aa r X i v : . [ m a t h . C V ] A ug LELONG NUMBERS AND VECTOR BUNDLES.
BO BERNDTSSONA
BSTRACT . We study Lelong numbers and integrability indices for S -invariant singular metricson vector bundles over the disk. Dedicated to the memory of G. M. Henkin.
1. I
NTRODUCTION .The notion of a singular metric on holomorphic line bundles has proved to be very useful incomplex analysis and algebraic geometry. The analogous notion for holomorphic vector bundles,which appears in[7], [5], [17] and [15], see also [11] for a clear discussion, has been much lessstudied. In this paper we will focus on a very particular situation: Vector bundles over the unitdisk, ∆ , with metrics that satisfy a condition of S -invariance. Although this is a very specialsituation we will show in two examples that such metrics occur naturally and can be useful inapplications. In both these examples the principle idea is that one embeds a certain problem, orestimate, into a scale of problems depending on a real parameter t . Thinking of t as the real partof a complex variable ζ , the norms one wants to study can be seen as defining a hermitian metricon a (trivial) vector bundle of positive or negative curvature over a half plane, or equivalently, viaa logarithmic map, the disk. Of particular interest is the behaviour of the norms as the parametertends to infinity, or the origin in the disk picture, where a singularity of the metric appears.As a bundle, E , over the disk is necessarily trivial, we can write E = V × ∆ , where V is an n -dimensional complex vector space. We fix one such global trivialization, and it is should bestressed that our later discussion depends on the choice of trivialization. A singular metric on E is then basically a measurable map z → h· , ·i z , where h· , ·i z is an hermitian form on the fiber E z = V . We want to allow that our quadratic forms are degenerate and also that they may attainthe value + ∞ . To avoid a detailed discussion of what this means, we will here use the followingdefinition of a singular metric. Definition:
A singular metric on the holomorphic vector bundle E is a map from ∆ to thespace of positive definite quadratic forms on V , which is defined almost everywhere , and satisfiesthe condition that for any local holomorphic section u of E , log h u, u i = log k u k is measurableand locally integrable. (cid:3) (In practice, since we will discuss only metrics that have either positive or negative curvature,they will have much stronger regularity properties than just being measurable.)Following the articles cited above we say that h is negatively curved if for any holomorphicmap z → u ( z ) ∈ V , the norm function k u ( z ) k z is subharmonic as a function of z . We refer to[17], for a detailed discussion of this. Here we just stress that the condition is equivalent to the seemingly stronger condition that log k u ( z ) k z is subharmonic. (This follows since, if k u ( z ) k z is subharmonic for any choice of u , then k ue p ( z ) k z = k u k z e p ( z ) is also subharmonic for anyholomorphic function p , and this implies that log k u k z is subharmonic.) We also say that ourmetric is positively curved, if the induced metric on the dual bundle E ∗ = V ∗ × ∆ is negativelycurved. It is a standard fact that for smooth and strictly definite metrics, these notions mean thatthe curvature operators Θ h = ¯ ∂h − ∂h are positive. These definitions extend to bundles over ahigher dimensional base.One important feature of the definitions is that they define negativity and positivity of thecurvature, without defining any sort of curvature form or current. Indeed, as shown by a pertinentexample of [17], the curvature of a singular metric, defined in a way analogous to how it is definedin the smooth case, will in general not be a current with measure coefficients, even if the metrichas negative curvature. This is in stark contrast to the line bundle case ( n = 1 ), where h = e − φ has the curvature current ∂ ¯ ∂φ , which has measure coefficients if h is positively or negativelycurved, i e if φ is subharmonic or superharmonic.We will introduce and study a notion of Lelong number and integrability index for metrics onvector bundles. Recall that for a plurisubharmonic function φ in the ball of C m , the (classical)Lelong number of φ at the origin can be defined as γ φ (0) = lim inf z → φ ( z ) / log | z | . On the other hand, the integrability index, ι φ (0) , is defined as the infimum of all numbers c > such that Z e − φ/c < ∞ . As is well known, when m = 1 , the integrability index equals the Lelong number and thiscommon value is also equal to µ ( { } ) , where µ = ( i/π ) ∂ ¯ ∂φ , i e the ’curvature’ of e − φ at theorigin.If φ is (pluri)subharmonic in D , h = e − φ is a (singular) metric of positive curvature on thetrivial line bundle C × D , whereas e φ is a metric of negative curvature on the same (or actuallydual) bundle, dual to e − φ . The Lelong numbers are defined in terms of the asymptotic behaviourof log k k e φ as we approach zero, while the integrability index is defined in terms of k k e − φ .Similarily, our definition of Lelong numbers for vector bundles applies to metrics of negativecurvature, while the ’integrability indices’ for vector bundles will be defined for metrics of posi-tive curvature. Definition 1.
