Invariance of Plurigenera and Torsion-Freeness of Direct Image Sheaves of Pluricanonial Bundles. Yum-Tong Siu 1

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1 Invariance of Plurigenera and Torsion-Freeness of Direct Image Sheaves of Pluricanonial Bundles Yum-Tong Siu Abstract. The deformational invariance of plurigenera proved recently by the author is a special case of the torsion-free property of the first direct image sheaf of the pluricanonical line bundle when the target space of the proper surjective holomorphic map is the open unit -dis with nonsingular fibers. This article discusses the adaptation of the techniques used in the proof of the deformational invariance of plurigenera to the general problem of proving the torsion-free property of the first direct image sheaf of the pluricanonical line bundle. The discussion covers also the more general case of the torsion-free property of the first direct image of the pluricanonical line bundle after twisting by a Hermitian holomorphic line bundle with semipositive curvature current and by its multiplier ideal sheaf. A number of results are obtained for the torsion-free problems by the adaptation of the techniques of the proof of the deformational invariance of plurigenera. Table of Contents. 0. Introduction.. Multiplier ideal sheaves and their global generation.. Normal ideal sheaf of hypersurface. 3. Extension theorem of Ohsawa-Taegoshi type. 4. Induction argument and extension of twisted sections. 5. Extension from singular fiber without twisting. 6. Induction technique on multiplicity of multiple point. Partially supported by a grant from the National Science Foundation. AMS 000 Mathematics Subject Classification: 3J5, 3G05, 3C35, 3L0, 3L0, 3U05, 3U5. Published in Finite or infinite dimensional complex analysis and applications, pp.45 83, Adv. Complex Anal. Appl.,, Kluwer Acad. Publ., Dordrecht, 004.

2 0. Introduction. The deformational invariance of plurigenera proved in [S98, S0] is a special case of the torsion-free property of the first direct image sheaf of the pluricanonical line bundle. The special case is when the target space of the proper surjective holomorphic map is the open unit -dis with nonsingular fibers. Kollar in [Ko86] proved, among a number of other related results, the following torsionfree property of the direct image sheaves of the canonical line bundle. Let X and Y be projective varieties and assume that X is smooth. Let f : X Y be a surjective map. Then R i f O X K X is torsion-free for i 0, where K X is the canonical line bundle of X. [Ko86, Th.. i]. In this article we discuss the problem of the torsion-free property of the first direct image of the pluricanonical line bundle. Since the verification of the nonexistence of torsion involves only the use of a single holomorphic function, without loss of generality we can confine ourselves to the case when the target manifold is the open unit - dis. Let be the open unit -dis { t C t < }. Unless explicitly mentioned to the contrary, t will be used as the coordinate for C and, in particular, for. The canonical line bundle of a complex manifold Y is denoted by K Y. The structure sheaf of a complex space Z is denoted by O Z. We use the terminology of a ringed space to highlight the structure sheaf of a possibly unreduced complex space, i.e., a complex space whose structure sheaf may contain nonzero nilpotent elements. A ringed space Z, O Z, used to denote a complex space, is a pair whose first component is the underlying space Z and whose second component is the structure sheaf O Z which may admit nonzero nilpotent elements. Problem0.. Let X be a complex manifold and π : X be a proper surjective holomorphic map whose fibers may be singular and may have unreduced complex structure. Suppose that there exists some positive line bundle over X i.e., π : X is a family of projective algebraic complex spaces. Let m be an integer. Is the first direct image sheaf R π O X mk X of the m-th power of the canonical line bundle of X torsion-free on? An equivalent formulation is whether the restriction map from Γ X,mK X to Γ X, / O X OX w mk X is surjective for any holomorphic function w on. The special case of Problem0. where the fibers of π : X are all nonsingular with multiplicity is equivalent to the following result on the deformational invariance of the plurigenera proved in [S0].

3 Theorem0.. Let π : X be a holomorphic family of compact complex projective algebraic manifolds over with fiber X t t. Let m be any positive integer. Then the restriction map Γ X,mK X Γ X t,mk Xt is surjective for t and, in particular, the complex dimension of Γ X t,mk Xt is independent of t for t. Theorem0. is a special case of the following theorem proved in [S0] which allows twisting in the deformational invariance of the plurigenera. Theorem0.3. Let π : X be a holomorphic family of compact complex projective algebraic manifolds over with fiber X t t. Let L be a holomorphic line bundle over X with a possibly singular metric e ϕ whose curvature current ϕ is semi-positive on X such π that e ϕ X0 is locally integrable on X 0. Let m be any positive integer. Then any element s Γ X 0,mK X0 + L with s e ϕ locally bounded on X 0 can be extended to an element s Γ X,mK X + L in the sense that s X0 = s π dt. Theorem0.3 motivates the following more general version of Problem0.. Problem0.4. Let X be a complex manifold and π : X be a proper surjective holomorphic map. Suppose that there exists some positive line bundle over X. Let L be a holomorphic line bundle over X with a possibly singular metric e ϕ whose curvature current ϕ π is semi-positive on X. Let I ϕ be the multiplier ideal sheaf for the metric e ϕ in the sense that I ϕ consists of all holomorphic function germs f on X with f e ϕ locally integrable on X. Let m be an integer. Is the first direct image sheaf R π I ϕ mk X + L torsion-free on? Theorem0.3 is not yet strong enough to imply even the special case of Problem0.4 where the fibers of π : X are all nonsingular with multiplicity. That special case under the additional assumption of I ϕ t O X = t I ϕ would be a consequence of the following more general conjecture which is still open. Conjecture0.5. Let π : X be a holomorphic family of compact complex projective algebraic manifolds over with fiber X t t. Let L be a holomorphic line bundle over X with a possibly singular metric e ϕ whose curvature current π ϕ is semi-positive on X. Let 3

