinations and Jacobian determinants which occur in this overview willnot be explained here but will be presented later in (III.3), (III.4), and (III.5).We start out with an ideal generated by holomorphic function germs F , · · · , F N on C n at the origin whose common zero-set is the origin. Themultiplicity q of the ideal generated by F , · · · , F N at the origin is what we usefor effectiveness statements. For n generic C -linear combinations g , · · · , g n of F , · · · , F N the multiplicity of the function f defined by dg ∧ · · · ∧ dg n = f ( dz ∧ · · · ∧ dz n ) is no more than m q at the origin, where m q is some positiveinteger depending effectively on q (see (III.5)). The main idea is to use theprocedure of replacing C n by the subspace V defined by the multiplier f tocut down successively on the dimension of the zero-set of multipliers whilemaintaining effectiveness.There are two difficulties here. One difficulty is that the subspace definedby f is in general not regular. The other difficulty is that we are allowedonly to form Jacobian determinants of C -linear combinations g , · · · , g n of F , · · · , F N and not allowed to form the Jacobian determinants of the restric-tions of such C -linear combinations g , · · · , g n − to V . The two difficultiesare related. If V is nonsingular, we could compute the Jacobian determi-nant of g | V , · · · , g n − | V by computing the coefficient of dz ∧ · · · ∧ dz n in dg ∧ · · · ∧ dg n − ∧ df .When V is singular at the origin, we have to differentiate f not just onceto form df but as many times as the multiplicity of V . To enable us to do16t by using Jacobian determinants, we construct a Weierstrass polynomial ˜ f in z n whose coefficients are functions of g , · · · , g n − so that ˜ f vanishes onthe subspace V and therefore contains f as a factor. We then differentiate˜ f as many times as its multiplicity at the origin by applying the operator dg ∧ · · · ∧ dg n − ∧ d ( · ) to ˜ f and making use of the fact that ˜ f is a Weierstrasspolynomial of the type described above. To continue applying the operator dg ∧ · · · ∧ dg n − ∧ d ( · ) to ˜ f , we need to modify first the result from theprevious differentiation by comparing on V the Jacobian determinant ∂ ( g , · · · , g n − ) ∂ ( z , · · · , z n − )with an appropriate polynomial p ( g , · · · , g n − ) of g , · · · , g n − and using theReal Radical Property of Kohn’s algorithm in (I.1)(B) to replace ∂ ( g , · · · , g n − ) ∂ ( z , · · · , z n − )by p ( g , · · · , g n − ). The final result of differentiating ˜ f this way as manytimes as the multiplicity of ˜ f at the origin produces a new multiplier whichdefines on V a subspace with effective multiplicity at the origin. This wayof cutting down on the dimension of the subspace defined by such effectivelyconstructed multipliers gives the effective termination of Kohn’s algorithmfor special domains.In the details of the proof for special domains given below in (III.6),(III.7), (III.8), and (III.9), we actually do not carry out completely the in-duction of cutting down on the dimension of the zero-set of effectively con-structed multipliers. A short-cut is used to simplify the process to reach thesame goal (see (III.9)).(II.3) Modification for Effectiveness for Real-Analytic Case.
Before we givethe rigorous details of the proof of the effective termination of the Kohn al-gorithm for special domains, we would like to discuss how the techniques inthe above sketch for special domains in (II.2) can be modified for the gen-eral real-analytic case. We consider the following real-analytic case wherethe weakly pseudoconvex domain of finite type is defined by r < r ( z , · · · , z n , z , · · · , z n ) being real-analytic and vanishing at the origin (whichis the boundary point we consider). The main idea is to let w j = z j for1 ≤ j ≤ n and let R be the ring of convergent power series in w , · · · , w n and17onsider the n + 1 holomorphic function germs H , H , · · · , H n on C n (withcoordinates z , · · · , z n ) at the origin with coefficients in the ring R definedas follows. H ( z , · · · , z n ) = r ( z , · · · , z n , w , · · · , w n ) ,H j ( z , · · · , z n ) = ∂r∂w j ( z , · · · , z n , w , · · · , w n ) for 1 ≤ j ≤ n, where the coefficients of the power series expansion of H j in z , · · · , z n areall elements of R for 0 ≤ j ≤ n . For the complex Euclidean space C n withcoordinates z , · · · , z n we denote by m C n , the maximum ideal of C n at theorigin. Finite type condition for the domain { r < } at the origin impliesthe statement that(II.3.1) there exists some effective positive integer q such that R ( m C n , ) q iscontained in the ideal generated by H , H , · · · , H n in the ring R { z , · · · , z n } of convergent power series in z , · · · , z n with coefficients in R .The statement (II.3.1) simply follows directly from the definition of finitetype. It can be regarded as the real-analytic analog of condition (I.2.3) fora special domain. Note that finite type is actually much stronger than thestatement (II.3.1).In a way analogous to applying condition (I.2.3) to do an inductivemultiplier-construction process to obtain a nonzero constant as a multiplierfrom the Kohn algorithm for a special domain as described in (II.2), we nowapply statement (II.3.1) to do the same inductive multiplier-constructionprocess with the difference that now the coefficients of the power series ofthe function germs involved are elements of R = { w , · · · , w n } instead ofjust C . One modification is needed for the inductive multiplier-constructionprocess. When we are in the case of a special domain, we use n generic C -linear combinations g , · · · , g n of F , · · · , F N , but here in the real-analyticcase when we choose n R -linear combinations g , · · · , g n of H , · · · , H n , oneof g , · · · , g n must be chosen to be H . The reason for this modification isthat we are not in the special case where the domain is of the formRe w + r ( z , · · · , z n , z , · · · , z n ) < ∂r to definethe tangent space of type (1 ,
0) for the boundary of the domain.18ote that when we take ∂G ∧ · · · ∧ ∂G n − ∧ ∂H for generic C -linearcombinations G , · · · , G n − of H , · · · , H n , we are simply using (I.1)(A)(ii)and (I.1)(B)(ii) in Kohn’s algorithm.The inductive multiplier-construction process in the real-analytic casenow gives us a nonzero element f of R instead of a nonzero element of C inthe case of a special domain. The main point is that, because of the finitetype condition the multiplicity of this element f ( w , · · · , w n ) of R at 0 isbounded effectively by a constant depending on n and the order of the finitetype. Now we consider the anti-holomorphic function germ ˜ f on C n at theorigin defined by ˜ f = f ( z , · · · , z n ) and consider the complex conjugate g of˜ f . We let V be the subspace germ defined by the holomorphic functiongerm g on C n at the origin. We then consider V × V in C n × C n insteadof the full 2 n -dimensional complex Euclidean space C n × C n itself (with z , · · · , z n , z , · · · , z n being the variables of C n × C n ). Let R be the ring ofholomorphic function germs on V at 0 when V is considered as a subspacegerm of 0 in C n at the origin with coordinates w , · · · , w n . We now applythe inductive process to obtain a holomorphic function germ f on V at 0(which is a subspace germ at 0 of C n with variables w , · · · , w n ).Now we consider the function germ ˜ f obtained from f by replacing w , · · · , w n by z , · · · , z n . Let g be the complex-conjugate of ˜ f . Let V bea complete intersection of codimension two in C n at the origin defined bytwo holomorphic functions which belong to the radical of the ideal generatedby g and the ideal of V . We then consider V × V in C n × C n instead of C n × C n itself (with z , · · · , z n , z , · · · , z n being the variables of C n × C n ).Let R be the ring of holomorphic function germs on V at 0 when V isconsidered as a subspace germ of 0 in C n with coordinates w , · · · , w n . Wenow can continue with this inductive subspace-construction process whichso far yields for us the subspace of complete intersection V and V . Wecontinue with this inductive subspace-construction process to get V ℓ +1 from V ℓ for 1 ≤ ℓ ≤ n − V n of C n which consistsonly of the origin. This then immediately gives us the effective terminationof Kohn’s algorithm for the real-analytic case. Again, instead of carryingout completely the inductive argument of cutting down the dimension of thesubspace described above, it is also possible to use the analog of the short-cuttechnique given in (III.9). 19nother way of describing this modification is to redo the algebraic-geometric argument for the case of a special domain but to do it over aparameter space defined by the ring R . The coordinates for R are thecomplex-conjugates of the coordinates for the ambient space C n . We candescribe the modification as redoing the algebraic-geometric argument forthe case of a special domain over Spec( R ) instead of over the single pointSpec ( C ). While the case of a special domain yields effectively a nonzeroelement of C as a multiplier, the real-analytic case would yield effectively anonzero element of R . Then we replace C n or by the subspace defined bythis nonzero element of R and repeat the argument to get down to lowerand lower dimensional subspaces until we get to a single point, or we use theanalog of the short-cut technique given in (III.9).(II.4) Modification for Effectiveness for Smooth Case.
We are going to haveyet another discussion, this time about modifying further the techniques inthe above sketch for special domains in (II.2) in order to handle the generalsmooth case, before going into the rigorous details of the proof of the effectivetermination of the Kohn algorithm for special domains. Now suppose thatwe have a smooth bounded weakly pseudoconvex domain Ω of finite typegiven by r < r defined on some neighborhood ofthe topological closure ¯Ω of Ω in C n and that the origin 0 of C n is a boundarypoint of Ω.Let q be the positive integer which is the order of the finite type of theorigin as a boundary point of Ω. Let r N be the N -th partial sum of the formalpower series expansion of r at the origin with respect to the coordinates z , · · · , z n of C n . We choose N effectively large enough so that the type of r N = 0 at the origin is also q .We apply Kohn’s algorithm for the real-analytic case to r N . From the ef-fectiveness for the real-analytic case (II.3), we can find some positive integer N q which depends only on q and n such that the assigned order of subel-lipticity ε for the final nonzero constant multiplier from the effective Kohnalgorithm for r N satisfies ε > N q .When we choose N effectively large enough, for example, N > N p , theeffective termination of Kohn’s algorithm for r N also gives the effective ter-mination of Kohn’s algorithm for r with precisely the same steps and thesame assigned order of subellipticity for each step. Note that this process ofapproximating r by r N is very different from the approximation of a bounded20mooth weakly pseudoconvex domain of finite type by a real-analytic smoothweakly pseudoconvex domain of finite type, which is in general not possible.The N -th partial sum r N is simply used as an algebraic-geometric compari-son guide to guarantee the effective termination of Kohn’s algorithm for theoriginal smooth defining function r .Note that when we do the approximation of r by r N , we are doing thisapproximation only at the boundary point under consideration and not usingthe approximation along the normal directions of the zero-sets of multipli-ers from Kohn’s algorithm for r . The reason is that the purpose of theapproximation is to use the effective termination of Kohn’s algorithm forthe real-analytic function r N to conclude for a sufficiently large N that thecorresponding steps result in the effective termination of Kohn’s algorithmfor the smooth function r . The motivation for choosing this procedure ofapproximation is twofold. One is that the notion of finite type at a bound-ary point of the weakly pseudoconvex domain depending only on the formalpower series expansion of the defining function r at that point. The otheris that the zero-sets of multipliers from Kohn’s algorithm for r are definedby the vanishing of smooth functions and it is not clear how one can do areal-analytic approximation along the normal directions of such zero-sets. Inour use of the approximation of r by r N , the zero-sets of multipliers fromKohn’s algorithm for r are different from the zero-sets of multipliers fromKohn’s algorithm for r N . When we use the “real radical property” to pro-duce multipliers from Kohn’s algorithm for r N , we simply perform the sameoperation for the corresponding but different zero-set in Kohn’s algorithmfor r . 21 art III. Details of Proof of Effective Termination of Kohn’s Algo-rithm for Special Domains (III.1) Precise Formulation.
Let F , · · · , F N be holomorphic function germson C n at the origin 0. Assume that s := dim C O C n , , N X j =1 O C n , F j ! < ∞ so that the subscheme of C n defined by F , · · · , F N is an Artinian subscheme.We will call s the multiplicity of the ideal generated by F , · · · , F N . This def-inition agrees with that given in (III.3) below for ideals generated by k holo-morphic function germs whose common zero-set is of complex codimension k . Let A = N X j =1 O C n , ( dF j )be the O C n , -submodule of the O C n , -module O C n , ( T C n , ) ∗ of all germs ofholomorphic (1 , C n at 0. Take a sequence of positive integers q ν for any positive integer ν . By induction on the positive integer ν we defineas follows the ideals I ν and J ν of O C n , and the O C n , -submodule A ν +1 ofthe O C n , -module O C n , ( T C n , ) ∗ .For ν ≥ J ν of O C n , is generated over O C n , by all holomorphicfunction-germs f on C n at 0 satisfying g ∧ · · · ∧ g n = f ( dz ∧ · · · ∧ dz n )with g , · · · , g n ∈ A ν . The ideal I ν is defined by the set of all holomorphicfunction germs f on C n at 0 so that f q ∈ J ν for some 1 ≤ q ≤ q ν .For ν ≥ O C n , -submodule A ν of the O C n , -module O C n , ( T C n , ) ∗ isgenerated by all df for f ∈ I ν − and all elements of A ν − .(III.2) Main Theorem.
There exists an explicit sequence { q ν } ν ∈ N and anexplicit number m depending only on n and s such that I m = O C n , .To prepare for the proof of the Main Theorem, we put together some lem-mas about selecting C -linear combinations of F , · · · , F N to generate idealswith effective multiplicity and about estimating the multiplicity of Jacobiandeterminants. 22III.3) Lemma (on Selection of Linear Combinations of Holomorphic Func-tions for Effective Multiplicity).
Let 0 ≤ q ≤ n . Let f , · · · , f q be holomor-phic function germs on C n at the origin whose common zero-set W q is of purecodimension q in C n as a subvariety germ, with the convention that W = C n and P j =1 O C n , f j = 0 for the case q = 0. Let m be the multiplicity of theideal P qj =1 O C n , f j at the origin in the sense thatdim C O C n , , q X j =1 O C n , f j + n − q X j =1 O C n , L j ! ! = m for any n − q generic C -linear functions L , · · · , L n − q on C n . Let F j ( z , · · · , z n )(1 ≤ j ≤ N ) be holomorphic function germs on C n at the origin which vanishat the origin. Let p be a positive integer and A be a positive number suchthat(III . . | z | p ≤ A N X j =1 | F j ( z ) | for all z in the domain of definition of F j ( z , · · · , z n ) (1 ≤ j ≤ N ). Then forgeneric choices of complex numbers { c j,k } ≤ j ≤ n − q, ≤ k ≤ N the C -linear combinations˜ F j = N X k =1 c j,k F k (1 ≤ j ≤ n − q )of F , · · · , F N satisfy the property thatdim C O C n , , q X j =1 O C n , f j + n − q X j =1 O C n , ˜ F j ! ! ≤ mp n − q . That is, the multiplicity of the ideal generated by f , · · · , f q , ˜ F · · · , ˜ F n − q − is ≤ mp n − q at the origin. Proof.