Let h be a (singular) metric of negative curvature on the vector bundle E = V × ∆ .Then the Lelong number at 0 of h in the direction u ∈ V is defined as γ h ( u,
0) := lim inf z → log k u k z / log | z | = γ log k u k (0) . Later it will be more natural to consider the negative of the Lelong number, so we also put α ( u ) = − γ ( u, (where we have suppressed the dependence on h ). Thus, α ( u ) = lim sup z → log k u k z / log(1 / | z | ) . Since for any norm k u + u k ≤ k u k , k u k ) , we see that for any α , V α := { u ∈ V ; α ( u ) ≤ α } is a linear subspace of V .Since V has finite dimension n , the dimension of the spaces V α can only jump at (at most) n places. In other words, there are numbers α ≤ α ≤ ...α n such that V α ⊆ V α ⊆ ...V α n = V, and if α j ≤ α < α j +1 , then V α = V α j , and V α j has dimension j . We also let α = −∞ and V α = { } .This filtration of V gives rise to a dual filtration of the dual space V ∗ =: F . Letting F α = V ⊥ α be the space of vectors in F that are annihilated by V α , we have F α n ⊆ ...F α ⊆ F α ⊂ F α = F. At this point we remark that if we did have a curvature form of the metric with measurecoefficients, Θ , then the Hermitian operator Θ( { } ) would give us a decomposition of V intoeigenspaces. As we have seen, such a curvature operator does not always exist, but what we getinstead is a filtration of V .The general idea is that while the spaces V α are defined in terms of Lelong numbers, thespaces F α are characterized by (a version of ) the integrability index. We will now try to makethis precise under the additional assumption that our metric h is S -invariant. By this we meanthat if u ∈ V , then k u k z = k u k ze iθ for any θ in the circle S .Before we state our result, we transfer our problem to the right half plane H = { Re ζ > } via the exponential map z = e − ζ . Changing notation, we call E = V × H , and consider metricson E that only depend on t = Re ζ . A metric on the bundle E over H corresponds under theexponential map to a metric over the punctured disk, and in case the metric has negative curvatureit extends to metric of negative curvature over the full disk if and only if the metric over the halfplane is bounded as t tends to infinity. It will be convenient to allow also a slightly more generalsituation. We say that a negatively curved metric over H has moderate growth (at infinity) if forsome constants a and C , and all u in V , k u k t ≤ C ( u ) e at . Equivalently, the metric has moderate growth if for some a , the corresponding metric over thepunctured disk is such that k u k z | z | a extends to a negatively curved metric over the full disk.If u is a vector in V , then log k u k t is now a convex function, and the negative of our Lelongnumbers are α ( u ) = lim t →∞ (1 /t ) log k u k t . Notice that, by convexity, (1 /t )(log k u k t − log k u k ) is increasing, so the limit exists.Our main result is as follows. Theorem 1.1.