4 4 m be any positive integer. Let s Γ X t,mk Xt + L with s e ϕ locally integrable on X t. Then s can be extended to s Γ X,mK X + L with s e ϕ locally integrable on X. In Theorems 0., 0., 0.3, Problems 0., 0.4, and Conjecture 0.5, we assume the existence of a positive line bundle on X. When we replace that assumption by the assumption that X is Kähler, the situation becomes much more difficult and the analogs of Theorems 0., 0., 0.3 for the Kähler setting still remain open. We will not discuss the Kähler case here and also we will not discuss the higher direct image sheaves R q π O X mk X and R q π I ϕ mk X + L for q. In this article we will present partial results toward the solutions of Problems0. and 0.4 which we can obtain by refining and adapting the methods of [S98, S0]. The methods of [S98, S0], though useful and powerful for the problem of the deformatonal invariance of plurigenera, have their limitations when applied to Problems0. and 0.4. Theorem0.6 and Theorem0.7 below respectively give our partial results toward the solutions of Problem0. and Problem0.4. Some crucial techniques for the complete solutions of Problems0. and 0.4 are still lacing. Theorem0.6. Let π : X be a holomorphic family of compact complex projective algebraic manifolds whose fibers are of complex dimension n. Let A be a holomorphic line bundle on X such that for every point P of X there are a finite number of elements of ΓX,E which all vanish to order at least n + at P and which do not simultaneously vanish outside P. Let a be a positive integer such that ak X A admits a non identically zero global holomorphic section s D on X which is non identically zero on X 0 and whose divisor in X is D. Let L be a holomorphic line bundle over X with a possibly singular metric e κ with semi-positive curvature current on X. Let l be an integer such that e l a l l+a κ s D l is locally integrable on X. Let be any positive integer. Then any s Γ X 0, K X0 + L with s e κ p e lκ locally integrable on X 0 for p l can be extended to an element of Γ X, K X + L. Theorem0.7. Let X be a complex manifold of complex dimension n+ and π : X be a proper surjective holomorphic map. Let t be the coordinate of. Assume that the set of points of X 0 := X {t = 0} where dt vanishes is of complex codimension at least in X 0. Let J be the ideal sheaf generated by all germs of holomorphic n + forms on

5 X of the form ω dt such that ω is a local holomorphic n-form defined on some open subset U of X whose pullbac to the intersection of U and the regular part of X 0 is square integrable. Then every element of Γ X 0, J m/ J m O X t mk X can be extended to an element of Γ X,mK X. Remar0.8. Theorem0.6 is an improvement over Theorem0.3 only in the following two points. The improvement is obtained at the expense of the additional assumption of general type. The first point is the removal of the assumption of local integrability of e ϕ from Theorem0.3. The second point is the replacement of the assumption of the local supremum bound of s e ϕ by the local integrability of s e lϕ for some finite positive numbers and l. The significance of the difference between the the supremum bound and the local integrability is that the latter condition defines a coherent ideal sheaf for the holomorphic function germs s while the sheaf defined by the former is in general not coherent see below. Remar0.9. Twisting by a holomorphic line bundle L on X with metric e ϕ and with ϕ locally plurisubharmonic can be added to Theorem0.7 in the same way that twisting is added to Theorem0. to yield Theorem0.3. We will not give a proof of the statement with the twisting for Theorem0.67, because the modifications needed for it are quite straightforward and could be readily seen from [S0] and the arguments presented in this article. Remar0.0. The ideal sheaf J introduced in Theorem0.7 is coherent. A proof of its coherence as well as the definitions for it from other perspectives will be given in. For the invariance of plurigenera the main argument is in [S98]. The argument by induction on m extends m-canonical sections from a fiber to the family and it relies on the two following ingredients: i the global generation of multiplier ideal sheaves on a compact complex manifold after twisting by a sufficiently ample line bundle A which is independent of the multiplier ideal sheaves; ii an extension theorem of L holomorphic top-degree forms with respect to a weight e ϕ, with ϕ plurisubharmonic, from the fiber to the family when e ϕ is defined on the family. The induction argument on m wors because an m-canonical section can be regarded as a top-degree form with coefficient in the m - canonical bundle. It is based on the following naive motivation. If one could write an element s m of Γ X 0,mK X0 as a sum of terms, each of which is the product of an element s of Γ X 0,K X0 and 5

6 6 an element s m of Γ X 0, m K X0, then one can extend s m to an element of Γ X,mK X by induction on m. While in general it is clearly impossible to so express s m as a sum of such products, one could successfully implement a modified form of the naive motivation, in which s is only a local holomorphic section and s m is an element of Γ X 0, m K X0 + A instead of Γ X 0, m K X0, where A is an ample line bundle on X independent of m and is sufficiently ample for the global generation of multiplier ideal sheaves. The argument by induction first enables one to extend a holomorphic m-canonical section from a fiber to the family after multiplying it by a holomorphic section of A. To get rid of the contribution from A, one uses the technique of first raising the section s to a power l for some large l before extension and then uses the l-th root of the absolute value of s to get the correct weight e ϕ for the application of the extension theorem. To avoid the limiting process of l, the technical assumption of general type is used in [S98]. The limiting process of l poses a convergence problem for the construction of the weight e ϕ which is needed for the application of the extension theorem. In [S98] the convergence problem of the limiting process of l is overcome by i introducing a Hermitian metric h as singular possible for the canonical line bundle of the fiber with respect to which the m- canonical section on the fiber to be extended is L and ii using the concavity of the logarithm in the estimate. The metric h as singular as possible is used to guarantee the uniform bound of the dimension of the space of all L holomorphic lm-canonical sections with respect to h on the fiber as l. Such a uniform bound is essential for the estimate by the concavity of the logarithm. It is the need to use the supremum bound in the handling of the convergence problem that maes impossible a direct adaptation of the method for the proof of Theorem0.3 to the proof of Conjecture0.5. The additional assumption of general type in Theorem0.6 enables us to avoid the limiting process so that we can replace the use of the supremum bound by an L p bound for some finite p. However, the improvement is still not enough to enable us to stay with the L norm to get Conjecture0.6. In his recent preprint [T0] posted on the web Tsuji introduced a new generalized Bergman ernel for the pluricanonical line bundle to handle the problem of convergence in the deformational invariance of plurigenera in the argument of [S98] when the assumption of general

7 type is removed. Tsuji s generalized Bergman metric is an elegant and natural approach to the convergence problem. Unfortunately one crucial estimate is still lacing and seems unliely to be establishable, as explained in [S0]. Since the arguments in this article are adaptations of those in [S98, S0], the emphasis is on the presentation of the adaptations and reformulations. Many details which are in [S98, S0] are therefore not provided here in their entirety. Prior to [S98, S0] which uses induction on m to prove the surjectivity of Γ X,mK X Γ X 0,mK X0, Levine [L83] used the degeneracy of the Hodge to DeRham spectral sequences to extend, in the Kähler setting, elements of Γ X 0,mK X0 to over a double point at the origin of. By induction on the multiplicity of the multiple point at the origin of over which elements of Γ X 0,mK X0 can be extended, he also verified the deformational invariance of the plurigenera for the Kähler setting, provided that the general section of the m-canonical bundle defines a nonsingular divisor or a divisor with only mild singularities. In applying the methods of [S98, S0] to the invariance of twisted plurigenera and to Problem0.4 the main difficulty is the sufficiently ample line bundle A which is required for the global generation of multiplier ideal sheaves. It is possible to combine the methods of multiplier ideal sheaves and the process, introduced by Levine [L83], of induction on the multiplicity of multiple points in the base over which extension of the section is possible. Such a combination has the advantage that no sufficiently ample line bundle A is needed. However, there are still obstacles beyond the stage of the double points in this combination of techniques. The reason is that the fibers over the multiple points of the base are unreduced complex spaces and there are still no vanishing theorems of Kawamata-Viehweg-Nadel [Ka8, V8, N89] available for such unreduced complex spaces. At the end of this article 7 we will discuss the combination of the methods of multiplier ideal sheaves and the process of induction on the multiplicity of the multiple point. We will illustrate this combination of techniques in the proof of a result on the extension of twisted pluricanonical sections on a reduced fiber to the unreduced fiber over the double point of the base.. Multiplier Ideal Sheaves and Their Global Generation. For a plurisubharmonic function ϕ on an open subset Ω of C n, 7