We use induction on 1 ≤ ν ≤ n − q to show that for generic complexnumbers { c j,k } ≤ j ≤ ν, ≤ k ≤ N V ν of the f , · · · , f q andthe C -linear combinations˜ F j = N X k =1 c j,k F k (1 ≤ j ≤ ν )of F , · · · , F N is precisely n − q − ν and the multiplicity of the ideal q X j =1 O C n , f j + ν X j =1 O C n , ˜ F j is no more than mp ν .We introduce the case of ν = 0 and the convention that P j =1 O C n , ˜ F j =0 for the case ν = 0. With this convention, we start out our inductionassumption with the case ν = 0 which is trivially true.Suppose the induction process has been carried out for some 0 ≤ ν < n − q and we would like to verify it for the next step when ν is replaced by ν + 1.We now already have ˜ F , · · · , ˜ F ν . Let I ν = q X j =1 O C n f j + ν X j =1 O C n ˜ F j . The zero-set of I ν is the subvariety V ν of pure dimension n − q − ν . Let E ν be a generic linear subspace of C n of codimension n − q − ν − n − q − ν − G , · · · , G n − q − ν +1 so that the subvariety V ν ∩ E ν is of pure dimension 1. Let J ν = n − q − ν − X j =1 O C n G j + q X j =1 O C n f j + ν X j =1 O C n ˜ F j . Let J ν = Λ \ λ =1 L λ be the primary decomposition of the ideal sheaf J ν . Note that, since thezero-set of J ν is of pure complex dimension 1 and J ν is generated by n − G , · · · , G n − q − ν +1 , f , · · · , f q , ˜ F , · · · , ˜ F ν , it follows that all the associated prime ideals of J ν are isolated and none areembedded [ZS60, p.397, Theorem 2]. 24et C λ be the complex curve-germ which is the zero-set of the ideal sheaf L λ . Let µ λ be the multiplicity of the curve C λ at the origin. Let ˆ µ λ be themultiplicity of the ideal sheaf L λ at a generic point Q ∈ C λ , which can becharacterized as the dimension over C of O C n ,Q .(cid:16) ( L λ ) Q + O C n ,Q L (cid:17) , where L is a generic polynomial of degree 1 on C n vanishing at Q and ( L λ ) Q is the stalk of the ideal sheaf L λ at the point Q .Without loss of generality we can assume that the coordinates ( z , · · · , z n )of C n are chosen so that C λ is defined by ( z = ζ µ λ ,z j = g λ,j ( ζ ) for 2 ≤ j ≤ n for ζ in some open neighborhood of the origin in C , where the initial termof g λ,j ( ζ ) is a nonzero complex number times ζ N λ,j for some N λ,j ≥ µ λ for2 ≤ j ≤ n . Let π λ : ˜ C λ → C λ be the normalization of C λ defined by π λ : ζ z = ( ζ µ λ , g λ, ( ζ ) , · · · , g λ,n ( ζ )) , where ˜ C λ is an open neighborhood of 0 in C with ζ as coordinate. Thepullback π ∗ λ m C λ , to ˜ C λ of the maximum ideal m C λ , of C λ at the origin isgenerated by ζ µ λ , g λ, ( ζ ) , · · · , g λ,n ( ζ ). Since π ∗ λ m C λ , is a principal ideal, itmust be generated by ζ µ λ .The inequality (III.3.1), when pulled back by π λ , becomes(III . . ν | ζ | pµ λ ≤ A λ N X j =1 | ( F j ◦ π λ ( ζ )) | for 1 ≤ λ ≤ Λ, where A λ is a positive number. Take a generic point( c ν +1 , , · · · , c ν +1 ,N ) ∈ C N and let ˜ F ν +1 = N X k =1 c ν +1 ,k F k .
25y (III . . ν , for each 1 ≤ λ ≤ Λ the vanishing order of (cid:16) ˜ F ν +1 ◦ π λ (cid:17) ( ζ ) at ζ = 0 is some number ˜ µ λ which is no more than pµ λ . For a small genericnonzero η ∈ C the number of zeros of η + (cid:16) ˜ F ν +1 ◦ π λ (cid:17) ( ζ ) on ˜ C λ is precisely˜ µ λ with multiplicity 1 for each 1 ≤ λ ≤ Λ. Since the map π λ : ˜ C λ → C λ isone-to-one, it follows that for any small generic nonzero η ∈ C the numberof zeroes of η + ˜ F ν +1 on C λ is precisely ˜ µ λ with multiplicity 1.Since the multiplicity of the ideal sheaf L λ at a generic point Q ∈ C λ isˆ µ λ , it follows from ˜ µ λ ≤ pµ λ that the dimension over C of the vector space O C n , , n − q − ν − X j =1 O C n , G j + q X j =1 O C n , f j + ν +1 X j =1 O C n , ˜ F j ! is no more than p P Λ λ =1 µ λ ˆ µ λ . By induction hypothesis the multiplicity of I ν = q X j =1 O C n f j + ν X j =1 O C n ˜ F j is no more than mp ν at the origin. The multiplicity of J ν at the ori-gin, which can be computed from I ν by adding generic C -linear functions G , · · · , G n − q − ν − on C n , is also no more than mp ν . We can compute themultiplicity of J ν at the origin by adding to J ν a generic C -linear function L on C n and considering the sum of the multiplicities at points of intersectionof the zero-set with L + η for some small generic η ∈ C . From the decompo-sition J ν = T Λ λ =1 L λ and the multiplicity ˆ µ λ of L λ at the origin we concludethat Λ X λ =1 µ λ ˆ µ λ ≤ mp ν . Thusdim C O C n , , n − q − ν +1 X j =1 O C n , G j + q X j =1 O C n , f j + ν +1 X j =1 O C n , ˜ F j ! ! ≤ mp ν +1 . Since G , · · · , G n − q − ν +1 are generic linear functions on C n , it follows that themultiplicity of the ideal q X j =1 O C n , f j + ν +1 X j =1 O C n , ˜ F j
26t the origin is no more than mp ν +1 . This finishes the induction process.Q.E.D.(III.4) Corollary.
Let F j ( z , · · · , z n ) (1 ≤ j ≤ N ) be holomorphic functiongerms on C n at the origin which vanish at the origin. Let p be a positiveinteger and A be a positive number such that | z | p ≤ A N X j =1 | F j ( z ) | for all z in the domain of definition of F j ( z , · · · , z n ) (1 ≤ j ≤ N ). Then forgeneric choices of complex numbers { c j,k } ≤ j ≤ n, ≤ k ≤ N the C -linear combinations˜ F j = N X k =1 c j,k F k (1 ≤ j ≤ n )of F , · · · , F N satisfy the property thatdim C O C n , , n X j =1 O C n , ˜ F j ! ≤ p n . Proof.
Introduce one more complex variable w and consider F j as a holo-morphic function germ on C n +1 at 0 in the variables z , · · · , z n , w thoughit is independent of the variable w . Add the function w to the functions F , · · · , F N . Let f = w and apply Lemma (III.3) on Selection of LinearCombinations of Holomorphic Functions for Effective Multiplicity to the case m = 1 with C n replaced by C n +1 . Q.E.D.(III.5) Lemma (Multiplicity Estimate for Jacobian Determinant).
Let g , · · · , g n be holomorphic function germs on C n at the origin such thatdim C O C n , , n X j =1 O C n , g j ! ≤ m. Let dg ∧ · · · ∧ dg n = f ( dz , · · · , z n ). Then the multiplicity of f at the originis ≤ m . 27 roof. We can find a connected open neighborhood U of 0 in C n and anopen ball neighborhood W of 0 in C n such that the map π : U → W definedby g , · · · , g n is a proper holomorphic map. This is possible, because thecommon zero-set of g , · · · , g n consists only of the origin in a sufficientlysmall neighborhood of the origin in C n . The number of sheets in the analyticcover map π : U → V is ≤ m . Let Y be the divisor of f in U and Z be theimage of Y in W . Let Z be the set of regular points of (the reduction of Z ).Let L be a generic complex line in the target space C n such that L ∩ Z is asingle point P in Z and L intersects Z transversally at P . For a sufficientlysmall neighborhood D of P in W the map U ∩ π − ( D ) → D induced by π isjust a cyclic branched cover on each topological component of U ∩ π − ( D ).Thus the multiplicity of the intersection of the regular curve π − ( L ) and thedivisor Y is no more than the number of sheets of π : U → W . Since the line L is generic, it follows that the multiplicity of the divisor Y is more than m .Q.E.D.(III.6) Preparatory Remarks on Proof of Main Theorem.
We now start thesetting for the proof of the Main Theorem (III.2). Let F , · · · , F N be holo-morphic function germs on C n at the origin whose common zero-set is theorigin. Let q be a positive integer. Assume that, for some positive number A ,(III . . | z | q ≤ A N X j =1 | F j ( z ) | for all z in the domain of definition of F j ( z , · · · , z n ) (1 ≤ j ≤ N ). Be-cause of the discussion in (II.1), for the case of special domains we need onlyconsider multipliers which are holomorphic and we need only consider vector-multipliers which are holomorphic (1 , F , · · · , F N are notmultipliers, their differentials dF , · · · , dF N are vector-multipliers and, in or-der to form Jacobian determinants to generate multipliers, we can also use ℓ C -linear combinations of F , · · · , F N and n − ℓ multipliers for 0 ≤ ℓ ≤ n toform a Jacobian determinant which will then be a multiplier. We will referto any C -linear combination of F , · · · , F N and multipliers as pre-multipliers so that the (1 , C -vector space but do not form an ideal. Theproduct of a multiplier and a holomorphic function germ is again a multi-plier, but the product of a pre-multiplier and a holomorphic function germ28n general is not a pre-multiplier. In our proof of the Main Theorem (III.2)we will not use vector-multipliers, because we will directly form the Jacobiandeterminants of the holomorphic pre-multipliers to generate new multipli-ers to bypass the process of forming vector-multipliers by differentiation andthen using Cramer’s rule.In order not to be encumbered by complicated expressions of constants,we will not explicitly keep track of the various effective bounds occurring inthe proof. We introduce the following terminology. A multiplier is called effectively constructed if there is an effective upper bound for its multiplicityand there is an effective positive lower bound for its assigned order of subel-lipticity. Effective means some explicit function of the multiplicity of theideal generated by the pre-multipliers F , · · · , F N , which means an explicitfunction of q given in (III.6.1). The goal is to show that the function-germwith constant value 1 can be effectively constructed.To make the argument more transparent and to minimize notational clut-ters, we start out with the proof of the simple case where n = 2.(III.7) Proof of Main Theorem for Dimension Two.
We now assume that n = 2 and we have holomorphic function germs F , · · · , F N on C at theorigin whose zero-set is the origin of C n . The multiplicity of the ideal gener-ated by F , · · · , F N is the number used to express effectiveness. By applyingCorollary (III.4) and (III.5) to get two C -linear combinations of F , · · · , F N and form their Jacobian determinant, we get an effectively constructed mul-tiplier ˜ h ( z , z ) at the origin, which vanishes at the origin. Because themultiplicity of ˜ h ( z , z ) is effectively bounded at the origin, by replacing˜ h ( z , z ) by the product of holomorphic function germs defining the branchgerms of the reduction of the subspace defined by ˜ h ( z , z ), we can assumewithout loss of generality that the subspace germ C defined by ˜ h is a re-duced curve germ in C at the origin with effectively bounded multiplicity.Note that in general the curve germ C is not irreducible, though C is areduced curve. A reduced curve means that its structure sheaf does not con-tain any nonzero nilpotent elements. For example, it means that h does notvanish to order higher than one at any regular point of C .Now the ideal generated by the functions ∂F i ∂z j for 1 ≤ i ≤ N and j =1 , ≤ i ≤ N , the function germ ( F i ) at the origin belongs to the ideal generated by ∂F i ∂z j for j = 1 ,
2. We consider29he pre-multiplier h = P Nj =1 c j F j with generic c j ∈ C for 1 ≤ j ≤ N andconsider a new generic linear coordinate system ( w , w ) which is related to( z , z ) by w i = P j =1 b ij z j with generic b ij ∈ C for 1 ≤ i, j ≤
2. By (III.3)we can find generic c j ∈ C for 1 ≤ j ≤ N and generic b ij ∈ C for 1 ≤ i, j ≤ h and ˜ h has effectively bounded multiplicityat the origin,(ii) the ideal generated by ∂h ∂w and ˜ h has effectively bounded multiplicityat the origin, where the partial derivative ∂h ∂w is computed with w being kept constant,(iii) the projection P g ( P ) makes C an analytic cover over C locally atthe origin as germs, and(iv) the projection ( w , w ) ( h , w ) makes C an analytic cover over C locally at the origin as germs.Without loss of generality we can assume that the coordinate system ( w , w )is just the coordinate system ( z , z ). Note that h is only a pre-multiplierand in general may not be a multiplier. The function germ ∂h ∂z is in generalnot a multiplier and not even a pre-multiplier.Consider the image ˆ C of C under the projection ( z , z ) ( h , z ) andlet h = z λ + λ − X j =0 a j ( h ) z j be the Weierstrass polynomial in C with coordinates ( h , z ) whose vanish-ing defines the curve-germ ˆ C at the origin in C . This is possible, becausethe projection P h ( P ) makes C an analytic cover over C locally at theorigin as germs. When regarded as a function-germ in the variables ( z , z )the function-germ h contains ˜ h as a factor, because the inverse image ofˆ C under the projection ( z , z ) ( h , z ) contains C and C is a reducedcurve. Since ˜ h is a multiplier, it follows that h is also a multiplier and is,in fact, an effectively constructed multiplier. The multipliers in the effectiveprocedure presented here and also in (III.8) and (III.9) are all effectivelyconstructed multipliers (unless explicitly pointed out otherwise) and we will30rop the description “effectively constructed” when we mention these multi-pliers here and in (III.8) and (III.9). Sometimes, to highlight certain aspectsof effectiveness, we may mention “the assigned order of subellipticity havingan effective positive lower bound” or “the multiplicity having an effectiveupper bound” in conjunction with such multipliers, though according to theconvention given here such multipliers are all effective constructed unlessexplicitly pointed out otherwise.Since the ideal generated by ∂h ∂z and ˜ h has effectively bounded multi-plicity at the origin and since h vanishes at the origin, it follows that, forsome effectively bounded positive integer s , the function germ ( h ) s belongsto the ideal generated by ∂h ∂z and ˜ h . In particular,(III . . | ( h ) s | < ∼ (cid:12)(cid:12)(cid:12)(cid:12) ∂h ∂z (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ˜ h (cid:12)(cid:12)(cid:12) . Here and for the rest of this note the symbol < ∼ means “less than someconstant times” and is being used to avoid introducing new symbols forconstants. We now form dh ∧ dh and get dh ∧ dh = dh ∧ λz λ − dz + λ − X j =1 ja j ( h ) z j − dz + λ − X j =0 a ′ j ( h ) z j dh ! = dh ∧ λz λ − dz + λ − X j =1 ja j ( h ) z j − dz ! = (cid:18) ∂h ∂z dz + ∂h ∂z dz (cid:19) ∧ λz λ − dz + λ − X j =1 ja j ( h ) z j − dz ! = ∂h ∂z λz λ − + λ − X j =1 ja j ( h ) z j − ! dz ∧ dz , where a ′ j ( h ) is the derivative of a j ( h ) as a function of h . Since h is apre-multiplier, the coefficient of dz ∧ dz in dh ∧ dh is a multiplier. Thus ∂h ∂z λz λ − + λ − X j =1 ja j ( h ) z j − !