Let k · k t be a metric of negative curvature and moderate growth on the bundle E = V × H over H , depending only on t = Re ζ for ζ ∈ H . Let k · k − t be the dual metricon E ∗ = V ∗ × H , and let V α , F α be the filtrations of V and V ∗ = F described above. Let α j ≤ α < α j +1 . Then the following are equivalent for v ∈ V ∗ : a. v ∈ F α j , b. Z ∞ k v k − t e tα dt < ∞ , c. lim sup t →∞ (1 /t ) log k v k − t ≤ − α j +1 . The equivalence between conditions (a) and (b) here says that the spaces F α can be defined interms of the integrability properties of the dual norms. In the next section we will give a proof ofthis somewhat technical looking result, the only non trivial part being the implication from a. toc. After that we will give two examples of how metrics satisfying our very particular assumptionsactually arise in ’practice’.Notice however already now that it follows from Theorem 1.1 that forany v in V ∗ , the set of α such that Z ∞ k v k − t e tα dt < ∞ , is always an open interval. 2. T HE PROOF OF THEOREM E = V × H be a trivial vector bundle over the right half plane. We suppose givena metric k u k t defined for u in V and t > , such that the bundle metric k u k Re ζ has negativecurvature. We also assume that the metric has moderate growth at infinity, so that k u k t ≤ C ( u ) e at for some constants C and a . It will be convenient to have the metric defined also for t = 0 . By the curvature assumption, log k u k t is convex, hence also continuous and we assume itextends continuously to t = 0 . Then it follows from the convexity that (1 /t ) log( k u k t / k u k ) is an increasing function. Hence u lies in V α = { u ; α ( u ) ≤ α } if and only if k u k t ≤ k u k e tα forall t ≥ .At several occasions we will have use for the following lemma (see [8], [1], [13]) Lemma 2.1.
Let E be a vector bundle over a bounded domain D in C (or C n ). Let h and h betwo metrics on E that extend continuously to the closure of D . Assume that h has semipositivecurvature and h has seminegative curvature and that h ≥ h over ∂D . Then h ≥ h in D . Take t > . By the spectral theorem we can find a basis e j = e j ( t ) which is orthonormal for t = 0 and diagonalizes k · k t . Hence there are numbers λ j = λ j ( t ) , such that if u = P c j e j then k u k = X | c j | , and k u k t = X | c j | e tλ j ( t ) . Reordering, we assume that λ ≤ λ ≤ ...λ n . Given t we now define a new metric(2.1) k u k s,t = X | c j | e sλ j ( t ) for ≤ s ≤ t . This metric is clearly flat and coincides with our original metric for s = 0 and s = t . By the lemma it follows that k u k s,t ≥ k u k s for ≤ s ≤ t . The min-max formula foreigenvalues then gives that λ j ( s ) ≤ λ j ( t ) , so the λ j ( t ) :s are increasing functions of t . Since ourmetric has moderate growth they are also bounded from above and therefore have limits λ j ( ∞ ) .We shall prove that λ j ( ∞ ) = α j .If φ is a subharmonic function in the disk, bounded from above, we can write it as φ = v + g ,where v is harmonic and g = G [∆ φ ] is the Green potential of ∆ φ . Replacing µ = ∆ φ by a Diracmass at 0, µ ( { } ) δ , we get a subharmonic function φ which is larger that φ , harmonic in thepunctured disk, with the same Lelong number at 0 as φ . The next proposition generalizes thisconstruction to the vector valued case. Proposition 2.2.