8 8 the multiplier ideal sheaf I ϕ is defined as the sheaf of all holomorphic function germs f on Ω such that f e ϕ is locally integrable on Ω. The multiplier ideal sheaf I ϕ is coherent and was first introduced in [N89]. Multiplier ideal sheaves are defined by L conditions with respect to e ϕ. One can generalize the notion to L p multiplier ideal sheaves for other values of p. The L p multiplier ideal sheaf I ϕ,l p is defined as the sheaf of all holomorphic function germs f such that f p e ϕ is locally integrable on Ω. The multiplier ideal sheaves I ϕ introduced by Nadel are the L multiplier ideal sheaves I ϕ,l. When p <, the L p multiplier ideal sheaf I ϕ,l p is also coherent. Since we will not use the coherence of I ϕ,l p for p < in this article, we will only very briefly setch here the arguments for the coherence without providing all the details. We first consider the special case when ϕ = α log g for some α > 0 and some holomorphic function g on Ω. For that special case, we tae a resolution π : Y Ω of the embedded singularity of the zero-set of g and a collection of nonsingular hypersurfaces {E j } j in Y in normal crossing so that the divisor of π g is of the form j a je j and K Y π K Ω = j b je j. Then I ϕ,l p is the zeroth direct image, under π, of the ideal sheaf on Y for the divisor j max 0, α a p j b j, where u respectively u means the largest integer not exceeding u respectively the smallest integer no less than u. For the general case, one uses the coherence criterion of generation by global sections. Tae a local section f. We argue that the local integrability f p e ϕ is an open condition in the sense that f p e +εϕ is still locally integrable for some sufficiently small ε > 0. Locally one finds a sequence of real numbers α ν and a sequence of holomorphic functions g ν such that the difference of the Lelong number of α ν log g ν and the Lelong number of + εϕ goes to zero uniformly on compact subsets as ν see e.g., [D93, p.346, Proposition 9.]. Tae 0 < ε < ε. Then locally for ν sufficiently large, I +εϕ,l p I αν+ε log g ν,l I p ϕ,l p. When p =, we can also define I ϕ,l as the sheaf of all holomorphic function germs f on Ω such that f e ϕ is locally bounded on Ω. However, the ideal sheaf I ϕ,l in general is not coherent. An easy example is the plurisubharmonic function ϕ = j= ε j log z c j on Ω = C with c j C and ε j > 0 such that c j 0 and ε j 0 sufficiently rapidly. Then the stal of I ϕ,l at 0 is zero, but its stal and at any

9 c 0 near 0 is nonzero. The difference between the case of p < and the case of p = is that the condition for f e ϕ to be locally bounded is not an open condition and we cannot conclude the local boundedness of f e +εϕ for any sufficiently small ε > 0 from the local boundedness of f e ϕ. For the proof of Theorem0.6 we need only the following original statement on the global generation of multiplier ideal sheaves in [S98]. Proposition.Global Generation of Multiplier Ideal Sheaves. Let L be a holomorphic line bundle over an n-dimensional compact complex manifold Y with a Hermitian metric which is locally of the form e ξ with ξ plurisubharmonic. Let I ξ be the multiplier ideal sheaf of the Hermitian metric e ξ. Let A be an ample holomorphic line bundle over Y such that for every point P of Y there are a finite number of elements of ΓY,A which all vanish to order at least n + at P and which do not simultaneously vanish outside P. Then ΓY, I ξ L + A + K Y generates I ξ L + A + K Y at every point of Y. For the proof of Theorem0.7 we need the effective version of the global generation of multiplier ideal sheaves. To eep the notations as simple as possible, we restate the effective version in [S0, p.34, Theorem.] in a simpler form with less precise constants of estimation but still sufficient for our purpose. We state it here for a general line bundle L rather just for the line bundle m K X in [S0, p.34, Theorem.]. The arguments for a general line bundle is completely analogous to the special case m K X. Actually for the proof of Theorem0.7 only the special case L = m K X is needed. We still state our effective version here for the general line bundle L, because the notations are easy to follow for the general case. 9 Proposition.Effective Global Generation of Multiplier Ideal Sheaves Let L be a holomorphic line bundle over an n-dimensional compact complex manifold Y with a Hermitian metric which is locally of the form e ξ with ξ plurisubharmonic. Let I ξ be the multiplier ideal sheaf of the Hermitian metric e ξ. Let A be an ample holomorphic line bundle over Y such that for every point P of Y there are a finite number of elements of ΓY,A which all vanish to order at least n + at P and which do not simultaneously vanish outside P. Let C be a finite open cover of Y. Let N be the complex dimension of the subspace of all elements s Γ Y,K Y + L + A

10 0 such that Y s e ξ h A <. Then there exists a constant C and there exist open subsets U λ, Uλ of Y λ Λ such that U λ is relatively compact in Uλ and Λ λ= U λ = Y and each Uλ is contained in some member of C and the following property is satisfied. There exist with σ,,σ N Γ Y,K Y + L + A Y σ e ξ h A N such that, if λ J and if s Γ Uλ,mK Y + A with s e ξ h A = C s <, U λ then one can find holomorphic functions b on U λ such that N s = b σ m on U λ and sup U λ = N b C C s. = Definition.3. To facilitate the description of the global generation of multiplier ideal sheaves, we introduce the following definition to describe the condition on the ample line bundle A in Propositions. and.. Let Y be a complex manifold of complex dimension n. A positive holomorphic line bundle A on Y is said to be sufficiently ample for the global generation of multiplier ideal sheaves on Y if for every point P of Y there are a finite number of elements of ΓY,A which all vanish to order at least n + at P and which do not simultaneously vanish outside P.. Normal Ideal Sheaf of Hypersurface. We now describe the ideal sheaf J of Theorem0.7 from other perspectives and prove its coherence. Let Z be a complex submanifold of complex codimension in a complex manifold Y of complex dimension n defined by w = 0 for some holomorphic function w on Y so that dw is nonzero at every point of