31s a multiplier. Since ˜ h is a multiplier, it follows that˜ h λz λ − + λ − X j =1 ja j ( h ) z j − ! is a multiplier. From (III.7.1) it follows that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( h ) s λz λ − + λ − X j =1 ja j ( h ) z j − !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ∼ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂h ∂z λz λ − + λ − X j =1 ja j ( h ) z j − !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ h λz λ − + λ − X j =1 ja j ( h ) z j − !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Hence ( h ) s λz λ − + λ − X j =1 ja j ( h ) z j − ! is a multiplier. Let h (0)2 = h and h (1)2 = ( h ) s λz λ − + λ − X j =1 ja j ( h ) z j − ! and for 1 < ν ≤ λ define h ( ν )2 = ( h ) sν λ !( λ − ν )! z λ − ν + λ − X j = ν j !( j − ν )! a j ( h ) z j − ν ! . We are going to verify by induction on ν that h ( ν )2 is a multiplier. We knowthat both h (0)2 and h (1)2 are multipliers. Assume that we have already verifiedthat h (0)2 , · · · , h ( ν − are multipliers. Then dh ∧ dh ( ν − is equal to dh ∧ d ( h ) s ( ν − λ !( λ − ν + 1)! z λ − ν +12 + λ − X j = ν − j !( j − ν + 1)! a j ( h ) z j − ν +12 !! = dh ∧ ( h ) s ( ν − λ !( λ − ν )! z λ − ν + λ − X j = ν j !( j − ν )! a j ( h ) z j − ν ! dz ! ∂h ∂z ( h ) s ( ν − λ !( λ − ν )! z λ − ν + λ − X j = ν j !( j − ν )! a j ( h ) z j − ν !! dz ∧ dz Since the coefficient of dz ∧ dz in dh ∧ dh ( ν − is a multiplier, it followsthat ∂h ∂z ( h ) s ( ν − λ !( λ − ν )! z λ − ν + λ − X j = ν j !( j − ν )! a j ( h ) z j − ν !! is a multiplier. Since ˜ h is a multiplier, it follows that˜ h ( h ) s ( ν − λ !( λ − ν )! z λ − ν + λ − X j = ν j !( j − ν )! a j ( h ) z j − ν !! is a multiplier. From (III.7.1) it follows that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( h ) s ( h ) s ( ν − λ !( λ − ν )! z λ − ν + λ − X j = ν j !( j − ν )! a j ( h ) z j − ν !!(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ∼ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂h ∂z ( h ) s ( ν − λ !( λ − ν )! z λ − ν + λ − X j = ν j !( j − ν )! a j ( h ) z j − ν !!(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ h ( h ) s ( ν − λ !( λ − ν )! z λ − ν + λ − X j = ν j !( j − ν )! a j ( h ) z j − ν !!(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Hence ( h ) s ( h ) s ( ν − λ !( λ − ν )! z λ − ν + λ − X j = ν j !( j − ν )! a j ( h ) z j − ν !! is a multiplier and h ( ν )2 is a multiplier. When ν = λ , we end up with h ( λ )2 = ( h ) sλ λ !being a multiplier. 33ote that this step of forming Jacobian determinants λ times to construct h ( λ )2 from ˜ h is the step of differentiating a multiplier as many times as itsmultiplicity to form a new multiplier, which is referred to at the end of theIntroduction of this note. Also note that though this step only requires h to be a pre-multiplier, yet ˜ h has to be a multiplier instead of just a pre-multiplier, otherwise we cannot conclude that h is a pre-multiplier, becausein general the set of all pre-multipliers do not form an ideal.Since the multiplicity of the ideal generated by h and ˜ h is effectivelybounded, there exists some positive integer σ which is effectively boundedsuch that z σ and z σ both belong to the ideal generated by h ( λ )2 and ˜ h . Henceboth z σ and z σ are multipliers. We take the σ -th roots of both z σ and z σ to produce multipliers z and z . We finally form the Jacobian determinantof the two holomorphic function germs z and z to conclude that Kohn’salgorithm effectively produces the function F ≡ Construction of a New Multiplier in Higher Dimensional Case byFiberwise Differentiating a Given Multiplier as Many Times as its Multi-plicity.
We now look at the higher dimensional case. As a preparation forthe proof of the Main Theorem for the higher dimensional case, we do the ar-gument here for the construction a new multiplier in higher dimensional caseby fiberwise differentiating a given multiplier as many times as its multiplic-ity. The argument is the same as the 2-dimensional case with correspondingmodifications in notations.We have holomorphic function germs F , · · · , F N on C n at the originwhich generate an ideal of multiplicity q whose zero-set is the origin of C n .By applying Corollary (III.4) and (III.5) to get n C -linear combinations of F , · · · , F N and form their Jacobian determinant, we get an effectively con-structed multiplier ˜ h n ( z , · · · , z n ) at the origin, which vanishes at the origin.The divisor of ˜ h n is a subspace germ V n of codimension 1 in C n at the originwith effectively bounded multiplicity. Because V n has effectively boundedmultiplicity, by replacing ˜ h n by the product of holomorphic function germsdefining the branch germs of the reduction of V n , we can assume withoutloss of generality that V n is a reduced hypersurface germ in C n at the originwith effectively bounded multiplicity. Again this does not mean that V n isirreducible. It only means that the divisor of ˜ h n has coefficient 1 for every34ne of its irreducible components.By Proposition (A.3) in Appendix A, the ideal generated by ∂ (cid:0) F i , · · · , F i n − (cid:1) ∂ (cid:0) z j , · · · , z j n − (cid:1) for 1 ≤ i < · · · < i n − ≤ N and 1 ≤ j < · · · < j n − ≤ n contains aneffective power of the maximum ideal m C n , of C n at the origin. Just likethe argument given in the 2-dimensional case in (III.7), after a generic C -linear coordinate change and after taking n − C -linear combinations h , · · · , h n − of F , · · · , F N we have the following situation.(i) The ideal generated by h , · · · , h n − and ˜ h n has effectively boundedmultiplicity at the origin,(ii) The ideal generated by ˜ h n and ∂ ( h , · · · , h n − ) ∂ ( z , · · · , z n − )has effectively bounded multiplicity at the origin.(iii) The projection P ( h ( P ) , · · · , h n − ( P )) makes V n an analytic coverover C n − locally at the origin as germs.(iv) The projection ( z , · · · , z n ) ( h , · · · , h n − , z n ) makes C n an analyticcover over C n locally at the origin as germs.Consider the image ˆ V n of V n under the projection ( z , · · · , z n ) ( h , · · · , h n − , z n )and let h n = z λn + λ − X j =0 a j ( h , · · · , h n − ) z jn be the Weierstrass polynomial in the target space C n with coordinates ( h , · · · , h n − , z n )whose vanishing defines the subspace germ ˆ V n at the origin in C n . This ispossible, because the projection P ( h ( P ) , · · · , h n − ( P )) makes V n ananalytic cover over C n − locally at the origin as germs. When regardedas a function-germ in the variables ( z , · · · , z n ) the function-germ h n con-tains ˜ h n as a factor, because the inverse image of ˆ V n under the projection35 z , · · · , z n ) ( h , · · · , h n − , z n ) contains V n and because V n which is de-fined by ˜ h n is reduced. Since ˜ h n is a multiplier, it follows that h n is also amultiplier. Since the ideal generated by ˜ h n and ∂ ( h , · · · , h n − ) ∂ ( z , · · · , z n − )has effectively bounded multiplicity at the origin and since h · · · , h n − allvanish at the origin, it follows that there exists some polynomial p ( h , · · · , h n − )such that(i) the ideal generated by p ( h , · · · , h n − ) and ˜ h n has effectively boundedmultiplicity at the origin, and(ii) p ( h , · · · , h n − ) belongs to the ideal generated by ˜ h n and ∂ ( h , · · · , h n − ) ∂ ( z , · · · , z n − ) . In particular,(III . . | p ( h , · · · , h n − ) | < ∼ (cid:12)(cid:12)(cid:12)(cid:12) ∂ ( h , · · · , h n − ) ∂ ( z , · · · , z n − ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ˜ h n (cid:12)(cid:12)(cid:12) . One way to obtain the polynomial p ( h , · · · , h n − ) is to use the direct imageof the ideal generated by generated by ˜ h n and ∂ ( h , · · · , h n − ) ∂ ( z , · · · , z n − )under the local projection P ( h ( P ) , · · · , h n − ( P )) from C n to C n − andobtain p ( h , · · · , h n − ) from the zero-set of this direct image by taking aneffective power.We now form dh ∧ · · · ∧ dh n and get dh ∧· · ·∧ dh n = dh ∧· · ·∧ dh n − ∧ λz λ − n dz n + λ − X j =1 ja j ( h , · · · , h n − ) z j − n dz n ! = ∂ ( h , · · · , h n − ) ∂ ( z , · · · , z n − ) λz λ − n + λ − X j =1 ja j ( h , · · · , h n − ) z j − n ! dz ∧ · · · ∧ dz n . dz ∧ · · · ∧ dz n in dh ∧ · · · ∧ dh n is a multiplier, itfollows that ∂ ( h , · · · , h n − ) ∂ ( z , · · · , z n − ) λz λ − n + λ − X j =1 ja j ( h , · · · , h n − ) z j − n ! is a multiplier. Since ˜ h n is a multiplier, it follows that˜ h n λz λ − n + λ − X j =1 ja j ( h , · · · , h n − ) z j − n ! is a multiplier. From (III.8.1) it follows that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ( h , · · · , h n − ) λz λ − n + λ − X j =1 ja j ( h , · · · , h n − ) z j − n !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ∼ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ ( h , · · · , h n − ) ∂ ( z , · · · , z n − ) λz λ − n + λ − X j =1 ja j ( h , · · · , h n − ) z j − n !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ h n λz λ − n + λ − X j =1 ja j ( h , · · · , h n − ) z j − n !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Hence by the Real Radical Property of Kohn’s algorithm in (I.1)(C), p ( h , · · · , h n − ) λz λ − n + λ − X j =1 ja j ( h , · · · , h n − ) z j − n ! is a multiplier. Let h (0) n = h n and for 1 ≤ ν ≤ λ define h ( ν ) n = p ( h , · · · , h n − ) ν λ !( λ − ν )! z λ − νn + λ − X j = ν j !( j − ν )! a j ( h , · · · , h n − ) z j − νn ! . We are going to verify by induction on ν that h ( ν ) n is a multiplier. We knowthat both h (0) n and h (1) n are multipliers. Assume that we have already verifiedthat h (0) n , · · · , h ( ν − n are multipliers. Then dh ∧ · · · ∧ dh n − ∧ dh ( ν − n is equalto dh ∧ · · · ∧ dh n − ∧ d (cid:18) p ( h , · · · , h n − ) ν − (cid:18) λ !( λ − ν + 1)! z λ − ν +1 n λ − X j = ν − j !( j − ν + 1)! a j ( h , · · · , h n − ) z j − ν +1 n !! = dh ∧ · · · ∧ dh n − ∧ (cid:18) p ( h , · · · , h n − ) ν − (cid:18) λ !( λ − ν )! z λ − νn + λ − X j = ν j !( j − ν )! a j ( h , · · · , h n − ) z j − νn ! dz n ! = ∂ ( h , · · · , h n − ) ∂ ( z , · · · , z n − ) (cid:18) p ( h , · · · , h n − ) ν − (cid:18) λ !( λ − ν )! z λ − νn + λ − X j = ν j !( j − ν )! a j ( h , · · · , h n − ) z j − νn !! dz ∧ · · · ∧ dz n . Since the coefficient of dz ∧ · · · ∧ dz n in dh ∧ · · · ∧ dh n − ∧ dh ( ν − n is amultiplier, it follows that ∂ ( h , · · · , h n − ) ∂ ( z , · · · , z n − ) (cid:18) p ( h , · · · , h n − ) ν − (cid:18) λ !( λ − ν )! z λ − νn + λ − X j = ν j !( j − ν )! a j ( h , · · · , h n − ) z j − νn !! is a multiplier. Since ˜ h n is a multiplier, it follows that˜ h n p ( h , · · · , h n − ) ν − λ !( λ − ν )! z λ − νn + λ − X j = ν j !( j − ν )! a j ( h , · · · , h n − ) z j − νn !! is a multiplier. From (III.8.1) it follows that (cid:12)(cid:12)(cid:12)(cid:12) p ( h , · · · , h n − ) (cid:18) p ( h , · · · , h n − ) ν − (cid:18) λ !( λ − ν )! z λ − νn + λ − X j = ν j !( j − ν )! a j ( h , · · · , h n − ) z j − νn !!(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ∼ (cid:12)(cid:12)(cid:12)(cid:12) ∂ ( h , · · · , h n − ) ∂ ( z , · · · , z n − ) (cid:18) p ( h , · · · , h n − ) ν − (cid:18) λ !( λ − ν )! z λ − νn + λ − X j = ν j !( j − ν )! a j ( h , · · · , h n − ) z j − νn !!(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) ˜ h n (cid:18) p ( h , · · · , h n − ) ν − (cid:18) λ !( λ − ν )! z λ − νn + λ − X j = ν j !( j − ν )! a j ( h , · · · , h n − ) z j − νn !!(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Hence by the Real Radical Property of Kohn’s algorithm in (I.1)(C), p ( h , · · · , h n − ) (cid:18) p ( h , · · · , h n − ) ν − (cid:18) λ !( λ − ν )! z λ − νn + λ − X j = ν j !( j − ν )! a j ( h , · · · , h n − ) z j − νn !! is a multiplier and h ( ν ) n is a multiplier. When ν = λ , we end up with h ( λ ) n = p ( h , · · · , h n − ) λ λ !being a multiplier. Since the multiplicity of the ideal generated by p ( h , · · · , h n − )and ˜ h n is effectively bounded at the origin, it follows that the multiplicity ofthe ideal generated by p ( h , · · · , h n − ) λ and ˜ h n is effectively bounded at theorigin. We can conclude that p ( h , · · · , h n − ) is a multiplier admitting anorder of subellipticity with an effective positive lower bound.(III.9) Effective Termination of Kohn’s Algorithm in the Higher Dimen-sional Case.
Recall that in (III.8) we have the multiplier h n constructedfrom F , · · · , F N and ˜ h n by choosing n − C -linear combinations of F , · · · , F N . Now we enhance the construction of h n by choosing n good C -linear combinations of F , · · · , F N so that any subset of n − C -linear combinations for our purpose. More precisely, as in (III.8) wechoose n generic C -linear combinations H , · · · , H n of F , · · · , F N such that(i) the map π : C n → C n defined by H , · · · , H n is an analytic cover maplocally at the origin whose number of sheets is effectively bounded, and(ii) for any 1 ≤ j ≤ n we can use H , · · · , H j − , H j +1 , · · · , H n as h , · · · , h n − for the argument in (III.8) to produce a polynomial p j ( H , · · · , H j − , H j +1 , · · · , H n )of H , · · · , H j − , H j +1 , · · · , H n which is a multiplier and whose multi-plicity at the origin is effectively bounded and whose assigned order ofsubellipticity has an effective positive lower bound.39he argument in (III.8) shows that each p j ( H , · · · , H j − , H j +1 , · · · , H n ) isan effectively constructed multiplier for 1 ≤ j ≤ n .We introduce coordinates z , · · · , z n in the domain space C n of the map π : C n → C n . We use coordinates w , · · · , w n in the target space C n of themap π : C n → C n . Since the polynomial p j ( w , · · · , w j − , w j +1 , · · · , w n ) asa function of w , · · · , w j − , w j +1 , · · · , w n has effectively bounded multiplicityat the origin. it follows that an effectively bounded positive power of themaximum ideal of the target space C n of π at the origin is contained in theideal of the target space C n of π at the origin generated by the n polynomials p j ( w , · · · , w j − , w j +1 , · · · , w n ) for 1 ≤ j ≤ n .Since the map π : C n → C n defined by H , · · · , H n is an analytic covermap locally at the origin whose number of sheets is effectively bounded,it follows that an an effectively bounded positive power of the maximumideal of the domain space C n of π at the origin is contained in the idealof the domain space C n of π at the origin generated by the n holomorphicfunction germs p j ( H , · · · , H j − , H j +1 , · · · , H n ) for 1 ≤ j ≤ n . Since each p j ( H , · · · , H j − , H j +1 , · · · , H n ) is an effectively constructed multiplier for1 ≤ j ≤ n , it follows that each of the coordinates z , · · · , z n of the domainspace C n of π is a multiplier with effective assigned order of subellipticity. Byforming the Jacobian determinant of the multipliers z , · · · , z n , we concludethat the function F ≡ Remark on the Need to Fiberwise Differentiate as Many Times asthe Multiplicity of the Given Multiplier.