Let E = V × H have a negatively curved metric h of moderate growth, thatdepends only on t = Re ζ . Then there is a unique negatively curved metric h ∞ ≥ h , such that h ∞ depends only on t , h ∞ = h for t = 0 , which is flat and satisfies (2.2) lim t →∞ (1 /t ) log k u k t, ∞ = lim t →∞ (1 /t ) log k u k t for any u in V . Hence h ∞ defines the same jumping numbers α j and filtration V α as h .Proof. Fix first t > and define a metric h t by k u k s,t which is equal to k u k s if s ≥ t and flat for < s < t . For s < t it is given by (2.1) and we haveseen that k u k s ≤ k u k s,t for all s > .Since h t is given by a max construction, log k u ( z ) k Re z,t is subharmonic for any holomorphicsection u ( z ) of E , so h t is negatively curved. It therefore follows from the lemma again that h t increases with t . Hence h t has a limit, h ∞ . By the subharmonicity criterion, this limit hasseminegative curvature and the dual metric on E ∗ also has seminegative curvature. Hence themetric is in fact flat.If u ∈ V α then k u k t ≤ k u k e tα . Since log k u k s,t is convex and log( k u k e sα ) is linear it followsthat k u k s,t ≤ k u k e sα for s < t . Hence k u k s, ∞ ≤ k u k e sα so lim s →∞ (1 /s ) log k u k s, ∞ ≤ α. Hence the ’negative Lelong numbers’ of k · k s, ∞ are not larger than the negative Lelong numbersof k · k s , which proves the existence part of the theorem since the opposite inequality is evident.To prove the uniqueness part we first note that if we apply the first part of the argument toa metric that is already flat, then k · k s,t = k · k s for s ≤ t , which means that λ j ( s ) = λ j ( t ) is independent of s . Moreover, since the span of { e j ( s ) } j ≤ k is equal to the span of { e j ( t ) } j ≤ k (or can be chosen equal in case some of the eigenvalues are multiple), it follows that the unitarychange of basis from e j ( s ) to e j ( t ) is diagonal. This is because the corresponding unitary matrixis lower triangular, which implies that it is diagonal if it is unitary. It follows that we can choosethe basis e j independent of s , so the norms can be written X | c j | e tλ j for fixed λ j and a fixed basis if k · k t is flat.Now let | u | t be a flat metric satisfying the assumptions of the theorem. Then k u k t, ∞ ≤ | u | t .By the discussion above, there are two fixed bases, orthnormal for k · k , f j and ˜ f j such that if u = P a j f j = P b j ˜ f j , then k u k t, ∞ = X | a j | e tλ j ( ∞ ) , | u | t = X | b j | e t ˜ λ j . The jumping numbers for both these norms are the same since both coincide with the jumpingnumbers for k u k t . Hence λ j ( ∞ ) = ˜ λ j . It also follows that the spans of { f j } j ≤ k and { ˜ f j } j ≤ k arethe same, so by the same argument as above the unitary base change matrix is diagonal. Hencethe norms are identical, which is what we wanted to prove. (cid:3) From the proof above we have that there is one fixed basis such that if u = P a j f j , k u k s, ∞ = X | a j | e sλ j ( ∞ ) . Since for fixed s the norms k · k s,t increase to k · k s, ∞ , the corresponding eigenvalues (with respectto k · k ) converge. Hence λ j ( ∞ ) = lim t →∞ λ j ( t ) . On the other hand, λ j ( ∞ ) are the jumpingnumbers for h ∞ , which are the same as the jumping numpers of h , so lim t →∞ λ j ( t ) = α j .The next lemma is the crucial step in the proof of Theorem 1.1. Lemma 2.3.
Assume that v ∈ V ∗ and that v ∈ F α m , where F α m = V ⊥ α m and α m < α m +1 . Then lim sup t →∞ (1 /t ) log k v k − t ≤ − α m +1 . Proof.