11 Z. Let T Z respectively T Y be the tangent bundle of Z respectively Y and N Z Y be the normal bundle of Z in Y. The sequence.0. 0 T Z T Y Z N Z Y 0 is exact over Z. The bundle-homomorphism T Y Z Z C defined by evaluation by dw induces a bundle-isomorphism Ψ : N Z Y Z C between N Z Y and the trivial line bundle Z C over Z. From the exact sequence.0. and the isomorphism Ψ we have an isomorphism Φ : K Z K Y Z defined as follows. For a local n -form ω on Z we extend it to a local n -form ω on Y. Then Φ ω = ω dw. We denote Φ s by s dw for any local section s of K Y over Z. Another description of s dw is as follows. Let s Γ Z, O Y / OY w K Y. Then there exists a unique holomorphic n -form on ω on Z such that ω dw = s on Z in the sense that every point Q of Z admits an open neighborhood U in Z and ω Γ U, n TY which is pulled bac to ω on U and which satisfies ω dw = s at every point of Z U. Here T Y is the dual bundle of T Y and n TY is the exterior product of n copies of TY. For, if ω, ω Γ U, n TY with ωj dw = s at every point of Z U for j =,, then ω ω = τ dw at every point of U for some open neighborhood U of Q in U and for some τ Γ U, n TY, as one can readily see by expressing ωj in terms of a local holomorphic basis of TY with dw as a member. We can now define s dw as ω. Definition.. Let Ω be an open subset of C n and w be a holomorphic function on Ω. Let E be the set of points of Ω where w = 0 and dw 0. Assume that E is of complex codimension in Ω. Let I dw be the ideal sheaf such that its stal at Q Ω consists of all of holomorphic functions f on Ω such that f dz dz n dw < {w=0} U E for some open neighborhood U of Q in Ω. We call the ideal sheaf I dw the normal ideal sheaf of the hypersurface defined by w. Remar.. It is by a considerable abuse of language and for lac of a better name that the ideal sheaf I dw is called the normal ideal sheaf of the hypersurface defined by w. When Y = Ω E and Z = Y {w = 0}

12 with the continuation of the above notations, the normal bundle N Z Y is naturally isomorphic to Hom K Y,K Z through the pairing map ω,g ω g dw, where ω is a local n -form on Z and g is a local holomorphic function on Z so that g dw defines a local section of the dual N Z Y of N Z Y and ω g dw is a local n-form on Y. Since K Y has a global basis dz dz n, the natural isomorphism between N Z Y and Hom K Y,K Z identifies a local section of N Z Y as a local section of K Z. The ideal sheaf I dw on Ω is the subsheaf of all holomorphic function germs g defined on U such that ω is a square integrable n -forms on Z U when ω g dw = dz dz n. The designation of I dw as the normal ideal sheaf of w = 0 is motivated by the fact that the equation ω g dw = dz dz n maes the sheaf of holomorphic function germs g naturally isomorphic with O Ω NZ Y on Ω E when the transition functions of the two underlying bundles on Ω E are compared. Lemma.3. Let w = z z z ν + + zνn n in C n, where ν j for j n. Then normal ideal sheaf I dw on C n of the hypersurface defined by w is equal to the ideal sheaf on C n generated by z z l z l+ z z ν + + zνn n l. Proof. For l let E l = {z l = 0} j l {z j = 0}. Consider a holomorphic n-form ω := f z,,z n dz dz n, where f z,,z n is a local holomorphic function germ on C n at 0. Let n ω dw = g j z,,z n dz dz j dz j+ dz n. j= Then on E l we have = f z,,z n dz dz n n g j dz dz j dz j+ dz n dw j= = g l dz dz l dz l+ dz n z z l z l+ z z ν + + zνn n dz l and g l = l n f z z l z l+ z z ν + +. zνn n

13 For ω dw to be locally square integrable on E l at the origin, one must have the divisibility of f El by z z l z l+ z z ν + + zνn n on E l at the origin as holomorphic function germs. For this to hold for every l, it is necessary and sufficient f belongs to the ideal generated by Q.E.D. z z l z l+ z z ν + + zνn n l. Remar.4. The normal ideal sheaf I dw can also be described by the finiteness of the limit of integrals of tubular neighborhoods. Let Ω be an open subset of C n and w be a holomorphic function on Ω. Let E be the set of points of Ω where w = 0 and dw 0. Assume that E is of complex codimension in Ω. If f O Ω Q is a holomorphic function germ on Ω at Q Ω which is defined on some open neighborhood U of Q in Ω, then f belongs to the stal of I dw at Q if and only if lim ε 0 + ε U { w <ε} for some open neighborhood U of Q in U. f dz dz n < According to Lemma.3, even in the case n = and w = z z where the singularity of the zero-set of w in C is a singularity of normal crossing at the origin, the ideal sheaf I dw at the origin is different from the unit ideal and is equal to the maximum ideal of C. When one uses the description of I dw by the finiteness of the limit of integrals of tubular neighborhoods, one can understand why I dw is different from the unit ideal at the origin by computing the limiting behavior of the volume V ε of the regular part of { z z < ε, z <, z < } as a function of ε > 0 when ε 0. Let η j = z j for j =,. Then max ε, V ε = π η ε dη dη = π max, dη η =0 η =0 η =0 η = πε ε + π dη = πε + log ε. η =ε η Hence lim ε 0 + ε V ε = lim πε + log ε =. ε 0 + ε 3