An earlier version of this paperputs in the proof only one fiberwise differentiation for the given multiplierinstead of the number of fiber differentiations equal to the multiplicity of themultiplier. This version adds the required number of differentiations. Let usexplain the need to fiberwise differentiate as many times as the multiplicityof the multiplier by considering the following simple situation in complexdimension 2.Let f ( z, w ) be a Weierstrass polynomial of degree q in w , which is a monicpolynomial in w whose coefficients, except the leading one, are holomorphicfunction germs in z vanishing at the origin. Denote by f w ( z, w ) the derivativeof f ( z, w ) with respect to w . Let D ( z ) be the discriminant of f ( z, w ) as a40olynomial in w . Then Euclid’s algorithm gives D ( z ) = a ( z, w ) f ( z, w ) + b ( z, w ) f w ( z, w ), where a ( z, w ) and b ( z, w ) are holomorphic function germson C at the origin.Note that if q is small, we can only conclude that the multiplicity q of f ( z, w ) at the origin is small and we cannot conclude that the coefficients ofpowers of w , other than the leading one, have low vanishing order in z at z = 0.Suppose f ( z, w ) is a multiplier and z is a pre-multiplier. When we applythe operator dz ∧ d ( · ) to f ( z, w ) to get f w ( z, w ) dz ∧ dw , we conclude that f w ( z, w ) is a multiplier. From D ( z ) = a ( z, w ) f ( z, w ) + b ( z, w ) f w ( z, w ) itfollows that the discriminant D ( z ) is a also multiplier which in general isnot effectively constructed. The vanishing order of D ( z ) in z at z = 0 ingeneral does not have anything to do with q and certainly in general cannotbe bounded by an effective function of q . Thus the ideal generated by themultipliers f ( z, w ) and D ( z ) may have high multiplicity at the origin if D ( z )has high vanishing order in z at z = 0. This function germ D ( z ) is obtainedby one single fiberwise differentiation of the multiplier f ( z, w ).The discriminant D ( z ) is given by Q i = j ( w i ( z ) − w j ( z )) , where { w ( z ) , · · · , w q ( z ) } (without any well-defined ordering) is the collection of the q roots of f ( z, w )in w with the multiplicities of the roots counted. If the minimum distanceof two points in { w ( z ) , · · · , w q ( z ) } as a function of z vanishes to high orderin z at z = 0, we would have high vanishing order for D ( z ). The process ofgetting D ( z ) by differentiating once does not help in our goal of achievingan effective termination of Kohn’s algorithm when two of the roots from theset { w ( z ) , · · · , w q ( z ) } are becoming close very fast as z approaches 0. Sincewe have no control over how fast some of the roots { w ( z ) , · · · , w q ( z ) } aregetting close as z →
0, we need to differentiate q times in order to achieveour goal of an effective termination of Kohn’s algorithm. This explains whywe need to fiberwise differentiate as many times as the multiplicity of themultiplier.(III.11) Motivation of the Proof of Termination of Kohn’s Algorithm fromthe Fundamental Theorem in Multivariate Calculus for Fubini’s Iterated In-tegration.
We would like to remark that the motivation for the above proofof the termination of Kohn’s algorithm for special domains comes from thefundamental theorem in multivariate calculus for the following theorem of41ubini on iterated integration. The reason for this motivation is that Jaco-bian determinants occur in the change-of-variables formula for integrals ofseveral variables and that an induction process can be used when we convertan integral of several variables to an iterated integral by Fubini’s theorem.(III.11.1)
Fubini’s Iterated Integration.
Let y , · · · , y n − be functions defininga projection from an n -space G with coordinates x , · · · , x n to an ( n − D with coordinates y = ( y , · · · , y n − ) so that x n can be used to bea local coordinate for the fiber L y of the projection over the point y ∈ D .Then for a function f on G the formula Z G f = Z y ∈ D Z L y f ! holds with the use of appropriate measures.Like the fundamental theorem of calculus of a single real variable, thefundamental theorem in multivariate calculus for the above theorem of Fubinion iterated integration changes integration to differentiation. If we write thefunction f in the form f ( y , · · · , y n − , x n ), then dy ∧ · · · ∧ dy n − ∧ df = dy ∧ · · · ∧ dy n − ∧ (cid:18) ∂f∂x n (cid:19) dx n so that fiberwise integration over L y with respect to x n in (III.11.1) changesover to fiberwise differentiation on L y with respect to x n .When we use a multiplier as f and pre-multipliers as y , · · · , y n − to formthe Jacobian determinant with respect to x , · · · , x n , we get dy ∧ · · · ∧ dy n − ∧ df = ∂ ( y , · · · , y n − ) ∂ ( x , · · · , x n − ) (cid:18) ∂f∂x n (cid:19) ( dx ∧ · · · ∧ dx n ) . The occurrence of the factor ∂f∂x n enables us to reduce the vanishing order of f by differentiation and the occurrence, as a factor, of the Jacobian determinant ∂ ( y , · · · , y n − ) ∂ ( x , · · · , x n − )involving one fewer variable makes it possible to use an induction process.42 art IV. Geometric Formulation of Kohn’s Algorithm in Termsof Frobenius Theorem on Integral Submanifolds and the Rˆole ofReal-Analyticity Kohn’s conjecture for the real-analytic case without effectiveness wasproved by Diederich-Fornaess [DF78]. We are going to formulate Kohn’salgorithm geometrically in terms of the theorem of Frobenius on integralsubmanifolds and present a proof of the real-analytic case of the ineffectivetermination of Kohn’s algorithm from the geometric viewpoint. This geo-metric formulation of Kohn’s algorithm in terms of the theorem of Frobeniusenables one to see clearly how the procedures of Kohn’s algorithm come aboutnaturally in the geometric context. Moreover, the proof of the real-analyticcase of the ineffective termination of Kohn’s algorithm from the geometricviewpoint gives a better understanding of the rˆole played by the real-analyticassumption and of the hurdles standing in the way of generalizing the inef-fective real-analytic case to the ineffective smooth case.(IV.1)
Usual Theorem of Frobenius on Integral Submanifolds for R m . Thesetting of the usual Frobenius theorem on integral submanifolds of real di-mension k starts out with a domain U in R m and a distribution x W x ⊂ T R m ,x = R m for x ∈ U which is smooth, where W x is a k -dimensional R -linear subspace of the tan-gent space T R n ,x of R n at x .The vector-field version of Frobenius’s theorem states that the distribu-tion x W x is locally integrable (in the sense that locally U is foliated bysmooth real submanifolds of real dimension k whose tangent space at thepoint x is precisely W x ) if and only if [ W x , W x ] ⊂ W x for all x ∈ U (in thesense that for all x ∈ U the value at x of the Lie bracket of two local vectorfields whose values at y in their domains of definition are in W y for each y belongs to W x ).The equivalent differential-form version of Frobenius’s theorem statesthat the distribution x W x is locally integrable if and only if for anylocal smooth differential 1-form ω , · · · , ω m − k whose common kernel is W x there exist local smooth differential 1-forms η , · · · , η m − k such that dω j = P m − kℓ =1 ω ℓ ∧ η ℓ for 1 ≤ j ≤ m − k . 43he vector-field version of Frobenius’s theorem is related to its differential-form version by Cartan’s formula relating Lie brackets of vector fields andthe exterior differentiation of differential forms (see, for example, [He62, p.21,Formula (9)]).(IV.2) Setting of CR Hypersurface for the Theorem of Frobenius Theorem.
In the formulation of Kohn’s algorithm in terms of Frobenius’s theorem thesetting is the boundary S of a bounded smooth domain Ω in C n and thedistribution on S is P T R S,P ∩ J (cid:0) T R S,P (cid:1) , where T R S,P is the space of all realtangent vectors in S at P and J is the almost-complex operator of C n .In this setting the condition of the theorem of Frobenius is equivalentto S being Levi-flat, in which case S is locally foliated by local complexsubmanifolds of complex dimension n − S which are tangential to the distribution x W x of S . The underlyingpoint set of an Artinian subscheme is just a single point, but its structuresheaf may be more than the complex number field C and can be an Artinianring ( i.e. a ring of finite dimension) which is the quotient of the structuresheaf of S .We will not go into the precise definition an Artinian subscheme here.Its definition depends on the structure sheaf of S which in the real-analyticcase is the sheaf of germs of all real-analytic functions and in the smoothcase is the sheaf of germs of all smooth functions. As an illustration wegive here the following two examples of Artinian subschemes A of the ringedspace ( C n , O C n ) supported at the origin of C n , where O C n is the sheaf of allholomorphic function germs on C n .Let m C n , be the maximum ideal at the origin of C n . Then the ringed space( { } , O C n /( m C n , ) q ) for any positive integer q is an Artinian subscheme ofthe ringed space ( C n , O C n ) supported at the origin of C n . For any ideal I of O C n , with ( m C n , ) N ⊂ I for some positive integer N , the ringed space( { } , O C n / I ) is also an Artinian subscheme of the ringed space ( C n , O C n )supported at the origin of C n .(IV.3) Steps of Kohn’s Algorithm from Constructing Integral Complex Curves.
We are going to see how the steps of Kohn’s algorithm naturally arise in the44tudy of conditions necessary for the construction of integral complex curvesin the boundary of a smooth bounded domain. Again the setting is a weaklypseudoconvex smooth bounded domain Ω with boundary S and again on S we consider the distribution P T R S,P ∩ J (cid:0) T R S,P (cid:1) for P ∈ S . We will laterspecialize to the case where the boundary S of the bounded domain Ω isreal-analytic and will investigate precisely the rˆole played by the assumptionof real-analyticity of S . To anticipate the later specialization into the caseof the boundary S being real-analytic, we would like to explore conditionswhich give as a consequence the existence of some local complex curve in S . What we would like to do is to assume that Kohn’s algorithm does notterminate and seek to produce geometrically a local complex curve in S inthe real-analytic case. For this purpose, in our discussion, from time to timewe will restrict ourselves to some appropriate open subsets of S in order toexclude the singularity of real-analytic subsets which arise in our discussion.Let N (1 , S be the set of all (1 , S which is in the null spaceof the Levi form of S . Let N be the real part of N (1 , S in the sense that ata point P of S the space N consists of all Re ξ with ξ ∈ N (1 , S at the point P . Let T R S be the vector bundle of all real tangent vectors of S . One keyproperty of N is the following.(IV.3.1) Let P be a point of S and U be an open neighborhood of P in S . Let ξ and η be smooth sections of T R S over U . That is, ξ and η are real tangentvector fields of S defined on U . Assume that both ξ and η belongs to N at P . Then the value of the Lie bracket [ ξ, η ] at P belongs to N .Another way to state (IV.3.1) is the following.(IV.3.2) The real part of the null space of (1 , S the first derivative of theLevi form for a (1 , G be a nonempty open subset of S where the real dimension of N is constant, say ℓ . For the case ℓ ≥
1, it follows from Frobenius theoremand condition (IV.3.1) that over G we can integrate N to get local integral45ubmanifolds M of G of real dimension ℓ so that the tangent space of M atany point P of M is equal to the real vector space N at P .Since at every point P of S the space N (1 , S is a vector space over thecomplex number field C , we know that its real part N must be invariant underthe almost-complex-structure operator J . Thus we can conclude that thetangent space N of each local integral submanifold M is invariant under thealmost-complex-structure operator J . This implies that each M is complex-analytic. As a consequence, one has the following trivial remark.(IV.4) Remark.
Suppose S is a local smooth weakly pseudoconvex hyper-surface in C n . If at each point of S the null space of the Levi form of S is nontrivial, then some nonempty open subset G of S is foliated by localcomplex submanifolds of positive dimension.(IV.5) Natural Occurrence of the Steps of Kohn’s Algorithm.
The algorithmof Kohn comes into the picture only when we do not have a nonempty opensubset G of S where the real dimension of N is some positive constant ℓ .We consider the set E of points of S where the real dimension of N is somepositive constant ℓ . The case of interest is when E does not contain anopen subset of S . This step of introducing E corresponds to introducingthe coefficients of the ( n, n − ∂r ∧ (cid:0) ∂ ¯ ∂r (cid:1) n − as multipliers in Kohn’salgorithm. We are going to assume that E is a smooth submanifold of realdimension m ≥ N | E is a smooth vector bundle over E . In thereal-analytic case because of the stratification of real-analytic subvarietieswe can always get to a real-analytic submanifold and a real-analytic bundleby replacing the point under consideration by another point nearby. In thegeneral smooth case there is no such stratification and the situation becomescomplicated and calls for other techniques than those discussed here.We want to apply Frobenius’s theorem to E with the distribution of vectorspaces N over it. The trouble is that the vector space N at a point P of E may not be inside the real tangent space T R E,P = (cid:0) T R E (cid:1) P of E at P . To applyFrobenius’s theorem to E we must work with a distribution of subspaces ofthe tangent spaces of E . We are forced to replace N by N ∩ T R E at each point P of E . We also want to keep the smaller new vector space N ∩ T R E invariantunder the almost-complex-structure operator J , because we are interestedin producing local complex curves inside S . We use the even smaller vectorspace N ∩ T R E ∩ J T R E . (Note that N is invariant under J .) Then we considerthe new subset E of E where the real dimension of N ∩ T R E ∩ J T R E is positive.46or the real-analytic case this step corresponds to introducing real-valuedreal-analytic function germs f vanishing on E as multipliers and also ∂f asvector-multipliers. The reason is that taking intersection with T R E is the sameas considering the kernel of the differential df of local real-valued functions f on S which vanish identically in E . Taking the further intersection with J T R E is to consider also the kernel of the J -image J df of the differential df of local real-valued functions f on S which vanish on E . Taking bothintersections together is the same as considering the kernel of ∂f for localreal-valued functions f on S which vanish on E . Th use of all local real-analytic function germs vanishing on E tells us how the step (I.1)(C) ofKohn’s algorithm naturally arises from the geometric viewpoint. The use of ∂f tells us how the step (I.1)(B)(i) of Kohn’s algorithm naturally arises fromthe geometric viewpoint.As the initial steps of an inductive process we set E = E and N (0) = N and N (1) = N (0) ∩ T R E ∩ J T R E . Then we inductively define N ( ν +1) = N ( ν ) ∩ T R E ν ∩ J T R E ν = N ∩ T R E ν ∩ J T R E ν and define E ν +1 to be the subset of E where the real dimension of N ( ν ) ispositive. We obtain the limiting common intersection E ∞ defined by E ∞ = ∩ ν E ν . By replacing E ∞ by a nonempty open subset in the real-analyticcase, we can assume that E ∞ is regular and N ( ∞ ) := N ∩ T R E ∞ ∩ J T R E ∞ is a real-analytic vector bundle over E ∞ . Note that, though we go to theregular part of E ν in order to describe more easily the tangent bundle T E ν of E ν , when we take the intersection E ∞ = ∩ ν E ν we have to make sure thatthe intersection E ∞ = ∩ ν E ν is defined in the real-analytic case as a real-analytic subvariety, which forces us to consider T E ν also at singular pointsof E ν where it is defined as the common kernel of differentials of all localreal-analytic functions vanishing on E ν .Note that the definition of E ν +1 as the subset of E where the real di-mension of N ( ν ) is positive involves the existence of a nontrivial solution ina system of homogeneous linear equations or equivalently the vanishing ofthe determinant of the coefficient matrix or equivalently the vanishing of thecorresponding exterior product of co-vectors. This tells us how the step ofKohn’s algorithm described in (I.1)(B)(ii) naturally arises from the geometricviewpoint.Now the distribution of vector spaces N ( ∞ ) = N ∩ T R E ∞ ∩ J T R E ∞ is con-tained in the tangent space of E ∞ and each N ( ∞ ) = N ∩ T R E ∞ ∩ J T R E ∞ is47 -invariant. For the purpose of understanding how the procedures of Kohn’salgorithm come about naturally in the geometric context, we assume thateach fiber of N ( ∞ ) = N ∩ T R E ∞ ∩ J T R E ∞ is of positive dimension and we alsoassume that we are in the real-analytic case so that we have the benefit ofstratification. Under such assumptions and after restriction to a dense opensubset if necessary, E ∞ is a CR manifold and has holomorphic dimension atleast 1. However, for N ( ∞ ) = N ∩ T R E ∞ ∩ J T R E ∞ we may not have the invo-lutive condition of the theorem of Frobenius (which means closure under Liebracket). In order to apply the theorem of Frobenius we generate a largerlinear subspace of the tangent space of E ∞ by taking iterated Lie bracketsof local sections of the vector bundle N ( ∞ ) = N ∩ T R E ∞ ∩ J T R E ∞ to generatea new distribution ˜ N . This new distribution ˜ N now satisfies the followingthree conditions.(i) ˜ N is contained in the tangent space of E ∞ .(ii) ˜ N is involutive in the sense that it is closed under Lie bracket.(iii) ˜ N belongs to the real part of the null space of the Levi form of S .Note that Condition (iii) is a consequence of (IV.3.1). However, in general ˜ N is no longer J -invariant. An integral submanifold M of ˜ N has the followingproperty. At each point of M the holomorphic dimension of S is at least1. An open dense subset of M is a CR manifold, but in general M is notcomplex-analytic. We are going to show, with our present assumption ofreal-analyticity, that when the Kohn algorithm does not terminate, we areable to produce some local complex curve inside M . One key point here isthat the tangent space of M is contained in the null space N of the Levi-formwhich is J -invariant.Since we have assumed that we are in the real-analytic case, at a genericpoint of M we can consider the smallest complex submanifold germ V in C n which contains the germ of M at that point. We then have the followingsituation. At a generic point P of M there exist(i) an open neighborhood U of P in C n ,(ii) a complex submanifold V in U , and(iii) real-valued real-analytic functions ρ , · · · , ρ ℓ on V M ∩ U is the common zero-set of ρ , · · · , ρ ℓ ,(b) ∂ρ , · · · , ∂ρ ℓ are C -linearly independent at points of M ∩ U , and(c) at any point of M ∩ U the tangent space of V is contained in N .Condition (b) means that, besides the R -linear independence of dρ , · · · , dρ ℓ at points of M ∩ U , we also have the R -linear independence of( J dρ ) | T R M , · · · , ( J dρ ℓ ) | T R M at points of M ∩ U . The complex dimension of T R M ∩ J T R M is equal todim C V − ℓ , which is ≥
1. The reason why the smallest complex submanifoldgerm V of C n at P containing the germ of M at P satisfies condition (c)is that T R M is contained in the J -invariant vector space N at any point of M and we can determine V as the zero-set of holomorphic function germs on C n at P obtained by extending CR real-analytic functions on M by usingthe condition of their annihilation by ¯ ∂ to define the infinite jets of theirextensions.Condition (c) means that V is tangential to S at points of M . There aretwo possibilities. One is that V is contained in S , in which case S containsa local complex curve and we are done. The other possibility is that V is not contained in S . We are going to assume the second possibility andderive a contradiction for the real-analytic case so that we can conclude inthe real-analytic case that S must contain a local complex curve.For clarity in the later discussion we digress at this point to say somethingabout the well-known alternative description of the Levi form and also aboutthe process of polarization.(IV.6) Alternative Description of Levi Form.