We will carry out the proof assuming that all jumping numbers α j are different, leavingthe mostly notational changes for the general case to the reader.Take t >> and write for u ∈ V , u = P c j e j with respect to an orthonormal basis for k · k as above, so that k u k t = X | c j | e tλ j ( t ) . We identify V with F = V ∗ , by the conjugate linear isomorphism defined by k · k , so that if v ∈ F , v = P v j e j and the dual norm of v is k v k − t = X | v j | e − tλ j ( t ) . We can assume k v k = 1 . We also use the orthonormal basis f j , with respect to which u = P b j f j and k u k t, ∞ = X | b j | e tα j . Write f j = P d jk e k . Then e tα j = k f j k = X | d jk | e tλ k ( t ) . If t is large enough, λ k ( t ) is close to α k . Taking j = 1 , we see that d k is then (exponentially)small if k > , so | d | must be close to 1 (since the change of basis is unitary). We can thereforesubtract suitable multiples of f from all the f j with j > , to achieve d j = 0 . Carrying on inthis way and renormalizing we get a new basis g j of unit vectors, such that g = f and the span [ g , ...g j ] = V α j and g j = X g jk e k , where the matrix ( g jk ) is upper triangular so that g jk = 0 if k < j . We then have that v ⊥ g j for j ≤ m by hypothesis. We claim that for any ǫ > , and any k ,(2.3) | v k | e − tλ k ( t ) ≤ Ce − tλ m +1 e ǫt , if t is large enough. This is clearly true, with C = 1 and ǫ = 0 , if k > m , since λ k ≥ λ m +1 then,and | v k | ≤ . We next argue by induction, assuming the claim holds for all larger indices than k .Since v ⊥ g k ¯ v k g kk = − X j>k ¯ v j g kj . Since g k ∈ V α k and k g k k = 1 we have k g k k t ≤ e tα k , so X | g kj | e tλ j ( t ) ≤ e tα k . If j > k , λ j is close to α j > α k so g kj must be (exponentially) small, and then | g kk | must be closeto 1, since g kj = 0 if j < k . Hence | v k | ≤ C X j>k | v j | e − tλ j ( t ) X j>k | g kj | e tλ j ( t ) ≤ C X j>k | v j | e − tλ j ( t ) e tα k . By the induction hypothesis this is smaller than Ce − tλ m +1 e ǫt e tα k ≤ Ce − tλ m +1 e ǫt e tλ k ( t ) e ǫt , is t is large enough, which proves the claim.From the claim it follows immediately that k v k − t ≤ Ce − tα m +1 e ǫt , so the proposition is proved. (cid:3) We are now ready to prove Theorem 1.1. It is immediately clear that (b) implies that lim inf k v k − t e tα = 0 . Take u in V α j such that k u k = 1 . Then |h v, u i| ≤ k v k − t k u k t ≤ k v k − t e tα j , since u ∈ V α j and k u k = 1 implies k u k t ≤ e tα j . The liminf of the right hand side is zero.Hence v lies in F α j so we have proved (a). This implies (c) by the lemma, which in turn gives(b) again. Hence all the conditions are equivalent and we have proved Theorem 1.1.3. E XAMPLE
1: T
HE GLOBAL CASE OF THE ’ STRONG OPENNESS PROBLEM ’.The original ’openness conjecture’ concerns the local integrability of e − ψ where ψ is plurisub-harmonic and states that the interval of all positive numbers p such that e − pψ is integrable in someneighbourhood of the origin is open. This was proved in [3]. The strong openness conjectureis the same statement for functions | f | e − pψ , where f is holomorphic, and was first proved byGuan-Zhou in [10]. In this section we shall show how a (simpler) global version of strong open-ness follows from Theorem 1.1. The full strong openness would follow from an extension ofTheorem 1.1 to bundles of infinite rank – this is one reason why we think that such an extensionwould be interesting. (See the remark below.)We consider the following setting. X is a compact projective (or only Kähler) manifold and L is a semipositive holomorphic line bundle over X . Let e − φ be a smooth metric of semipositivecurvature on L . Let ω = i∂ ¯ ∂φ and let ψ be a function such that ψ is ω -plurisubharmonicfunction on X . This means that ψ is integrable and i∂ ¯ ∂ψ + ω ≥ . (The constant 2 here will beclear later: it could be replaced by any number greater than 1.)We consider the vector space H ( X, K X + L ) which we think of as the space of holomorphic ( n, -forms on X with values in L . It can be equipped with the L -norms c n Z X v ∧ ¯ ve − φ and c n Z X v ∧ ¯ ve − φ − ψ . ( Here c n = i n is the standard unimodular constant that makes the norms nonnegative.) Proposition 3.1.