14 4 Lemma.5. Let Ω be an open subset of C n and w be a holomorphic function on Ω. Let E be the set of points of Ω where w = 0 and dw 0. Assume that E is of complex codimension in Ω. Then the normal ideal sheaf I dw of the hypersurface defined by w is coherent on Ω. Proof. Let π : Y Ω be a resolution of the embedded singularities of {w = 0} such that i π Y π E : Y π E Ω E is a biholomorphism, ii the pullbac of the ideal O Ω w to Y is the ideal of a divisor j J a jv j, where a j is a positive integer and {V j } j J is a collection of nonsingular hypersurfaces in Y in normal crossing, and iii K Y π K Ω = j J b jv j, where b j is a nonnegative integer. Let J 0 be the subset of J such that j J 0 if and only if π maps V j biholomorphically onto a branch of the zero-set of w. Then a j = and b j = 0 for j J 0. Let c j = max 0,a j b j for j J. Then c j = for j J 0. Let ζ j be the local defining function of V j for j J. Let J be the ideal sheaf on Y generated by the local functions ζ j c j for l J 0. j J,j l By the argument in the proof of Lemma.3 the ideal sheaf I dw is equal to zeroth direct image of J under π and hence is coherent. Q.E.D. Remar.6. The normal ideal sheaf I dw in some cases is the unit ideal at points where dw vanishes. The Fermat hypersurface cone w = z + + zn for different values of and n is a good example for illustration. We blow up the origin of C n to get π : Y C n with exceptional fiber E = π 0. Then E is biholomorphic to P n and Y is biholomorphic to the total space of the line bundle O Pn over E = P n. Let θ : Y E be the bundle map O Pn P n. Let V be the Fermat hypersurface in P n consisting of all points whose homogeneous coordinates [z,,z n ] satisfy z + + zn = 0. Let Ṽ = θ V. Then the divisor of π w is equal to E + Ṽ. The two hypersurfaces E and Ṽ of Y intersect normally and K Y = n E. Let J be the ideal sheaf on Y for the divisor max 0, n E. From the argument in the proof of Lemma.3, we conclude that the ideal sheaf I dw is equal to zeroth direct image of J under π. So the ideal sheaf I dw is the unit ideal if and only if n. This example also shows that, when n, the holomorphic n -form dz dz n dw on the regular part of the affine hypersurface

15 z + + z n = 0 is not square integrable on the regular part of { z + + z n = 0 } U for any open neighborhood U of the origin in C n, even though the affine hypersurface z + + z n = 0 is normal when n Extension Theorem of Ohsawa-Taegoshi Type. Ohsawa- Taegoshi [OT87] and Ohsawa [O88] developed the theory of extension of holomorphic L sections from a submanifold to an ambient manifold with respect to a weight e ϕ with ϕ plurisubharmonic on the ambient manifold. The form of such an extension theorem which we need is the following. Theorem3. Extension Theorem of Ohsawa-Taegoshi Type. Let Y be a complex manifold of complex dimension n. Let w be a bounded holomorphic function on Y with nonsingular zero-set Z so that dw is nonzero at any point of Z. Let L be a holomorphic line bundle over Y with a possibly singular metric e κ whose curvature current is semipositive. Assume that there exists a hypersurface V in Y such that V Z is a subvariety of codimension at least in Z and Y V is the union of a sequence of Stein subdomains Ω ν of smooth boundary and Ω ν is relatively compact in Ω ν+. If f is an L-valued holomorphic n -form on Z with f e κ <, Z then fdw can be extended to an L-valued holomorphic n-form F on Y such that F e κ 8π e + sup w f e κ. e Y Z Y Remar3.. Theorem3. is given and proved in [S0, p.4, Theorem 3.]. In its proof there are some misprints on page 46, lines 7 and 8 from the bottom, in [S0] where the formula should correctly read u,g ε Ω,ψ = u,g ε e ψ Ω u, f dw ε Ω w ε d w e ψ Induction Argument and Extension of Twisted Sections. We give now the induction argument for the extension of

16 6 twisted pluricanonical sections from a fiber to the family which is used in the proof of Theorem0.6. Proposition4.. Let π : X be a holomorphic family of compact complex projective algebraic manifolds. Let m be an integer. Let E be a holomorphic line bundle over X with a possibly singular metric e ϕ E with semi-positive curvature current on X. Let e ϕp be a metric for p K X0 + E over X 0 with semi-positive curvature current on X 0 for 0 p < m. Let A be a holomorphic line bundle on X which is sufficiently ample for the global generation of multiplier ideal sheaves on X. Assume that i I ϕp I ϕp on X 0 for 0 < p < m, and ii I ϕe X0 agrees with I ϕ0 on X 0 where I ϕe X0 means the multiplier ideal sheaf on X 0 for the metric e ϕ X 0 of E X 0 on X 0. Let f be an element of Γ X 0,mK X0 + E + A which locally on X 0 belongs to the multiplier ideal sheaf I ϕm. Then f can be extended to an element f of Γ X,mK X + E + A. Proof. By the assumption on A, we can apply Proposition. to obtain the following open cover for X 0. We cover X 0 by a finite number of open subsets U λ λ Λ so that, for some nowhere zero element ξ λ Γ U λ, K X0, we can write ξ λ f U λ = N m = b m,λ s m, where b m,λ is a holomorphic function on U λ and s m Γ X 0, I ϕm m K X0 + E + A for N m. We can further assume that the open subsets U λ are chosen small enough so that, for p m, inductively we can write where b p,λ j, ξ λ s p+ j U λ = N p = b p,λ j, sp, is a holomorphic function on U λ and s p Γ X 0, I ϕp p K X0 + E + A for N p. We are going to verify the following claim by induction on p < m.

17 Claim4... can be extended to for j N p and p < m. s p j Γ X 0,pK X0 + E + A s p j Γ X,pK X + E + A The case p = follows from assumption on the agreement between I ϕe X 0 and I ϕ0 on X 0 and Theorem3.. Now we go from Step p to Step p +. Let N p χ p = log s p = so that e χp is a metric of p K X + E + A. Since s p j X 0 = s p j for p N p by Step p, the germ of s p+ j at any point of X 0 belongs to I χp X0 for j N p+. By Theorem3., can be extended to s p+ j Γ X 0, p + K X0 + E + A s p+ j Γ X, p + K X + E + A, finishing the verification of Step p + of Claim4... Since s m j X 0 = s m j for p N m from Claim4.., the germ of f at any point of X 0 belongs to I χm X0. By Theorem3., can be extended to Q.E.D. f Γ X 0,mK X0 + E + A f Γ X,mK X + E + A. 4. Proof of Theorem0.6. Let m = l. Let E = ll + D and ϕ E = lκ + log s D. Let ϕ p = log s p + m p κ + log s D for p m and ϕ 0 = ϕ E. Then e ϕp is a metric for pk X0 +ll+d = pk X0 + E. We are going to verify the following claim. Claim4... I ϕp I ϕp for p m. 7

18 8 To verify Claim4.., we tae a local holomorphic function g and assume that g belongs locally to I ϕp for some p m so that 4.. g e m p s D s p m κ 0 <. We have to verify that g belongs locally to I ϕp so that g s D s p e m p+ m κ 0 <. Since e m p κ s p is bounded from below by a positive number, it follows from 4.. that the local meromorphic function g s D is L and hence is locally equal to a holomorphic function G. So we need only show that implies G e m p s p m κ 0 < 4..3 G s p e m p+ m κ 0 <. We now tae any 0 < η < and rewrite the left-hand side of 4..3 as = G e p η m p p = κ s p η G s p G e p η m p p e s η κ s p η m p+ κ s η e m p+ p η m p κ p e ηm+ ηp p κ.