Recall the following formula ofCartan for exterior differentiation of differential forms( p + 1) ( dω ) ( X , · · · , X p +1 ) = p +1 X i =1 ( − i +1 X i (cid:16) ω (cid:16) X , · · · , ˆ X i , · · · , X p +1 (cid:17)(cid:17) + X i 12 ( J dρ ) ( ξ ) . When both ( dρ ) ( ξ ) ≡ J dρ ) ( ξ ) ≡ 0, we have J dρ ([ ξ, η ]) = 4 √− (cid:0) ∂ ¯ ∂ρ (cid:1) ( ξ, η ) . When we compute the Levi form of ρ we limit ourselves to vectors of type(1 , 0) which are tangential to ρ = 0. A vector ξ of type (1 , 0) means that J ( ξ ) = √− ξ . Tangency of ξ to ρ = 0 means that ( dρ ) ( ξ ) = 0, whichimplies automatically ( J dρ ) ( ξ ) = ( dρ ) ( J ξ ) = √− dρ ) ( ξ ) = 0, because bydefinition the operator J acting on 1-forms is the adjoint of the operator J acting on tangent vectors. Likewise, for a vector ¯ ξ of type (0 , 1) tangential to ρ = 0 we have ( dρ ) ( ¯ ξ ) = 0 and ( J dρ ) ( ¯ ξ ) = ( dρ ) ( J ¯ ξ ) = −√− dρ ) ( ¯ ξ ) = 0.Thus for vector fields ξ and η of type (1 , 0) or (1 , 0) tangential to ρ = 0 wehave J dρ ([ ξ, η ]) = 4 √− (cid:0) ∂ ¯ ∂ρ (cid:1) ( ξ, η ) . (IV.7) Polarization. Let Y be a CR submanifold of some open subset of C n .Let ξ , ξ be real-valued vector fields in T R Y ∩ J T R Y . The condition that ξ j isin T R Y ∩ J T R Y is equivalent to the condition that we can write ξ j = τ j + τ j forsome complex-valued vector fields τ j in T (1 , Y for j = 1 , 2. We have[ ξ , ξ ] = [ τ + τ , τ + τ ] = [ τ , τ ] + [ τ , τ ] − [ τ , τ ] + [ τ , τ ] . τ , τ ] in terms of [ τ, τ ]for some vector field τ of type (1 , 0) tangential to Y so that τ is expressedlinearly and explicitly in terms of ξ , ξ , J ξ , J ξ modulo C ⊗ R (cid:0) T R Y ∩ J T R Y (cid:1) = T (1 , Y ⊕ T (0 , Y . From[ τ + τ , τ + τ ] = [ τ , τ ] + [ τ , τ ] + [ τ , τ ] + [ τ , τ ]we subtract the expression with τ changed to − τ to get[ τ + τ , τ + τ ] − [ τ − τ , τ − τ ] = 2 [ τ , τ ] + 2 [ τ , τ ] . Then we add to it √− τ by √− τ and we get 4 [ τ , τ ] equal to[ τ + τ , τ + τ ] − [ τ − τ , τ − τ ]+ √− (cid:16)h τ + √− τ , τ + √− τ i − h τ − √− τ , τ − √− τ i(cid:17) . Since [ τ , τ ] is in T (1 , Y and [ τ , τ ] is in T (0 , Y , we conclude that modulo C ⊗ R (cid:0) T R Y ∩ J T R Y (cid:1) = T (1 , Y ⊕ T (0 , Y the Lie bracket [ ξ , ξ ] is equal to [ τ , τ ] − [ τ , τ ]which is in turn equal to times[ τ + τ , τ + τ ] − [ τ − τ , τ − τ ]+ √− (cid:16)h τ + √− τ , τ + √− τ i − h τ − √− τ , τ − √− τ i(cid:17) − (cid:26) [ τ + τ , τ + τ ] − [ τ − τ , τ − τ ]+ √− (cid:16)h τ + √− τ , τ + √− τ i − h τ − √− τ , τ − √− τ i(cid:17) (cid:27) = 2 √− (cid:16)h τ + √− τ , τ + √− τ i − h τ − √− τ , τ − √− τ i(cid:17) . Thus modulo C ⊗ R (cid:0) T R Y ∩ J T R Y (cid:1) = T (1 , Y ⊕ T (0 , Y the Lie bracket [ ξ , ξ ] isequal to √− (cid:16)h τ + √− τ , τ + √− τ i − h τ − √− τ , τ − √− τ i(cid:17) , where τ + √− τ = 12 ( ξ + J ξ ) + √− 12 ( ξ − J ξ ) , − √− τ = 12 ( ξ − J ξ ) − √− 12 ( ξ + J ξ ) , because τ j = (cid:0) ξ j − √− J ξ j (cid:1) for j = 1 , 2. Suppose ρ is a real-valuedfunction in some neighborhood of Y . Then by (IV.6) we have J dρ ([ τ , τ ]) = 4 √− (cid:0) ∂ ¯ ∂ρ (cid:1) ( τ , τ ) = 0 ,J dρ ([ τ , τ ]) = 4 √− (cid:0) ∂ ¯ ∂ρ (cid:1) ( τ , τ ) = 0and as a consequence( J dρ ) ([ ξ , ξ ]) = √− 12 ( J dρ ) (cid:16)h τ + √− τ , τ + √− τ i(cid:17) − √− 12 ( J dρ ) (cid:16)h τ − √− τ , τ − √− τ i(cid:17) . When | ( J dρ ) ([ ξ , ξ ]) | = C for some C > 0, we have | ( J dρ ) ([ τ, τ ]) | ≥ C forone of the following two values of τ . τ + √− τ = 12 ( ξ + J ξ ) + √− 12 ( ξ − J ξ ) ,τ − √− τ = 12 ( ξ − J ξ ) − √− 12 ( ξ + J ξ ) , (IV.8) Locating Holomorphic Direction at Which Precisely One Levi-Form IsNonzero. After the above digression on the alternative description of the Leviform and the process of polarization, we now go back to the situation of theCR submanifold M at the end of (IV.5). According to the construction of M as an integral submanifold of ˜ N the tangent bundle T R M of M is generatedby iterated Lie brackets of vector fields of N ∩ T E ∞ ∩ J T E ∞ defined on M .Moreover, we have N ∩ T E ∞ ∩ J T E ∞ ⊂ T R M ∩ J T R M = ℓ \ j =1 Ker (cid:16) ( J dρ j ) | T R M (cid:17) ⊂ T R M . When we take vector fields in N ∩ T E ∞ ∩ J T E ∞ defined on M and form theiriterated Lie brackets in order to generate T R M , there is a first time the vectorfield fails to be inside T R M ∩ J T R M = T ℓj =1 Ker (cid:16) ( J dρ j ) | T R M (cid:17) . Thus we canfind real-valued vector fields ξ , ξ in T R M ∩ J T R M = T ℓj =1 Ker (cid:16) ( J dρ j ) | T R M (cid:17) M such that their Lie bracket [ ξ , ξ ] is not in T R M ∩ J T R M = T ℓj =1 Ker (cid:16) ( J dρ j ) | T R M (cid:17) . There exists 1 ≤ j ≤ ℓ such that ( J dρ j ) ([ ξ , ξ ]) isnonzero. Without loss of generality we assume that j = 1 so that ( J dρ ) ([ ξ , ξ ])is nonzero. Since ξ , ξ are both in T R M ∩ J T R M , we can write ξ j = τ j + τ j forsome complex-valued vector fields τ j in T (1 , M for j = 1 , 2. As explained abovein (IV.7), the polarization process gives us( J dρ ) ([ ξ , ξ ]) = √− 12 ( J dρ ) (cid:16)h τ + √− τ , τ + √− τ i(cid:17) − √− 12 ( J dρ ) (cid:16)h τ − √− τ , τ − √− τ i(cid:17) , where τ + √− τ = 12 ( ξ + J ξ ) + √− 12 ( ξ − J ξ ) ,τ − √− τ = 12 ( ξ − J ξ ) − √− 12 ( ξ + J ξ ) , One of ( J dρ ) (cid:16)h τ + √− τ , τ + √− τ i(cid:17) and ( J dρ ) (cid:16)h τ − √− τ , τ − √− τ i(cid:17) must be nonzero at P . We can choose τ to be either τ + √− τ or τ − √− τ so that ( J dρ ) ([ τ, τ ]) is nonzero at P . Since ξ , ξ belong to T R M ∩ J T R M = T ℓj =1 Ker (cid:16) ( J dρ j ) | T R M (cid:17) , it follows that ( ∂ρ j ) ( τ ) = 0 at P for1 ≤ j ≤ ℓ .Now for 2 ≤ j ≤ ℓ we replace ρ j by ρ j − ∂ρ j ([ τ, τ ]) ∂ρ ([ τ, τ ]) ρ so that we can assume without loss of generality that0 ≡ ( ∂ρ j ) ([ τ, τ ]) = (cid:18) (cid:0) − √− J (cid:1) dρ j (cid:19) ([ τ, τ ])= − √− 12 ( J dρ j ) ([ τ, τ ]) for 2 ≤ j ≤ ℓ. We can write r | V = X ν + ··· + ν ℓ = k σ ν , ··· ,ν ℓ ( ρ ) ν · · · ( ρ ℓ ) ν ℓ + O ℓ X j =1 ( ρ j ) ! k +12 k ≥ 2, where σ ν , ··· ,ν ℓ is a real-analytic function on U (aftershrinking U as an open neighborhood of P in C n if necessary) and σ ν ∗ , ··· ,ν ∗ ℓ is nonzero at P for some ν ∗ + · · · + ν ∗ ℓ = k .(IV.9) Argument of Different Vanishing Orders for Complex Hessian on theComplex Tangent Space Along Vector Fields Tangential or Normal to theIntersection with the Weakly Pseudoconvex Boundary. To make the argumentmore transparent and more understandable, we will first consider the specialcase ℓ = 1 so that M = V ∩ { ρ = 0 } and V is a complex submanifold insome open neighborhood of some point P of M . For this special case, fornotational simplicity we drop the subscript 1 from ρ and simply denote ρ by ρ . By replacing ρ by its product with a local nowhere zero real-analyticfunction we can assume without loss of generality that r = ρ k on V .Let m be the complex dimension of V . We choose a local holomor-phic coordinate system ( z , · · · , z n ) on the open neighborhood U of P in C n centered at P (after shrinking U if necessary) such that S ∩ U ∩{ z m +1 = · · · = z n = 0 } is regular and V = { z m +1 = · · · = z n = 0 } ∩ U . Sinceour argument will be confined to an open neighborhood of P in C n , for nota-tional simplicity, by replacing C n by C m +1 and S by S ∩{ z m +1 = · · · = z n = 0 } we can assume without loss of generality that n = m + 1 and we have thefollowing setup.(i) dr = (0 , , · · · , , 1) at the origin so that the complex submanifold V of the neighborhood U of P in C n is an open subset of the complextangent space of S at the origin which is defined by z n = 0.(ii) r = x n + O ( | z | ) near the origin, where x n is the real part of thecoordinate z n .(iii) The intersection M = V ∩ S of V and S is a CR manifold whosecomplex tangent space T R M ∩ J (cid:0) T R M (cid:1) has positive complex dimension atevery point of M .(iv) M is defined by ρ = 0 in V for some real-valued real-analytic function ρ on V such that r | V = ρ k for some positive integer k and dρ is nowherezero on M .(v) For some nonzero tangent vector τ of type (1 , 0) tangential to M atthe origin the value of the Levi form of ρ at τ is nonzero.54e are going to derive a contradiction. First we sketch the main idea of theargument. On V we will introduce two vector fields of type (1 , M at points of M and the other is normal to M at points of M . when we compute the complex Hessian of ρ k at these two vector fieldsof type (1 , 0) on V , we get two different orders of vanishing as we approach M from V − M , one of order k − k − 2. Becausethe touching order between V and S is k along M , when we extend thesetwo vector fields of type (1 , 0) on M to an open neighborhood of P in C n so that the two extensions are tangential to S at points of S , the Levi formsof r with respect to the two extensions give again the two different ordersof vanishing as we approach M from S − M . Since one of the two ordersis odd, the weak pseudoconvexity of S is violated, yielding a contradiction.Now we give below the details of this argument of different vanishing ordersfor the complex Hessian of r on V along vector fields tangential or normalto its intersection M with the weakly pseudoconvex boundary S .There is some open neighborhood U of the origin 0 in U on which r ( z , · · · , z n ) = φ ( z , · · · , z n ) z n + φ ( z , · · · , z n ) z n + ρ ( z , · · · , z n − ) k for some smooth complex-valued function φ ( z , · · · , z n ) on U , because on V = { z n = 0 } the function r is of the form ρ k . Let ξ be any smooth vectorfield of type (1 , 0) on U whose n -th component is ξ n . Then ∂r = ∂φ z n + φdz n + (cid:0) ∂ ¯ φ (cid:1) z n + kρ k − ∂ρ. (IV . . h ∂r, ξ i = h ∂φ , ξ i z n + φξ n + (cid:10) ∂ ¯ φ, ξ (cid:11) z n + kρ k − h ∂ρ, ξ i . ¯ ∂r = ¯ ∂φ z n + (cid:0) ¯ ∂ ¯ φ (cid:1) z n + ¯ φ dz n + kρ k − ¯ ∂ρ.∂ ¯ ∂r = ∂ ¯ ∂φ z n − ¯ ∂φdz n + ∂ ¯ ∂ ¯ φ z n + ∂ ¯ φ dz n + k ( k − ρ k − ∂ρ ¯ ∂ρ + kρ k − ∂ ¯ ∂ρ. (IV . . (cid:10) ∂ ¯ ∂r, ξ ∧ ¯ ξ (cid:11) = (cid:10) ∂ ¯ ∂φ , ξ ∧ ¯ ξ (cid:11) z n − (cid:10) ¯ ∂φ, ¯ ξ (cid:11) ξ n + (cid:10) ∂ ¯ ∂ ¯ φ , ξ ∧ ¯ ξ (cid:11) z n + (cid:10) ∂ ¯ φ, ξ (cid:11) ¯ ξ n + k ( k − ρ k − h ∂ρ, ξ i (cid:10) ¯ ∂ρ, ¯ ξ (cid:11) + kρ k − (cid:10) ∂ ¯ ∂ρ , ξ ∧ ¯ ξ (cid:11) . At a point of r = 0 in U we have φz n + ¯ φz n + ρ k = 0 . A and B be respectively the real and imaginary parts of 2 ¯ φ . Then φ = A − Bi and ¯ φ = A + Bi so that φz n + ¯ φz n = Ax n + By n (where y n is the imaginary part of z n ) and at a point in U we have Ax n + By n + ρ k = 0 . Since dr = (0 , , · · · , , 1) at the origin, it follows that φ = at the originand A = 1 and B = 0 at the origin. Let Y be the set defined by y n = 0. Atany point of S ∩ Y ∩ U where A is nonzero, we have x n = − ρ k A , z n = − ρ k A , z n = − ρ k A . We can choose an open neighborhood U of the origin 0 in U of the form U = W × G with W ⊂ C n − and G ⊂ C such that(i) A is nowhere zero on U and for Q ∈ W the set G contains the point z n = − ρ ( Q ) k A and(ii) φ + ( ∂ n φ ) z n + (cid:0) ∂ n ¯ φ (cid:1) z n is nowhere zero on U .On S ∩ Y ∩ U the two functions z n and z n are of the order O (cid:0) ρ k (cid:1) .We now derive our contradiction by choosing ξ in two different ways. Thefirst way is to choose ξ equal to τ at the origin. Since τ (from (IV.8)) is avector of C n of type (1 , 0) at the origin which is tangential to E = V ∩ S and since V = { z n = 0 } , it follows that the n -th component of the n -vector τ is zero. Since the differential dρ of the real-valued function ρ on V ∩ U isnowhere zero at every point of E = V ∩ S , we can extend τ to some smooth(1 , ξ = ( ξ , · · · , ξ n − ) of W for some open neighborhood W of 0 in W such that h dρ, ξ i ≡ W .We regard ξ j = ξ j ( z , · · · , z n − ) as functions of ( z , · · · , z n − , z n ) ∈ W × G for 1 ≤ j ≤ n − ξ j the composite of ξ j and the natural projection W × G → W for 1 ≤ j ≤ n − φ + ( ∂ n φ ) z n + (cid:0) ∂ n ¯ φ (cid:1) z n is nowhere zero on U , we can define ξ n on W × G by(IV . . ξ n = − φ + ( ∂ n φ ) z n + (cid:0) ∂ n ¯ φ (cid:1) z n n − X j =1 ( ∂ j φ ) ξ j z n + n − X j =1 (cid:0) ∂ j ¯ φ (cid:1) ξ j z n ! 