Let v be an element of H ( X, K X + L ) . Assume that c n Z X v ∧ ¯ ve − φ − ψ < ∞ . Then there is a number p > such that c n Z X v ∧ ¯ ve − φ − pψ < ∞ . The proof of the proposition is mimicked on the arguments in [3]. The first ingredient is thefollowing calculus lemma which can be proved by direct computation.
Lemma 3.2.
Let x ≤ and < p < . Then Z ∞ e ps e − x + s, ds + 1 /p = C p e − px . We next let W := H ( X, K X + L ) and let F = W × H be the trivial vector bundle over theright half plane with fiber W . We equip W with the norms k v k − s := c n Z X v ∧ ¯ ve − φ − ψ s , where ψ s = max( ψ + s, . We normalize so that ψ ≤ . Since φ + 2 ψ s is plurisubharmonic on X × H ( s = Re ζ ) it follows from the results in [2] that k · k − Re ζ defines a positively curvedmetric on our bundle F . By Lemma 3.2 we have that C P c n Z X v ∧ ¯ ve − φ − pψ = Z ∞ k v k − s e ps ds − /p. By Theorem 1.1, the set of p < such that the right hand side here is finite, is open, whichproves Proposition 3.1.3.1. Local strong openness.
One would perhaps wish for a version of this proof for the localstrong openess. We would then have a negative plurisubharmonic function ψ in the unit ball, B , and v a holomorphic function in the ball, and study the integrability of | v | e − pψ over smallerballs. Approaching this problem in the same way we put W = A ( B ) , the Bergman space ofsquare integrable holomorphic functions in the ball, Z B | v | < ∞ . Given ψ , we define ψ s = max( ψ + s, as before and introduce the norms k v k − s := Z B/ | v | e − ψ s on W . One could then continue in the same way, given an extension of Theorem 1.1 to infinitedimensional spaces. This poses some obvious problems, but there is at least one feature of thisset up that may be helpful.We define V as the dual of W and consider the norms k u k t on V dual to k · k − t . We canthen define the negative Lelong numbers α ( u ) as before, but probably the possible values of α will no longer form a finite set. They will however satisfy a Noetherian property : All decreasingsequences V α j are stationary for j large. This means that if α = α ( u ) for some u in V , then thereis some ǫ > such that V α + ǫ = V α .To see this, we note that all elements u in V with k u k s < ∞ are bounded as functionals on W by the square norm of a function v on B/ . They therefore extend as linear functionals onthe space of functions that are holomorphic only in a neighbourhood of the closure of B/ . If h is holomorphic in a neighbourhood of ¯ B/ we can then define hu by duality. It follows that k hu k s ≤ sup B/ | h |k u k s , so the subspaces V α are stable under such multiplication. Hence thespace of functions holomorphic in a neighbourhood of the closure of B/ that are annihilatedby V α is a module over the ring H ( ¯ B/ , and our claim follows from the Noetherian property ofsuch modules. The statement of Theorem 1.1 still makes sense in the infinite rank case, if we add the as-sumption that the V α :s have the Noetherian property. If Theorem 1.1 holds in this setting, strongopenness can be proved in the same way as we have proved the global case.4. E XAMPLE
2: T HE L - EXTENSION PROBLEM FOR GENERAL IDEALS .First we recall the classical setting of the L -extension problem for domains in C n . We let D be a bounded pseudoconvex domain in C n , and φ a plurisubharmonic function in D . We alsosuppose given a linear complex subspace, M , of C n intersecting D . The next theorem is oneversion of the classical Ohsawa-Takegoshi extension theorem. Theorem 4.1.