19 We now apply Hölder s inequality with q = p and p η q = q = p so q η that + =. We get q q G e p η m p p G e p η m p p κ s p η G e m p κ s p κ s p η p p η p η p s η p η p e ηm+ ηp p κ G s η 9 e ηm+ ηp p η p κ η p sup G η p s e κ η p η p η e lκ. We set η = and get s e κ p η η m s 0 e lκ = e κ p e lκ. We now use our assumption that s e κ p e lκ is integrable for p l. This finishes the verification of Claim4... We now apply Proposition4.. Since clearly s l s D locally belongs to I ϕm, it follows from I ϕm I ϕm that locally s l s D locally belongs to I ϕm. Let so that s A,,,s A,qA Γ X,A s A, X0,,s A,qA X0 Γ X 0,A form a basis of Γ X 0,A over C. By Proposition4. we can extend each to an element for j q A. Let s l s D s A,j Γ X 0,mK X0 + ll + D + A s j Γ X,mK X + ll + D + A χ m = log q A j= s j so that e χm is a metric for mk X + ll + D + A = m + ak X + ll.

20 0 We now use Hölder s inequality. Let h D be a smooth metric for D without any curvature condition. Let h L be a smooth metric for L without any curvature condition. Let h KX0 be a smooth metric for K X0 without any curvature condition. Then dv X0 := h KX0 defines a smooth volume form of X 0. Let l = l so that + =. By l l l Hölder s inequality = X 0 s m0 e X 0 s sd l h l D e s l s D hd e X 0 = X 0 j m+aχ m m+aχ m h a m+a L m0 l m+aχ m h la m+a s l s D s A,j hd e l a m+a κ dv X0 e a m+a κ s D l l L dv X0 X 0 m+aχ m h la m+a L dv X0 j s A,j e sup m+aχ a m h D h la l m+a L X 0 j s A,j which is finite. Hence e sup X 0 m+aχ m+ h KX0 X 0 X 0 s e a m+a κ e l l X 0 m+a κ dv X0 h D s D h la m+a L m0 m+a χ m a X 0 s e j h l D h a e l l m+a L a dv X0 m+a κ dv X0 h D s D h la m+a L l X 0 e l l l a l m+a κ dv X0 h D s D h la m+a L l s l s D s A,j e χ m dv X0 l a l m+a κ dv X0 m0 m+aχ m a m+a κ h KX0 dv X0 <. Thus s can be extended to an element s Γ X, K X + L. This concludes the proof of Theorem0.6. l l

21 5. Extension from Singular Fiber Without Twisting. In this 5 we give the proof of Theorem0.7. We use all the notations in Theorem Choose a subvariety Z of complex codimension in X such that X Z is Stein and Z does not contain any branch of any fiber of π : X and Z X 0 contains the singular set of X 0. We can assume without loss of generality that Z is contained in the zero-set of some holomorphic section s A of some ample line bundle A over X. Let Z 0 = Z X 0. Let π 0 : X0 X 0 be a desingularization of X 0 so that X 0 is a compact complex manifold of complex dimension n and π 0 X0 π 0 Z 0 maps X 0 π 0 Z 0 biholomorphically onto X 0 Z 0. We identify X 0 Z 0 with X 0 π 0 Z 0 through this biholomorphic map π 0 X0 π 0 Z 0 so that X 0 Z 0 is also regarded as an open subset of X 0. Let Z 0 = π 0 Z In general the pullbac Ã0 of A X 0 to X 0 under π 0 is not ample any more on X 0. We have to loo at π 0 : X0 X 0 so that we have a collection of nonsingular hypersurfaces H j j J in normal crossing in X 0 so that Ã0 j J ε jh j is ample on X 0 for a certain collection of sufficiently small positive rational numbers ε j. Moreover, π 0 H j Z 0. We can assume that π 0 div s A j J ε jh j is effective by choosing ε j sufficiently small. We choose a sufficiently large positive integer q such that the line bundle B = q Ã0 j J ε jh j is sufficiently ample for the global generation of multiplier ideal sheaves on X 0. Let h A respectively h B be a smooth metric of A on X respectively B on X 0 with positive curvature. 5.3 Let be a positive integer. Tae f Γ X 0, J / J O X t K X. For the proof of Theorem0.7 we have to extend f to an element of Γ X, K X. The fibers of π X Z : X Z are all manifolds. Hence K X0 Z 0 can be naturally identified with K X X0 Z 0 by the map ω ω dt. This identification extends to a natural identification between p K X0 Z 0 and p K X X0 Z 0 for any positive integer p. Except when there is a specific need, we will mae this natural identification without any explicit mention. By this natural identification we regard f X0 Z 0 as an element of Γ X 0 Z 0, K X0 Z 0. Let f Γ X0 Z 0, K X0

22 be the pullbac of f X0 Z 0 Γ X 0 Z 0, K X0 Z 0 under π 0 X0 Z 0. First we verify that f can be extended to an element of Γ X0, K X 0. For every point Q of Z 0 there exists some open neighborhood U of Q in X such that f X0 U is induced by f f m0 with f j Γ U, J K X for j. Let g j = f j dt for j. Then g j X0 Z 0 U Γ X 0 Z 0 U,K X0 Z 0 is square integrable on X 0 Z 0 U as an n -form. Because of the square integrability of g j X0 Z 0 U on X 0 Z 0 U, the pullbac of g j X0 Z 0 U to X0 Z 0 π 0 U under π 0 X0 Z 0 is square integrable and therefore can be extended to an element of Γ X0 π 0 U,K X0. So the pullbac g of g g m0 X0 Z 0 U to X0 Z 0 π 0 U under π 0 X0 Z 0 can be extended to an element of Γ X0 π 0 U, K X0. Through the natural identification between K X0 Z 0 and K X X0 Z 0, the section f Γ X0 Z 0, K X0 agrees with g Γ X0 Z 0, K X 0 and hence can be extended to an element 5.4 We consider Since j J sqε j H j follows that s π 0 s q A j J s Γ X0, K X 0. s qε j H j Γ X0, K + B X0. is a holomorphic section of the line bundle qã0 B, it s qε j H j j J π 0h A q h B is a well-defined bounded nonnegative smooth function on X 0 and we denote this function by γ. Let C γ be the supremum of γ on X 0.