56o that the vector field ξ = ( ξ , · · · , ξ n − , ξ n ) on W × G satisfies h ∂r, ξ i ≡ S ∩ Y ∩ U the two functions z n and z n are ofthe order O (cid:0) ρ k (cid:1) , it follows from (IV.9.3) that ξ n is of the order O (cid:0) ρ k (cid:1) on S ∩ Y ∩ ( W × G ). By (IV.9.2) (cid:10) ∂ ¯ ∂r, ξ ∧ ¯ ξ (cid:11) = kρ k − (cid:10) ∂ ¯ ∂ρ , ξ ∧ ¯ ξ (cid:11) + O (cid:0) ρ k (cid:1) on S ∩ Y ∩ ( W × G ). Since at the origin (cid:10) ∂ ¯ ∂ρ, ξ ∧ ¯ ξ (cid:11) = (cid:10) ∂ ¯ ∂ρ, τ ∧ ¯ τ (cid:11) isnonzero and since S is weakly pseudoconvex at every point of S , it followsthat k must be odd.We now introduce our second way of choosing ξ with the goal of derivingfrom it the conclusion that k is even. We choose some smooth vector field( ξ , · · · , ξ n − ) of type (1 , 0) on some open neighborhood W of 0 in W suchthat h dρ, ( ξ , · · · , ξ n − ) i is nowhere zero on W . We now define ξ n on W × G by(IV . . ξ n = − φ + ( ∂ n φ ) z n + (cid:0) ∂ n ¯ φ (cid:1) z n n − X j =1 ( ∂ j φ ) ξ j z n + n − X j =1 (cid:0) ∂ j ¯ φ (cid:1) ξ j z n + kρ k − n − X j =1 ( ∂ j ρ ) ξ j ! so that the vector field ξ = ( ξ , · · · , ξ n − , ξ n ) on W × G satisfies h ∂r, ξ i ≡ S ∩ Y ∩ U the two functions z n and z n are ofthe order O (cid:0) ρ k (cid:1) , it follows from (IV.9.4) that ξ n is of the order O (cid:0) ρ k − (cid:1) on S ∩ Y ∩ ( W × G ). By (IV.9.2) (cid:10) ∂ ¯ ∂r, ξ ∧ ¯ ξ (cid:11) = k ( k − ρ k − h ∂ρ, ξ i (cid:10) ¯ ∂ρ, ¯ ξ (cid:11) + O (cid:0) ρ k − (cid:1) on S ∩ Y ∩ ( W × G ). Since at the origin h dρ, ( ξ , · · · , ξ n − ) i is nonzero andsince S is weakly pseudoconvex at every point of S , it follows that k must beeven. Thus we have a contradiction, because earlier we have the conclusionthat k must be odd.(IV.10) Another Special Case to Illustrate the Argument of Different Tan-gential and Normal Vanishing Orders for Complex Hessian When Approach-ing CR Submanifold of Higher Holomorphic Codimension. We now consideranother special case for the more general situation where locally M is de-fined by real-valued real-analytic functions ρ , · · · , ρ ℓ on V with ℓ > ∂ρ , · · · , ∂ρ ℓ are C -linearly independent at points of M . We use this specialcase to further illustrate the argument of different tangential and normalvanishing orders for the complex Hessian. We first explain what this specialcase is. 57s discussed above in (IV.8), there exist some τ ∈ T (1 , M such that( ∂ρ j ) ( τ ) = 0 at P for 1 ≤ j ≤ ℓ and (cid:0) ∂ ¯ ∂ρ (cid:1) ( τ, τ ) is nonzero but (cid:0) ∂ ¯ ∂ρ j (cid:1) ( τ, τ )is zero for 2 ≤ j ≤ ℓ . We can write r | V = X ν + ··· + ν ℓ = k σ ν , ··· ,ν ℓ ( ρ ) ν · · · ( ρ ℓ ) ν ℓ + O ℓ X j =1 ( ρ j ) ! k +12 for some integer k ≥ 2, where σ ν , ··· ,ν ℓ is a real-analytic function on U (aftershrinking U as an open neighborhood of P in C n if necessary) and σ ν ∗ , ··· ,ν ∗ ℓ is nonzero at P for some ν ∗ + · · · + ν ∗ ℓ = k . This special case which we nowconsider is when σ ν , ··· ,ν ℓ is nonzero at P for some ν + · · · + ν ℓ = k with ν = 0.For this special case, just as for the case of ℓ = 1 we can find a smoothvector field ξ of type (1 , 0) in some open neighborhood of P in C n which aretangential to ∂ Ω such that the value of ξ at P agrees with τ . By computingthe Levi form of r at the vector field ξ and its vanishing order at M by using r | V = X ν + ··· + ν ℓ = k σ ν , ··· ,ν ℓ ( ρ ) ν · · · ( ρ ℓ ) ν ℓ + O ℓ X j =1 ( ρ j ) ! k +12 , as in the case of ℓ = 1 we can conclude that k must be odd. Thus we have acontradiction. However, this argument depends on the additional assumptionthat σ ν , ··· ,ν ℓ is nonzero at P for some ν + · · · + ν ℓ = k with ν = 0 for aspecially chosen set of defining functions ρ , · · · , ρ ℓ .Note that to rule out the case of an odd k , we do not need this addi-tional assumption that σ ν , ··· ,ν ℓ is nonzero at P for some ν + · · · + ν ℓ = k with ν = 0. There is also another way to rule out the case of an odd k by using bounded strictly plurisubharmonic exhaustion functions for weaklypseudoconvex domains in the following way.(IV.11) Handling the Case of Odd Vanishing Order by Using Bounded StrictlyPlurisubharmonic Exhaustion Functions for Weakly Pseudoconvex Domains. First let us introduce the following trivial statement about the vanishingorder of a negative subharmonic function at a boundary segment, whichis related to Hopf’s lemma or the strong maximum principle [GT83, p.34,Lemma 3.4]. 58IV.11.1) Let D be a connected open subset of C and C is a smooth connectedcurve in D defined by ρ = 0 with dρ nowhere zero at points of C such that D − C consists of two nonempty components W and W with ρ < W .Let η > ϕ be a smooth negative subharmonic function on W . Thenit is impossible to write − ϕ = ( − ρ ) η on W .The reason is as follows. We compute¯ ∂ ( − ϕ ) = − η ( − ρ ) η − ¯ ∂ρ,∂ ¯ ∂ ( − ϕ ) = η ( η − 1) ( − ρ ) η − ∂ρ ¯ ∂ρ − η ( − ρ ) η − ∂ ¯ ∂ρ. Since ∂ ¯ ∂ϕ ≥ W , it follows that0 ≥ ∂ ¯ ∂ ( − ϕ ) = η ( η − 1) ( − ρ ) η − ∂ρ ¯ ∂ρ − η ( − ρ ) η − ∂ ¯ ∂ρ and ∂ ¯ ∂ρ ≥ ∂ ¯ ∂ ( − ϕ ) = η − − ρ ∂ρ ¯ ∂ρ, which is a contradiction, because the left-hand side evaluated at a point of W stays bounded as the point approaches some point of C but the right-hand side evaluated at the same point becomes ∞ as the point approachessome point of C .We now recall the following theorem of Diederich-Fornaess on boundedstrictly plurisubharmonic exhaustion functions for weakly pseudoconvex do-mains [DF77, p.133, Remark b].Let Ω be a domain in C n and P belong to the boundary of Ω so that forsome open neighborhood D of P in C n the boundary of Ω ∩ D in D is smoothand weakly pseudoconvex. Let δ be the distance function from a point of Ωto C n − Ω. Let ψ be a smooth strictly plurisubharmonic function on C n (orjust defined on some open neighborhood of some point of ∂ Ω in C n ). Thenfor any choice of 0 < γ < L > ∂ ¯ ∂ (cid:0) − δ γ e − Lψ (cid:1) is strictly positive onΩ ∩ D ′ for some open neighborhood D ′ of P in D .Suppose we have the case of an odd k in the following expansion whichwe would like to rule out. r | V = X ν + ··· + ν ℓ = k σ ν , ··· ,ν ℓ ( ρ ) ν · · · ( ρ ℓ ) ν ℓ + O ℓ X j =1 ( ρ j ) ! k +12 k ≥ 2, where σ ν , ··· ,ν ℓ is a real-analytic function on U (aftershrinking U as an open neighborhood of P in C n if necessary) and σ ν ∗ , ··· ,ν ∗ ℓ isnonzero at P for some ν ∗ + · · · + ν ∗ ℓ = k . Assume that k is odd. We can finda tangent vector η of V at the point P of M normal to M such that J η istangential to M and the k -derivative of r in the direction of η is nonzero. Let C be local complex curve in C n through P such that the complex tangentvector to C of type (1 , 0) at P is equal to η − √− J η and C ∩ ∂ Ω ⊂ M and C ∩ M is a regular curve in C . Since k is odd, after replacing C by anopen neighborhood of P in C we can assume without loss of generality that C − M consists of two nonempty connected components C ∩ Ω and C − Ω.Let κ = − δ γ e − Lψ and we restrict κ to C ∩ Ω. Let φ be a smooth functionon C whose zero-set is C ∩ M and which is negative on C ∩ Ω with dφ nowherezero on C ∩ M . Since − κ is equal to σ ( − r ) γ = ˜ σ ( − φ ) kγ on C ∩ Ω for somepositive-valued smooth functions σ and ˜ σ on C (after replacing C by anopen neighborhood of P in C if necessary), from the plurisubharmonicity of κ on Ω we have a contradiction to (IV.11.1) when 0 < γ < kγ > 1, because κ | C ∩ Ω is subharmonic on C ∩ Ω and − κ | C ∩ Ω is equal to (cid:16) − ˜ σ kγ φ (cid:17) kγ and ˜ σ kγ φ is smooth on C and is 0 at C ∩ M and d ˜ σ kγ φ is nowherezero on M . This argument avoids the process in (IV.10) of constructing theanalog of the second vector field, at the end of (IV.9), of type (1 , 0) in aneighborhood of P in C n tangential to S and not tangential to M at P .(IV.12) Handling the Case of Even Vanishing Order by Stratification Accord-ing to Iterated Lie Brackets. We now deal with the general case by choosingthe set of defining functions ρ , · · · , ρ ℓ by stratification according to iteratedLie brackets. Recall that iterated Lie brackets of vector fields on E ∞ withcoefficients in N ( ∞ ) = N ∩ T R E ∞ ∩ J T R E ∞ generate the distribution ˜ N and M is an integral submanifold of E ∞ whose tangent space at every point isequal to the subspace distribution ˜ N at that point. Because of the Jacobiidentity for the Lie brackets of three vector fields, we can select vector fields τ , τ , · · · , τ ℓ on E ∞ with values in N ( ∞ ) = N ∩ T R E ∞ ∩ J T R E ∞ such that in-ductively, ˜ τ = [ τ , τ ] and ˜ τ j = [˜ τ j − , τ j ] for 2 ≤ j ≤ ℓ and ˜ τ j ( P ) is notspanned by (cid:0) N ( ∞ ) (cid:1) P , ˜ τ ( P ) , · · · , ˜ τ j − ( P ) for 1 ≤ j ≤ ℓ . We now choose ρ , · · · , ρ ℓ such that, modulo (cid:0) N ( ∞ ) (cid:1) P , the 1-forms ( J dρ j ) ( P ) at P for1 ≤ j ≤ ℓ , when restricted to the tangent space T R M,P of M at P preciselyform a dual basis for ˜ τ ( P ) , · · · , ˜ τ ℓ ( P ). In other words, the R -linear func-tionals defined by ( J dρ j ) ( P ) at P for 1 ≤ j ≤ ℓ on the quotient space60 R M,P .(cid:0) N ( ∞ ) (cid:1) P form the dual basis of the elements in T R M,P .(cid:0) N ( ∞ ) (cid:1) P induced by ˜ τ ( P ) , · · · , ˜ τ ℓ ( P ).Let ξ j = (cid:0) ˜ τ j − − √− J ˜ τ j − (cid:1) and η j = (cid:0) τ j − √− J τ j (cid:1) on M for 1 ≤ j ≤ ℓ so that both ξ j and η j are of type (1 , 0) tangential to V with the realpart of ξ j being ˜ τ j and the real part of η j being τ j . Take 2 ≤ j ≤ ℓ . Asverified above in (IV.7), from ˜ τ j − = ξ j + ξ j and τ j = η j + η j we get[˜ τ j − , τ j ] = √− (cid:16)h ξ j + √− η j , ξ j + √− η j i − h ξ j − √− η j , ξ j − √− η j i(cid:17) modulo C ⊗ R (cid:0) T R M ∩ J T R M (cid:1) = T (1 , M ⊕ T (0 , M , where ξ j + √− η j = 12 (˜ τ j − + J τ j ) + √− 12 ( τ j − J ˜ τ j − ) ,ξ j − √− η j = 12 (˜ τ j − − J τ j ) − √− 12 ( τ j + J ˜ τ j − ) . At the point P we have 1 = ( J dρ j ) (˜ τ j )= √− 12 ( J dρ j ) (cid:16)h ξ j + √− η j , ξ j + √− η j i − h ξ j − √− η j , ξ j − √− η j i(cid:17) and at least one of( J dρ j ) (cid:16)h ξ j + √− η j , ξ j + √− η j i(cid:17) and ( J dρ j ) (cid:16)h ξ j − √− η j , ξ j − √− η j i(cid:17) has absolute value at least 1 and is nonzero at P . We set ζ j to be one of thetwo possibilities ξ j + √− η j = 12 (˜ τ j − + J τ j ) + √− 12 ( τ j − J ˜ τ j − ) ,ξ j − √− η j = 12 (˜ τ j − − J τ j ) − √− 12 ( τ j + J ˜ τ j − ) . so that (cid:12)(cid:12) ( J dρ j ) (cid:0)(cid:2) ζ j , ζ j (cid:3)(cid:1)(cid:12)(cid:12) ≥ P . From the way we define the 1-jet of ρ j at P we know that among the following vectors τ , τ , · · · , τ ℓ , J τ , J τ , · · · , J τ ℓ , τ , · · · , ˜ τ ℓ , J ˜ τ , · · · , J ˜ τ ℓ at P the only one at which dρ j is nonzero is J ˜ τ j where the value of dρ j is 1,because the vectors τ , τ , · · · , τ ℓ , J τ , J τ , · · · , J τ ℓ , ˜ τ , · · · , ˜ τ ℓ all belong to the tangent space T R M,P of M at P which is equal to (cid:0) N ( ∞ ) (cid:1) P and ρ j vanishes on M and because the R -linear functionals defined by( J dρ ) ( P ) , · · · , ( J dρ ℓ ) ( P )at P on the quotient space T R M,P .(cid:0) N ( ∞ ) (cid:1) P form the dual basis of the ele-ments in T R M,P .(cid:0) N ( ∞ ) (cid:1) P induced by ˜ τ ( P ) , · · · , ˜ τ ℓ ( P ). Hence ( dρ j ) ( ζ p )at P is 0 for j = p − A . From ˜ τ j − = ξ j + ξ j and τ j = η j + η j we get A ˜ τ j = [˜ τ j − , Aτ j ] = 12 (cid:16)(cid:2) ξ j + √− Aη j , ξ j + iAη j (cid:3) − h ξ j − √− Aη j , ξ j − √− Aη j i(cid:17) , where ξ j + √− Aη j = 12 (˜ τ j − + AJ τ j ) − √− 12 (˜ τ j − + AJ τ j ) ,ξ j − √− Aη j = 12 (˜ τ j − − AJ τ j ) − √− 12 (˜ τ j − − AJ τ j ) . At the point P we have A = A ( J dρ j ) (˜ τ j ) = ( J dρ j ) ( A ˜ τ j ) = ( J dρ j ) ([˜ τ j − , Aτ j ])= √− 12 ( J dρ j ) (cid:16)h ξ j + √− Aη j , ξ j + √− Aη j i − h ξ j − √− Aη j , ξ j − √− Aη j i(cid:17) and at least one of( J dρ j ) (cid:16)h ξ j + √− Aη j , ξ j + √− Aη j i(cid:17) and ( J dρ j ) (cid:16)h ξ j − √− Aη j , ξ j − √− Aη j i(cid:17) has absolute value at least A at P . We set ζ j,A to be one of the two possi-bilities ξ j + √− Aη j = 12 (˜ τ j − + AJ τ j ) + √− 12 ( Aτ j − J ˜ τ j − ) , j − √− Aη j = 12 (˜ τ j − − AJ τ j ) − √− 12 ( Aτ j + J ˜ τ j − )so that (cid:12)(cid:12) ( J dρ j ) (cid:0)(cid:2) ζ j,A , ζ j,A (cid:3)(cid:1)(cid:12)(cid:12) ≥ A at P . Note that when A = 1 we have ζ j,A = ζ j so that for any value of A > ζ j,A − ζ j is equal to ± (cid:18) ( A − J τ j + √− A − τ j (cid:19) . From the way we define the 1-jet of ρ j at P we know that among the followingvectors τ , τ , · · · , τ ℓ , J τ , J τ , · · · , J τ ℓ , ˜ τ , · · · , ˜ τ ℓ , J ˜ τ , · · · , J ˜ τ ℓ at P the only one at which dρ j is nonzero is J ˜ τ j where the value of dρ j is 1,because the vectors τ , τ , · · · , τ ℓ , J τ , J τ , · · · , J τ ℓ , ˜ τ , · · · , ˜ τ ℓ all belong to the tangent space T R M,P of M at P which is equal to (cid:0) N ( ∞ ) (cid:1) P and ρ j vanishes on M and because the R -linear functionals defined by( J dρ ) ( P ) , · · · , ( J dρ ℓ ) ( P )at P on the quotient space T R M,P .