There is a constant C , depending only on the diameter of D such that for anyholomorphic function f on M there is a holomorphic function F on D extending f which satisfiesthe estimate Z D | F | e − φ ≤ C Z M | f | e − φ (where on both sides we integrate with respect to Lebesgue measure). Next we sketch a proof of this theorem along the lines of [4]. Let G ( z ) be a plurisubharmonicfunction with logarithmic singularities on M , such that G ≤ in D . If M is defined by the linearequations l j ( z ) = 0 for j = 1 , , ..k we may take G ( z ) = log P | l j | − C where C is a suitableconstant. We next define the subdomains of D , D t = { z ∈ D ; G ( z ) < − t } for t ≥ . Then all the domains D t are pseudoconvex and moreover the domain D := { ( z, ζ ) ∈ D × H ; G ( z ) < − Re ζ } is a pseudoconvex subdomain of D × H . Note that the domains D t are just the vertical slices of D in the sense that D t = { z ; ( z, ζ ) ∈ D} if t = Re ζ .It is well known that any function in H ( M ) ( i e holomorphic on M ) can be extended to D asa holomorphic function. If we let J ( M ) be the ideal of functions in H ( D ) that vanish on M , thismeans that H ( M ) = H ( D ) /J ( M ) , and of course also H ( M ) = H ( D t ) /J ( M ) . This means that we get a scale of norms on H ( M ) as the quotient norms k f k − t = min F Z D t | F | e − φ , where the minimum is taken over all F that extend f . The Ohsawa Takegoshi theorem amountsto an estimate of k f k and the idea (first occuring in [6]) is to study the variation of the norms as t varies.The proof is based on the following lemma, which is a consequence of the main result in [2]. Lemma 4.2.
The norms k f k − Re ζ define a vector bundle metric of non negative curvature on thevector bundle F = H ( M ) × H . To study the norms we look first at the dual norms on the dual bundle E of F . By the lemmathis is a negatively curved metric and we use the following consequence of the lemma. Corollary 4.3.
Let u be an element of the dual space of H ( M ) which has finite norm for k · k .Then u is bounded for all the norms k · k t and if we denote by k u k t the dual norms, the function k ( t ) := log k u k t − kt is (convex and) decreasing.Proof. The convexity of k follows from the discussion in the introduction: Since k u k ζ hasnegative curvature and only depends on Re ζ , log k u k t is convex. In the situation at hand onecan also verify that k ( t ) is bounded from above as t goes to infinity, and therefore the convexityimplies that k is decreasing. (cid:3) Thus we have found that k u k t e − kt is decreasing and it follows that the dual norms k f k − t e kt are increasing. Hence k f k ≤ lim t →∞ k f k t e kt . If φ is smooth and extends to a neighbourhood of ¯ D it is not hard to estimate the limit in the righthand side by an absolute constant times Z M ∩ D | f | e − φ , so we get Theorem 4.1 under these additional assumptions. But, since the constant in the estimateis absolute, the general case follows by approximation.This proof suggests looking at a more general situation, where we consider more generalideals, J , than J ( M ) . This problem has been studied in [16] and recently in [9].( See also themore recent articles [12] and [14] that appeared after this paper was submitted to the JGEA.)The simplest such situation is when D = ∆ is the unit disk in C , and J = ( z n +1 ) . Then H (∆) / ( z n +1 ) is the space of jets of order n of holomorhic functions at the origin and we arriveat the problem to estimate the minimal weighted L -norm of all holomorphic functions f in thedisk with f (0) , f ′ (0) , ...f ( n ) (0) prescribed. In the more general situation, the ideal J has somezero locus M and one wants to extend functions on M together with their jets of different orderswith L -estimates.This seems to be a rather formidable problem, but it is clear that the general lines of theargument described above still apply. We can still find a plurisubharmonic function G withlogarithmic singularities on M as G ( z ) = log X | g j | − C if we