23 Let ψ = log s and m = l. For p < m let N p be the complex dimension of the subspace of all elements s Γ X0,pK + B X0 such that s e p ψ h B <. X 0 Let Ñ = sup p<m N p. Then Ñ is bounded independently of l. This supremum bound being independent of l is a ey point in this proof. 5.5 Let {U λ } λ Λ be a finite covering of X0 by Stein open subsets such that i for some U λ which contains U λ as a relatively compact subset, the assumptions and conclusions of Theorem. are satisfied when L with metric e ξ is replaced by p K X0 with metric e p ψ for 0 p m and when A is replaced by B, and ii for λ Λ there exist nowhere zero as a relatively com- for some open subset Uλ pact subset. τ λ,a Γ Uλ,A, ξ λ Γ Uλ, K X0 of X 0 which contains U λ 3 Let {ρ λ } λ Λ be a smooth partition of unity subordinate to the covering {U λ } λ Λ of X0. Let C % = s. X 0 Let C be the maximum of C C % max λ Λ sup U ξλ λ e ψ, C C % max λ Λ sup U λ π 0 s q A j J s qε j H j h B, where C is from Theorem.. Claim5.6. There exist i for p m, s p,,s p N p Γ X0,pK + B X0

24 4 ii holomorphic functions b p,λ j, on U λ, for p m, j N p+, N p, and λ Λ, and iii holomorphic functions b m,λ λ Λ such that, for λ Λ, a on U λ, b ξ λ s l π 0 s q A j J ξ λ s p+ j = s qε j H j = N p = on U λ for N m and N m = b p,λ j, sp on U λ for p m and j N p+, c d sup U λ N m sup U λ b m,λ N p b p,λ j, = C C, b m,λ s m for j N p+ and N p and p m, and e s p X 0 for N p and p m. e p ψ h B The proof of Claim5.6 follows immediately from Theorem. and descending induction on p. We have j s qε j H j e p ψ π 0h A q X 0 sp j J s p = j γ e p ψ h B sup γ = C γ. X 0 X By Theorem3. there exists a positive constant C such that, for any holomorphic line bundle E over X with a possibly singular metric h E of semi-positive curvature current and for any element s Γ X 0 Z 0,E + K X0 Z 0

25 5 with there exists X 0 Z 0 s h E <, s Γ X Z,E + K X such that s = s dt on X 0 Z 0 and s h E C X s h E. 0 Z 0 X Z We now verify the following claim. Claim5.8. The element s p j j J can be extended to s qε j H j Γ X 0 Z 0,pK X0 Z 0 + q A X 0 Z s p j Γ X Z,pK X + qa for p < m and j N p such that i X Z s p+ j p max Np s C C C γ for p m and j N p+, ii for j N, and iii X 0 Z s j X Z s l j J sqε j h q A C C γ H j s q A max Nm s m C C γ Λ X 0 Z λ= ρ λ ξ λ Λ X 0 Z λ= ρ λ ξ λ. The proof of Claim5.8 is by induction on p < m and by applying 5.7. With s p from the preceding induction step and using the

26 6 convention max N0 s X 0 Z 0 s p+ j =, from 5.6b we conclude that j J sqε j H j p max Np s C C γ Λ X 0 Z λ= ρ λ ξ λ for 0 p m and j N p+. We apply 5.7 to the line bundle E = p K X + q A with the metric p max Np s. The interpretation of the integrand on the left-hand side of 5.8. is as follows. The section s p+ j j J sqε j H j of p + K X0 Z 0 + q A over X 0 Z 0 is naturally identified with a p K X + q A-valued n -form on X 0 Z 0 so that s p+ j j J sqε j H j p max Np s is a possibly singular volume form of X 0 Z 0. By 5.7 we obtain s p+. The statement 5.8iii follows from 5.6a. The interpretation of the integrand on the left-hand side of 5.8iii is analogous to that of 5.8. and is as follows. The expression max Nm s m defines a metric for m K X + q A. The section s l s qε j H j s q A Γ X 0 Z 0,m K X0 Z 0 + q A j J is naturally identified with an m K X + q A-valued n -form on X 0 Z 0 so that s l j J sqε j H j s q A max Nm s m is a possibly singular volume form of X 0 Z 0. This finishes the verification of Claim5.8

27 By applying 5.7 and using 5.8iii, we conclude that the element s l s qε j H j Γ X 0 Z 0,m K X0 Z 0 + q A j J can be extended to such that s q A X 0 Z ŝ m Γ X Z,m K X + qa ŝ m X Z max Nm0 s m C C C γ Λ X 0 Z λ= ρ λ ξ λ. 5.9 We now prepare to pass to limit as l and will use the concavity of the logarithmic function for the estimate of the limiting process together with the fact from 5.4 that Ñ is bounded independently of l. Let C be the maximum of C C γ and Λ C C ρ λ s q A C γ X 0 Z λ= ξλ τ q. λ,a Let 0 < r 0 < r < r <. Choose a finite number of coordinate charts W λ with coordinates z λ,t = z λ,,z n λ,t for λ ˆΛ such that ii each W λ := { z λ < r,, } z λ < r, t < r is relatively compact in W λ for λ ˆΛ, ii X { t < r } = ˆΛ λ= W λ, iii X { t < r 0 } = ˆΛ λ= W λ, where { z λ W λ = < r 0,, } z λ < r 0, t < r 0 for λ ˆΛ, and iv there exist nowhere zero for λ ˆΛ. n n ˆτ λ,a Γ W λ,a, ˆξ λ Γ W λ, K X 7

28 8 Let dv z λ,t be the Euclidean volume form in the coordinates system z λ,t. Let { W λ z λ = < r,, } z λ < r, t < r. n From the concavity of the logarithmic function we conclude that πr n+ log max ˆτ q p+ λ,a ˆξ λ s p+ j dv z W λ j N λ,t p+ πr n+ log max ˆτ q ˆξ p λ,a λ W λ N sp dv z λ,t p max j Np+ ˆτ q p+ λ,a ˆξ λ s p+ j = πr n+ log W λ max Np ˆτ q ˆξ p λ,a λ sp dv z λ,t log max j Np+ ˆτ q p+ λ,a ˆξ λ s p+ j πr n+ W λ max Np ˆτ q ˆξ p λ,a λ sp dv z λ,t p+ max j Np+ s j log sup W λ πr n+ ˆξλ dvz λ,t W λ p max Np s log Ñ C sup W λ πr n+ ˆξλ dvz λ,t for p m. Liewise πr n+ W λ πr n+ log W λ log log max ˆτ q ˆξ m λ,a j N m Ñ C sup W λ max N m ˆτ q λ ŝm j ˆξ m λ,a λ s m πr n+ ˆξλ dvz λ,t dv z λ,t dv z λ,t