(cid:0) N ( ∞ ) (cid:1) P form the dual basis of the ele-ments in T R M,P .(cid:0) N ( ∞ ) (cid:1) P induced by ˜ τ ( P ) , · · · , ˜ τ ℓ ( P ). Hence ( dρ j ) ( ζ p )at P is 0 for j = p − 1. Moreover, at P ( dρ j ) ( ζ p,A − ζ p ) = ( dρ j ) (cid:18) ± (cid:18) ( A − J τ j + √− A − τ j (cid:19)(cid:19) = 0 . Thus ( dρ j ) ( ζ p,A ) = ( dρ j ) ( ζ p ) is independent of A for all p and j .Let 1 < q ≤ ℓ be the minimum such that ν q ≥ ν q + · · · + ν ℓ = k and σ , ··· , ,ν q , ··· ,ν ℓ ( P ) = 0. Since we have ζ q,A ρ j = 0 at P for j = q − 1, the termof lowest vanishing order at P which we can get is k − ζ q,A ζ q,A ρ j at P from X ν + ··· + ν ℓ = k σ ν , ··· ,ν ℓ ( ρ ) ν · · · ( ρ ℓ ) ν ℓ 63r come with the factor ζ q,A ρ q − ζ q,A ρ q − at P from X ν + ··· + ν ℓ = k +1 σ ν , ··· ,ν ℓ ( ρ ) ν · · · ( ρ ℓ ) ν ℓ in the expansion of r | V . The sum of all these terms of vanishing order k − P is identically zero only when X ν q + ··· + ν ℓ = k σ , ··· , ,ν q , ··· ,ν ℓ ν q (cid:0) ζ q,A ζ q,A ρ q (cid:1) ( ρ q ) ν q − ( ρ q +1 ) ν q +1 · · · ( ρ ℓ ) ν ℓ = − X ν + ··· + ν ℓ = k +1 σ ν , ··· ,ν ℓ ν q − ( ν q − − 1) ( ζ q,A ρ q − ) (cid:0) ζ q,A ρ q − (cid:1) ·· (cid:0) ( ρ ) ν · · · ( ρ q − ) ν q − ( ρ q − ) ν q − − ( ρ q +1 ) ν q +1 · · · ( ρ ℓ ) ν ℓ (cid:1) . Since the contradiction comes from the change of the sign of the Levi Form of S = ∂ Ω when we approach M from along some appropriate path in S = ∂ Ωwhich corresponds to a path in V up to order k , we have trouble only whenfor any choice of ν q ≥ , ν q +1 ≥ , · · · , ν ℓ ≥ ν q + · · · + ν ℓ = k we have σ , ··· , ,ν q , ··· ,ν ℓ ν q (cid:0) ζ q,A ζ q,A ρ q (cid:1) = − X ≤ i ≤ j 1) ( ζ q,A ρ q − ) (cid:0) ζ q,A ρ q − (cid:1) at the point P of M for any choice of ν q ≥ , ν q +1 ≥ , · · · , ν ℓ ≥ ν q + · · · + ν ℓ = k , which is the same as A σ , ··· , ,ν q , ··· ,ν ℓ ν q (cid:0) ζ q ζ q ρ q (cid:1) = − X ≤ i ≤ j 1) ( ζ q ρ q − ) (cid:0) ζ q ρ q − (cid:1) , because (cid:0) ζ q,A ζ q,A ρ q (cid:1) ( P ) = A (cid:0) ζ q ζ q ρ q (cid:1) ( P ) and ( ζ q,A ρ q − ) ( P ) = ( ζ q ρ q − ) ( P )and (cid:0) ζ q,A ρ q − (cid:1) ( P ) = (cid:0) ζ q ρ q − (cid:1) ( P ). Since (cid:0) ζ q ζ q ρ q (cid:1) ( P ) = 0, this trouble cansimply be handled with the choice of a sufficiently large A .Finally, in order to get a contradiction from the evenness of the vanishingorder k of r | V ∩ U at M ∩ U , we construct(i) a real-analytic curve Γ S in S ∩ U containing P which is transversal to M , 64ii) a real-analytic curve Γ V in V ∩ U containing P which is transversal to M ,(iii) a smooth bijection Ψ from Γ V to Γ S ,(iv) a vector field ζ S of type (1 , 0) tangential to S defined only at points ofthe curve Γ S and smooth along Γ S whose value at P is ζ q,A , and(v) a vector field ζ V of type (1 , 0) tangential to V defined only at points ofthe curve Γ V and smooth along Γ V whose value at P is ζ q,A such that(a) the distance between P ∈ Γ V and Ψ( P ) ∈ Γ S is of order of (dist Γ V ( P, P )) k ,where dist Γ V ( P, P ) is the distance between P and P along Γ V , and(a) the difference of the value of ζ V at P ∈ Γ V and the value of ζ S atΨ( P ) ∈ Γ S is of order of (dist Γ V ( P, P )) k .Then the Levi form of r in the direction ζ S at a point P in Γ S other than P will change sign as P moves along Γ S to pass P because the evenness of k implies that the Levi form of r in the direction ζ S at a point P vanishes ofodd order k − P along Γ S . This contradicts the weak pseudoconvexityof S . 65 ppendix A: Some Techniques of Applying of Skoda’s Theorem onIdeal Generation In this Appendix we give some techniques of applying Skoda’s theoremon ideal generation [Sk72, Th.1, pp.555-556] which involve derivatives andJacobian determinants. The significance is more in the techniques themselvesthan in the statements given here to demonstrate their use. Though thesetechniques are not directly used in this note (except the use of (A.2) in (III.7)and the use of (A.3) in (III.8)), they may be useful in reducing the vanishingorders of multiplier ideals in Kohn-type algorithms in the setting of moregeneral partial differential equations.(A.1) Proposition. Let Ω be a bounded Stein open subset of C n . Let g , · · · , g n , ρ be holomorphic functions on some open neighborhood ˜Ω of thetopological closure ¯Ω of Ω. Let Z be the common zero-set of g , · · · , g n in ˜Ω. Assume that ρ vanishes on Z . Let J be the Jacobian determi-nant of g , · · · , g n . Then there exist holomorphic h , · · · , h n on Ω such that ρJ = P nj =1 h j g j . Proof. For any 0 < γ < K of C n with coordinates w = ( w , · · · , w n ) the integral(A . . Z w ∈ K Q nj =1 (cid:0) √− dw j ∧ dw j (cid:1)(cid:16)P nj =1 | w j | (cid:17) γn is finite. Since ρ vanishes on Z , it follow that there exists some 0 < η < . . | ρ | (cid:16)P nj =1 | g j | (cid:17) ηn is bounded on some open neighborhood U of ¯Ω in ˜Ω. Let γ = 1 − η and α = 1 + η . Since J is the Jacobian determinant of g , · · · , g n , by pullingback (A.1.1) by w j = g j for 1 ≤ j ≤ n and using the uniform boundednessof (A.1.2) on U , we conclude that(A . . Z Ω | ρ J | (cid:16)P nj =1 | g j | (cid:17) αn < ∞ . 66y using (A.1.3) and applying Skoda’s theorem [Sk72, Th.1, pp.555-556] tothe Stein domain Ω to express ρ J as a linear combination of g , · · · , g n withholomorphic functions, we obtain h , · · · , h n satisfying the requirements ofthe Proposition. Q.E.D.(A.2) Proposition (Ideal Generated by Components of Gradient). Let f be aholomorphic function germ on C n at the origin which vanishes at the origin.Then f n +1 belongs to the ideal I generated by ∂f∂z j for 1 ≤ j ≤ n at theorigin, where z , · · · , z n are the coordinates of C n . Proof. Let π : ˜ U → U be the simultaneous resolution of singularities for theideal I and the ideal O C n f generated by f with exceptional hypersurfaces { E ℓ } ℓ in normal crossing in ˜ U , where U is an open neighborhood of the originin C n on which the holomorphic function germ f is defined. We claim that(A . . | f | P nj =1 (cid:12)(cid:12)(cid:12) ∂f∂z j (cid:12)(cid:12)(cid:12) is uniformly bounded in some relatively compact open neighborhood U ′ ofthe origin in U . Otherwise, when we write the divisor of π ∗ f of f as P ℓ a ℓ E ℓ and write π ∗ I as P ℓ b ℓ E ℓ with a ℓ and b ℓ being nonnegative integers, we have b ℓ > a ℓ for some ℓ with 0 ∈ π ( E ℓ ) and we can find a local holomorphic curve˜ ϕ : W → ˜ U with W being an open neighborhood of the origin in C and π ˜ ϕ (0) = 0 such that ϕ ( W ) is transversal to E ℓ and is disjoint from any E k with k = ℓ . Then d ( f ◦ ϕ ) vanishes at 0 to an order higher than that f ◦ ϕ ,which is a contradiction, because f ◦ ϕ vanishes at 0. This argument actuallygives a slightly higher vanishing order of | f | than that of P nj =1 (cid:12)(cid:12)(cid:12) ∂f∂z j (cid:12)(cid:12)(cid:12) alongeach E ℓ when they are pulled back to ˜ U so that Z U ′ | f n +1 | (cid:18)P nj =1 (cid:12)(cid:12)(cid:12) ∂f∂z j (cid:12)(cid:12)(cid:12) (cid:19) α ( n +1) < ∞ for some α > 1. The conclusion of the Proposition now follows from Skoda’stheorem [Sk72, Th.1, pp.555-556]. Q.E.D.(A.2.2) Remark on the Relation Between Proposition (A.2) and l’Hˆopital’sRule. The argument in the proof of Proposition (A.2) consists of the ver-ification of the uniform bound of (A.2.1) on some open neighborhood U ′ 67f the origin in C n and a straightforward application of Skoda’s theorem[Sk72, Th.1, pp.555-556]. The argument used in the verification of the uni-form bound of (A.2.1) on U ′ is actually the usual l’Hˆopital’s rule in calculusapplied to the pullback of the quotient (A.2.1) to the open unit 1-disk ∆in C by a holomorphic map g : ∆ → C n with g (0) = 0 when one appliesdifferentiation at the origin along any ray of ∆ until one ends up with anonzero derivative of the denominator. The uniformity of the bound comesfrom the fact that one needs only consider a compact holomorphic family ofsuch holomorphic maps g : ∆ → C n , as is easily seen, for example, by usinga resolution of singularities. Another simple way of looking at the boundof (A.2.1) is the trivial observation that the vanishing order of an analyticfunction at a point of its zero-set is no more than the vanishing order of itsgradient.(A.2.3) Remark on the Difference Between the Jacobian Determinants withRespect to All Variables and the Jacobian Determinants With Respect toAll Variables with Respect to a Proper Subset of Variables. Let F , · · · , F N be holomorphic function germs on C at the origin vanishing at the originsuch that the ideal I generated by F , · · · , F N contains an effective powerof the maximum ideal sheaf m C , of C . By Proposition (A.2) the idealgenerated by the components of the gradients of F , · · · , F N , namely by ∂F j ∂z k for 1 ≤ j ≤ N, ≤ k ≤ 2, contains an effective power of m C , . We canregard each ∂F j ∂z k for 1 ≤ j ≤ N, ≤ k ≤ F j with respect to the single variable z j . These first-orderpartial derivatives can be regarded as the Jacobian determinants with respectto a proper subset of all the variables. Proposition (A.2) can be restated asfollows. The ideal generated by all such Jacobian determinants with respectto a proper subset of all the variables contains an effective power of m C , .The situation is very different from the ideal I generated by all Jacobiandeterminants with respect to the full set of all the variables ∂ ( F j , F j ) ∂ ( z , z ) for 1 ≤ j , j ≤ N. In general, the ideal I does not contain an effective power of m C , , as onecan easily see in the special case where N = 2 and the ideal I is generatedby a single holomorphic function germ.In general, for the complex Euclidean space C n instead of C , when wehave holomorphic function germs F , · · · , F N on C n at the origin vanishing at68he origin such that the ideal generated by F , · · · , F N contains an effectivepower of the maximum ideal sheaf m C n , of C n , we can consider for 1 ≤ ν ≤ n the ideal I ν generated by the Jacobian determinants ∂ ( F j , · · · , F j ν ) ∂ ( z k , · · · , z k ν )for 1 ≤ j , · · · , j ν ≤ N and 1 ≤ k , · · · , k ν ≤ n − 1. As we see in Proposition(A.3) below, for 1 ≤ ν ≤ n − I ν contains an effective power of m C n , ,though in general the ideal I n does not contain an effective power of m C n , . Itmeans that the situation for the ideal generated by all Jacobian determinantswith respect to a proper subset of all the variables is very different from theideal I generated by all Jacobian determinants with respect to the full setof all the variables.(A.2.4) Remark on a Generalization of the Special Case of Proposition (A.2)for Dimension Two. The special case of Proposition (A.2) for dimension twois used in this note in (III.7) to prove the effective termination of Kohn’s algo-rithm for C . For the proof of the effective termination of Kohn’s algorithmfor C n the corresponding statement which has to be used is not Proposition(A.2) for dimension n , but the following Proposition (A.3).