assume the ideal J = ( g , ...g m ) is finitely generated. This gives again domains D t anddual vector bundles F and E , of positive and negative curvature respectively. The main difference, as compared to the situation in Theorem 4.1 is that there is no longer onefixed growth order. In the classical situation, we have that k u k t ∼ e kt for essentially all u in the dual space (at least for a dense subspace), but in the more generalcase, different u :s have different growth. In the model case of a ’fat’ point in the disk we have N linearly independent vectors in the dual of H (∆) / ( z n +1 ) , the sequence of derivatives of Diracmeasures at the origin, u j = δ ( j ) o for ≤ j ≤ n . It is easy to verify that k u j k t ∼ e jt , Thus we are precisely in the situation in Theorem 1.1. We have a negatively curved vectorbundle over H , E , and different ’negative Lelong numbers’ α for different vectors in the fiber of E . As in Theorem 1.1, let us denote by V the fiber of E , i e the dual of H ( D ) /J . Admittedly,this is not in general of finite dimension, but if we assume that the zero locus of J consist of onepoint (as in our model case), it has finite dimension. We will therefore now restrict to this case,even though the general set up of the problem still applies, even for bundles of infinite rank.We therefore get a filtration of V , ( V α j ) and a dual filtration ( F α j ) of V ∗ = H ( D ) /J . Itfollows, as in the corollary, that if u ∈ V α j , then k u k t e − tα j is decreasing. In particular, if the jumping numbers are α < ...α n we always have that k u k t e − tα n is decreasing, and hence k v k − t e tα n is increasing for any v in F . Hence we get the estimate k v k − ≤ lim t →∞ k v k − t e tα n . In general however, the right hand side here will be infinite. If it is finite, it is clear that v ∈ F α n − (compare the end of the proof of Theorem 1.1). In the model case in the disk, α j = j and thecondition that v ∈ F α n − means that all derivatives up to order n − of v vanish at the origin. Inthis case, the right hand side of our estimate is indeed finite, and the estimate is sharp.In order to proceed, we next look at V α n − . Then k u k t e − tα n − is decreasing for all u ∈ V α n − . The dual of this space is F (1) := F/F α n − , so k v k − t, e tα n − is increasing for v ∈ F (1) and k v k − , ≤ lim t →∞ k v k − t, e tα n − (where k · k − t, are the quotient norms). In particular k v k − ≤ lim t →∞ k v k − t, e tα n − if v is orthogonal to F α n − for the norm k · k − . Now, F α n − again defines an ideal in thespace of holomorphic functions in D ; p − ( F α n − ) , where p : H ( D ) → H ( D ) /J is the quotientmap. Hence we have in reality just repeated the first part of the argument with J replaced by p − ( F α n − ) , which in the model case means that we have replaced n by n − . Continuing inthis way, we decompose v in F , v = v n − + v n − + ... , where v j ∈ F α j and is orthogonal to F α j +1 , and can estimate k v k = k v n − k + ... by the procedure above.The main drawback with this is that the estimate we obtain depends on the orthogonal decom-position. Let us illustrate this with the model example, for n = 1 . If f is holomorphic in thedisk, the decomposition is f = f + f , where f = f (0) K φ ( z, /K φ (0 , , f = f − f (0) K φ ( z, /Kφ ( z, , with K φ ( z, w ) the Bergman kernel. The estimate we get is then k f k ≤ π ( | f (0) | + (1 / | f ′ (0) | ) e − φ (0) = π ( | f (0) | + (1 / | f ′ (0) − f (0)( ∂ log K φ ( z, z ) /∂z ) | z =0 | ) e − φ (0) . All of this can be compared with the recent work of Demailly, [9], who treats the L -extensionproblem for general ideals in a very general setting on compact manifolds. His work showsthe existence of L -extensions in very general circumstances, but shares the feature with thediscussion above that one gets an explicit estimate only for functions with the maximal vanishingorder, corresponding to the space F α n − here.R EFERENCES [1] B
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