29 Adding up, we get πr n+ W λ πr n+ + m log log max ˆτ q ˆξ m λ,a λ j N ŝm j dv z λ,t m log max ˆτ q ˆξ λ,a λ s dv z W λ N λ,t Ñ C sup W λ πr n+ ˆξλ dvz λ,t. 5.0 By the sub-mean-value property of plurisubharmonic functions sup log ˆτ q ˆξ m λ,a λ ŝm W λ π r r 0 n+ log ˆτ q ˆξ m λ,a λ ŝm dv z,t. λ W λ Choose a positive number C such that log C is no less than n+ r log Ñ C sup r r 0 W λ πr n+ ˆξλ dvz λ,t for every λ Λ and p m. Since Ñ is bounded independently of l, we can choose C to be independent of l. Let Ĉ be defined by log Ĉ = πr n+ log max ˆτ q ˆξ λ,a λ s dv z N λ,t. Then W λ sup sup ˆτ q ˆξ l λ,a λ ˆΛ W λ λ ŝ l Ĉ C l. 5. For λ ˆΛ let χ λ be the function on W λ which is the upper semi-continuous envelope of lim sup log ˆξ l λ ŝ l l l which is log C. From the definition of χ λ, we have 0 ˆξm e χ λ sup X 0 Z W λ for λ ˆΛ. Let e χ = λ s ˆξ λ e χ λ 9 be the metric of KX on X { t < r} so that the square of the pointwise norm of a local section σ

30 30 of K X on W λ is σˆξ λ e χ λ. Let {ˆρλ } λ ˆΛ be a smooth partition of unity on X 0 subordinate to the covering {W λ X 0 } λ ˆΛ of X 0. Then X 0 Z 0 s e m0 m χ 0 C λ Λ X 0 W λ ˆρ λ ˆξ <. λ Here the interpretation of the integrand on the left-hand side is as follows. The section s m0 of K X0 Z 0 over X 0 Z 0 is naturally identified with an K X -valued n -form on X 0 Z 0 so that s e m0 m χ 0 is a possibly singular volume form of X 0 Z 0. with Finally by applying 5.7, we can extend s X0 Z to an element s Γ X { t < r} Z, K X X { t <r} Z s e m0 m χ 0 <. By the removability of codimension-one singularities of L holomorphic functions, we can extend s to an element s Γ X { t < r}, K X. According to the general Stein theory for coherent sheaves on, there exists an element s # of Γ X, K X which induces the same element in Γ X, O X / OX t K X as s. Thus s # Γ K X extends f Γ X 0, J / J O X t K X. This concludes the proof of Theorem Induction Technique on Multiplicity of Multiple Point. In this 6 we combine the method of multiplier ideal sheaves with the induction process of Levine [L83] on the multiplicity of the multiple point of the base over which the sections can be extended. First we formulate the induction process of Levine [L83] in the following lemma. Lemma6. Let π : X be a holomorphic family of compact complex projective algebraic manifolds over with fiber X t t. Let {U i } i be an open cover of X such that each U i is a coordinate chart with coordinates z i,t = z i,,z n i,t. Let z i = g ij z j,t be the coordinate transformation from the coordinate chart U j to the coordinate chart U i. Let E be a holomorphic line bundle over X with transition functions Ξ ij z j,t on U i U j. Let l be a nonnegative integer. Let s be a global holomorphic section of E over the ringed

31 space X 0, O X / OX t l+ such that s is given by a holomorphic function s i z i,t on U i. Let Θ ji = n ν= s i z i ν g ij ν t Ξ ij t Ξ ij s i on U i U j. Then {Θ ji } i,j defines a cocycle for the covering {U i } with coefficients in E whose vanishing implies that there exists some global holomorphic section s of E over the ringed space X 0, O X / OX t l+ whose restriction to X 0, O X / OX t agrees with the restriction of s to X0, O X / OX t. Proof. By assumption on s and s i, we have and s i g ij z j,t,t Ξ ij z j,ts j z j,t mod t l+ 6.. t l+ ζ ji s i g ij z j,t,t Ξ ij z j,ts j z j,t mod t l+ for some ζ ji Γ U i U j,e. If we are able to write t l ζ ji = σ i σ j modulo t l+ on U i U j for some holomorphic function σ i on U i, then the collection {s j tσ j } j would define an element of X 0, O X / OX t l+ which extends s. For fixed z j we differentiate 6.. with respect to t and we get n l + t l s i g ν ij ζ ji + s i ν= z ν i t t Ξ ij t s s j j Ξ ij mod t l+ t n s i g ν ij Ξ ij Ξ z ν i ij t t s si i + t Ξ s j ij mod t l+. t Since ν= ν= { si t Ξ ij is a coboundary, it follows that { n s i g ν ij 6.. z ν i t ν= } s j t i,j Ξ ij t Ξ ij s i is a cocyle. If 6.. is a coboundary, we can write n s i g ν ij Ξ ij Ξ z ν i ij t t s i = τ i τ j mod t l+ } i,j 3

32 3 on U i U j for some holomorphic function τ j on U j and define σ j = τ j + s i l + t on U j to get t l ζ ji = σ i σ j modulo t l+ on U i U j. Q.E.D. Remar6.. In the notations of Theorem6., let R be the stal of the first direct image R π O X E at the origin of and if q is a positive integer chosen large enough such that the map t q R t q R defined by multiplication by t is injective, then any element s Γ X 0,mK X0 which can be extended to a global holomorphic section of L over the ringed space X 0, O X / OX t q+ can be extended to a global holomorphic section of E over X. This statement follows from the following argument. Let F = O X E. The commutative diagram 0 F ψ q+ F F / t q+ F 0 ψ q 0 F ψ F F / t F 0, where ψ is defined by multiplication by t, gives rise to the commutative diagram / R 0 π F R 0 π F t q+ F R π F ψ q / R 0 π F R 0 π F t F R π F ψ q+ ψ R π F R π F. Since the map from t q R to itself defined by multiplication by t is injective, it follows that in the commutative diagram above the map ψ q : R π F R π F maps the ernel of ψ q+ : R π F R π F to zero. This implies that in the commutative diagram above the image of R 0 π F / t q+ F R 0 π F / t q F is contained in the image of R 0 π F R 0 π F / t F. Let S be the ernel of the map from R to itself defined by multiplication by t. The ascending chain of submodules S S + R must stabilizes at some = q so that S q = S q+. Thus the integer q with the property that the map t q R t q R defined by multiplication by t is injective always exist. With such an integer q, Lemma6. gives the condition for the surjectivity of the restriction map Γ X,E Γ X 0,E when it is applied to l = q +. Lemma6.3. Let f be a holomorphic function on an open subset Ω of C n. Let α > be a positive number. Let z,,z n be the coordinates