(A.3) Proposition (Ideal Generated by Jacobian Determinants with Respectto a Proper Subset of Variables). Let F , · · · , F N be holomorphic functiongerms on C n at the origin vanishing at the origin such that the ideal generatedby F , · · · , F N contains an effective power of the maximum ideal of C n at theorigin. Let 1 ≤ ν < n . Let J ν be the ideal generated by ∂ ( F j , · · · , F j ν ) ∂ ( z k , · · · , z k ν ) . for 1 ≤ j < · · · < j ν ≤ N and 1 ≤ k < · · · < k ν ≤ n . Then the ideal J ν contains an effective power of the maximum ideal of C n at the origin. Proof. Let us first introduce some notations. For an ideal I of O C n , wedefine | s I | = k I X j =1 | s j,I | ! , where s ,I , · · · , s k I ,I form a set of generators of I . The expression | s I | isdefined up to a choice of the set of generators. We use this expression only69n the context of determining whether one such expression is dominated bya constant times another such expression | s J | for another ideal J of O C n , .For such a purpose the choices of generators in the definitions for | s I | and | s J | are immaterial. For our purpose, if λ ∈ N and ˆ I is I λ , then we can use | s ˆ I | = | s I | λ . For a holomorphic map ψ : ∆ → C n with ψ (0) = 0 and an ideal I of O C n , with generators s ,I , · · · , s k I ,I , by the vanishing order a I,ψ of I on ψ at 0 we mean the minimum of ord ( s j,I ◦ ψ ) for 1 ≤ j ≤ k I , where ord ( · )denotes the vanishing order on C at the origin. For an ℓ -jet ξ of C n at theorigin which can be represented by ψ we denote a I,ψ also by a I,ξ . (Here theconvention is that a 1-jet is a tangent vector.) If a I,ψ < ℓ , then a I,ξ = a I,ϕ for any holomorphic map ϕ : ∆ → C n with ϕ (0) = 0 which represents the ℓ -jet ξ .Note that for our purpose we could also use alternatively the conceptof the normalized vanishing order of I on ψ at 0 (instead of the vanishingorder a I,ψ ) by defining the normalized vanishing order of I on ψ at 0 as theminimum of ord ( s j,I ◦ ψ )ord ψ for 1 ≤ j ≤ k I , where ord ψ is the minimum of the vanishing orders of the n components of ψ on C at the origin.Since all the main arguments in this proof occur already in the proof of thespecial case where N = n = 3, for notational simplicity we will only presentthe proof of this special case. The general case is completely analogous butwith much more complicated notations. We break down the proof into thefollowing five steps. Step One. Let G , G be holomorphic function germs on C at the originvanishing at the origin such that the divisor Z of G is irreducible and ofmultiplicity 1. Assume that dG ∧ dG is not identically zero. Then thereexists some positive constant C such that X k ,k =1 | G ( dG ∧ dz k ∧ dz k ) | ≤ C X j =1 | dG ∧ dG ∧ dz j | on Z = { G = 0 } .Step One is verified by 70i) taking any holomorphic curve ϕ : ∆ → G with ϕ (0) = 0 and theimage of ϕ (∆) not contained in the zero-set of G ,(ii) using the fact that the vanishing order at the origin of the pullback G ◦ ϕ on ∆ is no more than the minimum of the vanishing orders ofits first-order partial derivatives at the origin, and(iii) observing that at a regular point of Z , where z k , z k are used as localcoordinates, the component of the gradient of the restriction of G to Z for the coordinate z k is equal to the quotient of dG ∧ dG ∧ dz k by dG ∧ dz k ∧ dz k as one can easily see by using the chain rule andthe implicit differentiation for functions defined on Z = { G = 0 } . Step Two. Let I and J be ideals in O C , contained in the maximum ideal m C , of O C , such that I contains ( m C n , ) q for some positive integer q . If | s I | is not dominated by a constant times | s J | , then there exists some ( q + 2)-jet ξ of C at the origin which is represented by some holomorphic map ψ : ∆ → C with ψ (0) = 0 such that a I,ψ ≤ q and a I,ψ < a J,ψ . Step Three. Let A be the ideal generated by elements F , F , F of m C , such that A contains ( m C , ) q for some positive integer q . Let p ∈ N . Thenthere exists a positive integer q depending only on q and there exists apositive integer m depending on q and p with the following property. Forany p -jet ξ of C at the origin, let P ( F , F , F ) be a generic homogeneouspolynomial of degree m in F , F , F whose divisor V contains a holomorphiccurve representing ξ and let ϕ : ∆ → C be a holomorphic curve germ with ϕ (0) = 0 whose image is a generic curve germ in V which represents ξ . Thenthe minimum vanishing order of ∂∂z ℓ P ( F , F , F ) on the holomorphic curvegerm ϕ : ∆ → C at the origin is no more than ( m − a A,ϕ + q for 1 ≤ ℓ ≤ V , as the divisor of a generic homogeneous polynomial of degree m in F , F , F which contains a holomorphic curve representing ξ , is a reducedand irreducible hypersurface germ in C at the origin. Moreover, the imageof ϕ : ∆ → C , as a generic curve germ in V which represents ξ , is containedin { } ∪ Reg( V ), where Reg( V ) is the regular part of V . Step Four. Let F , F , F be from Step Three. Let J be the ideal generatedby ∂ ( F j , F j ) ∂ ( z k , z k )71or 1 ≤ j < j ≤ ≤ k < k ≤ 3. Let λ ∈ N and I = ( m C , ) λ .Let p = λ + 2. Assume that | s I | is not dominated by any positive constanttimes | s J | . By Step Two, there exists a p -jet ξ of C at origin such that, forany holomorphic map ϕ : ∆ → C whose p -jet at the origin is equal to ξ ,the vanishing order a J,ϕ of J on ϕ at 0 is greater than the vanishing order a I,ϕ of I on ϕ at the origin. By Step Three we have positive integers q , m (with q depending only on p and with m depending only on p and q ) andwe have a polynomial P ( F , F , F ) homogeneous of degree m in F , F , F and a holomorphic curve germ ϕ : ∆ → C at the origin such that(i) the divisor of P ( F , F , F ) is a reduced and irreducible hypersurfacegerm of C at the origin,(ii) the image of the holomorphic curve germ ϕ : ∆ → C is contained inthe divisor of P ( F , F , F ),(iii) the holomorphic curve germ ϕ : ∆ → C represents the p -jet ξ of C atthe origin,(vi) the minimum vanishing order of ∂∂z ℓ P ( F , F , F ) on the holomorphiccurve germ ϕ : ∆ → C is no more than ( m − a A,ϕ + q for 1 ≤ ℓ ≤ P ( F , F , F ) of degree m in F , F and for 1 ≤ k 3, it follows from ∂P j ∂z k = X ℓ =1 ∂P j ∂F ℓ ∂F ℓ ∂z k by the chain rule that ∂ ( P , P ) ∂ ( z k , z k ) = X ℓ ,ℓ =1 ∂P ∂F ℓ ∂P ∂F ℓ ∂ ( F ℓ , F ℓ ) ∂ ( z k , z k ) . The vanishing order of ∂ ( P , P ) ∂ ( z k , z k )on the holomorphic curve germ ϕ : ∆ → C is at least 2( m − a A,ϕ + a J,ϕ .72pplying Step One to the case of G = P ( F , F , F ) and G = P ( F , F , F )with P ( F , F , F ) being any generic polynomial homogeneous of degree m in F , F , F , we get X k ,k =1 | P ( dP ∧ dz k ∧ dz k ) | ≤ C X j =1 | dP ∧ dP ∧ dz j | on P = 0, where C is a positive constant. We restrict this inequality to thecurve ϕ and conclude that ma A,ϕ + ( m − a A,ϕ + q ≥ m − a A,ϕ + a J,ϕ . By the choice of P ( F , F , F ) and the holomorphic curve germ ϕ : ∆ → C ,we have a J,ϕ > λ . Thus ma A,ϕ + ( m − a A,ϕ + q ≥ m − a A,ϕ + λ and we conclude that λ ≤ a A,ϕ + q ≤ q + q , because a A,ϕ ≤ q from the factthat A contains ( m C , ) q . Step Five. By setting λ = q + q + 1, we conclude from Step Four that | s I | is dominated by a constant times | s J | . As in the last part of the proof ofProposition (A.2), by Skoda’s theorem [Sk72, Th.1, pp.555-556] it followsfrom the local integrability of the quotient | s I | n +2) | s J | n +2) on C at the origin that ( m C , ) ( q + q +1)( n +2) is contained in the ideal J gen-erated by ∂ ( F j , F j ) ∂ ( z k , z k )for 1 ≤ j < j ≤ ≤ k < k ≤ 3. This finishes the proof.We would like to remark that the main point of this proof is to apply theargument for gradients given in Proposition (A.2) for C n to the divisor of P ( F , F , F ) in C instead of to C n . Q.E.D.73A.4) Proposition. Let h , · · · , h n be holomorphic function germs on C n atthe origin so that the origin is their only common zero. Let dh ∧ · · · ∧ dh n = J ( dz ∧ · · · ∧ dz n ). Then J does not belong to the ideal generated by h , · · · , h n . Proof. Suppose the contrary. Then there exist holomorphic function germs f , · · · , f n on C n at the origin such that J = P nj =1 f j h j . We let ω j = f j ( dz ∧ · · · ∧ dz n ) for 1 ≤ j ≤ n so that(A . . dh ∧ · · · ∧ dh n = n X j =1 h j ω j . Since the origin is the only common zero of h , · · · , h n , we can find connectedopen neighborhoods U and W of the origin in C n so that the map π : C n → C n defined by( z , · · · , z n ) ( w , · · · , w n ) = ( h ( z , · · · , z n ) , · · · , h n ( z , · · · , z n ))maps U properly and surjectively onto W and makes U a branched coverover W of λ sheets. By replacing U and W by relatively compact openneighborhoods U ′ and W ′ of the origin in U and W respectively, we canassume without loss of generality that R U | ω j | ≤ C < ∞ for 1 ≤ j ≤ n .We take the direct image of the equation (A.4.1) under π . The left-handside of the equation (A.4.1) yields λ ( dw ∧ · · · ∧ dw n ), because the map π is defined by w j = h j for 1 ≤ j ≤ n . Let θ j be the direct image of ω j under π for 1 ≤ j ≤ n . Let Z be the branching locus of π in W . Forany simply connected open subset G of W − Z , U ∩ π − ( G ) is the disjointunion of λ open subsets H , · · · , H λ of U and θ j ( Q ) = P λℓ =1 ω j (cid:16) ˜ Q j (cid:17) , where U ∩ π − ( Q ) = n ˜ Q , · · · , ˜ Q λ o with ˜ Q j ∈ H j . Now Z G | θ j | ≤ λ λ X j =1 Z H j | ω j | ≤ λC. Since W − Z can be covered by a finite number of simply connected opensubsets, it follows that Z G | θ j | < ∞ for 1 ≤ j ≤ n. θ j is a holomorphic n -form on G and λ dz ∧ · · · ∧ dz n = n X j =1 z j θ j on G , which gives a contradiction, because the left-hand side does not vanishat the origin whereas the right-hand side does. Q.E.D.(A.5) Remark. Proposition (A.4) uses only the direct images of top-degreeholomorphic forms and actually does not use Skoda’s theorem [Sk72, Th.1,pp.555-556]. The significance of Proposition (A.4) is that the coefficient J in dh ∧ · · · ∧ dh n = J ( dz ∧ · · · ∧ dz n )cannot be contained in ( m C n , ) p if( m C n , ) p ⊂ n X j =1 O C n , h j so that the vanishing order of J at 0 is no more than p .(A.5) Example to Show the Sharpness of the Exponent in Skoda’s Theorem. The exponent in the denominator of the assumption in Skoda’s theorem[Sk72, Th.1, pp.555-556] plays a rˆole in effective bounds. As stated in Skoda’stheorem [Sk72, Th.1, pp.555-556] it is sharp and cannot be lowered even inthe case of Riemann surfaces. Let X be the Riemann sphere P . Consider thehyperplane section line bundle H P . Take two holomorphic sections g , g of H P without common zeroes. Take the holomorphic section f of 2 H P + K P over P which corresponds to a constant function on P via the isomorphismbetween K P and − H P . If the exponent used in the denominator of theassumption in Skoda’s theorem [Sk72, Th.1, pp.555-556] can be lowered sothat α = 1, then p = 2 and n = 1 and q = min ( n, p − q ) = 1 and αq + 1 = 2and the assumption Z P | f | (cid:0) | g | + | g | (cid:1) αq +1 < ∞ is satisfied because g , g have no common zeroes. Note that when α > f ∈ Γ ( P , m H P + K P )75or some m > 2. If Skoda’s theorem [Sk72, Th.1, pp.555-556] holds withthe lower exponent in the denominator in its assumption, then we can write f = h g + h g with h , h ∈ Γ ( P , H P + K P )which is impossible, becauseΓ ( P , H P + K P ) = 0from the isomorphism between K P and − H P . References [Ca84] D. Catlin, Boundary invariants of pseudoconvex domains. Ann. ofMath. (1984), 529–586.[DA79] J. P. D’Angelo, Finite type conditions for real hypersurfaces. J. Diff.Geom. (1979), 59–66.[DF77] K. Diederich and J. E. Fornaess, Pseudoconvex domains: boundedstrictly plurisubharmonic exhaustion functions. Invent. Math. (1977),129–141.[DF78] K. Diederich and J. E. Fornaess, Pseudoconvex domains with real-analytic boundary. Ann. of Math. (1978), 371-384.[GT83] D. Gilbarg and N. Trudinger, Elliptic partial differential equationsof second order . Second edition. Grundlehren der Mathematischen Wis-senschaften, Volume . Springer-Verlag, Berlin, 1983.[He62] S. Helgason, Differential Geometry and Symmetric Spaces , AcademicPress 1962.[Ko79] J. J. Kohn, Subellipticity of the ¯ ∂ -Neumann problem on pseudo-convex domains: sufficient conditions. Acta Math. (1979), 79–122.[Ni07] Andreea C. Nicoara, Equivalence of types and Catlin boundary sys-tems, arXiv:0711.0429 (November 2007).[Sk72] H. Skoda, Application des techniques L `a la th´eorie des id´eaux d’unealg`ebre de fonctions holomorphes avec poids, Ann. Sci. Ec. Norm. Sup. (1972), 548-580. 76To72] J.-C. Tougeron, Id´eaux de fonctions diff´erentiables . Ergebnisse derMathematik und ihrer Grenzgebiete, Volume . Springer-Verlag, Berlin-New York, 1972.[ZS60] O. Zariski and P. Samuel, Commutative algebra . Vol. II. The Univer-sity Series in Higher Mathematics. D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York 1960. Author’s mailing address: Department of Mathematics, Harvard University,Cambridge, MA 02138, U.S.A.