Math. Ann. 305, 461-491 (1996)
tmalm
9 Springer-Verla81996
The Thullen type extension theorem for holomorphic vector bundles with L2-bounds on curvature Vsevolod V. Shevchishin Fakult~it fiir Mathematik, Ruhr-Universifiit Bochum, Universit~tsstrasse 150, D-44780 Bochum, Germany (e-mail:
[email protected]) Received: 10 February 1995/Revised version: 18 July 1995
Mathematics Subject Classification (1991)." 32L20, 32D10 0 Introduction The purpose of this paper is to prove the Thullen type extension theorem for holomorphic Hermitian vector bundles with L2-bounds on curvature. For a complex manifold X and a closed nowhere dense subset Q c X we consider a holomorphic vector bundle E over X\Q. We say that E extends to X through Q as a bundle (resp. as a sheaf) if there exists a bundle (resp. a coherent analytic sheaf) /~ over X such that the restriction J~lx\Q is holomorphically isomorphic to E. As usual we identify a holomorphic vector bundle with the sheaf of its holomorphic sections. In both cases we simply say that E extends through Q. If E extends to some sheaf/~, then its double dual J~** is a reflexive sheaf (see e.g. [OSS]) and is also an extension of E. Therefore we will consider only such extensions. If, in addition, rankc E = 1, then E** will be a bundle lOSS]. Thus for line bundles both definitions are equivalent. Our main result is the following Main Theorem. Let X be a complex manijbM and Q a closed subset of X of the .)Corm Q = Q\G where either i) 0 is an analytic set in X, G is an open subset of X having nonempty intersection with every irreducible component of Q of complex codimension 1; or
ii) Q is a connected Lipschitz submanijbM of real codimension 2 and G is an open subset of X having nonempty intersection with Q. Let E be a holomorphic Hermitian vector bundle over XkQ with the curvature F. I f F E L~oc(X), then E extends through Q as a reflexive sheaf E. P lf, in addition, F E Lloc(X) with some p > 3, then P~ is a bundle outside Qo, which is an analytic" set in X with codim C Q0 > p.
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Here the curvature F of a holomorphic Hermitian vector bundle E is the curvature of the uniquely defined connection D in E compatible with the holomorphic and Hermitian structures. The latter means that i) D ~ = 0, i.e. D~ = 0 iff ~ is a (local) holomorphic section of E; ii) d(~,t/)z = (D~,t/)E + (~,Dt/)~. The problem of extending of holomorphic vector bundles has been already considered in various contexts. For example, if Q is an analytic set in X and codim, Q > 3, then any holomorphic vector bundle over X \ Q extends to a reflexive sheaf. This result was originally proved by G. Trautmann [Tra] for the case dim~zX = 3. The book [Siul] contains the proof for the general case as well as other extension theorems for holomorphic vector bundle and coherent analytic sheaves. However, if Q is an analytic set in X of codimension 2 a similar statement is in general false. An example is given by a line bundle over r which is trivial over r x r and over r x r and has the transition function exp (--~1), where z a n d w are the complex coordinates on r One can obtain further extension theorems by imposing additional assumptions on a holomorphic vector bundle, such as an existence of a Hermitian metric with some natural conditions. In particular, B. Shiffman in [Shif] proved that a holomorphic Hermitian line bundle with the positive curvature extends through an analytic set of codimension two. This result was generalized by Y.-T. Siu [Siu2], who established the Thullen type extension theorem for holomorphic Hermitian vector bundles of arbitrary rank with positive curvature. Another natural assumption on the curvature F of a holomorphic Hermitian vector bundle is an LP-integrability condition. We say that F is locally Le-bounded and write F E L~oc(X ) if IFI is an L~-function on X. Here tFI = (tr FA * F) t/2 is computed via the Hermitian metric in the bundle E and some Hermitian metric ds z on X with the Hodge operator *. It is easy to see that this condition does not depend on the choice of ds 2. The condition F E L2o~(X) is often called the local boundedness of the energy. This take its origin from the Yang-Mills theory, where s176 is considered as an energy functional on the space of all unitary connections D in a fixed vector bundle E. The critical points of SavM are called Yang-Mills connections, whereas a bundle with a fixed Yang-Mills connection is called YangMills bundle. Moreover, the twistor transformation (see e.g. [At], [JaTa]) gives one-to-one correspondence between anti-self-dual bundles - a special case of Yang-Mills bundles - over an open subset in S 4 and certain holomorphic Hermitian bundles over the corresponding open subset in F 3. This correspondence preserves the energy, so extension theorems for holomorphic Hermitian vector bundles yield extension theorems for anti-self-dual bundles. The Thullen type extension theorem for holomorphic Hermitian line bundles with L2-bounds on curvature was obtained by Ivashkovich [Ivl]. He also conjectured [Iv2] that a similar result is true for bundles of arbitrary rank. We note also the result of Bando [Ba], who proved the extension theorem for holomorphic Hermitian vector bundles with L2-bounds on curvature through
IIFDIl~2
Thullen type extension theorem
463
subsets with locally finite Hausdorff measure of real codimension 4. This case includes analytic sets of complex codimension 2 and smooth submanifolds of real codimension 4. Our Main Theorem generalizes this result, in particular, it allows to handle the case of singularities having real codimension three. The organization of the paper is as follows. In Sect. 1 we discuss the phenomenon of the Limit Holonomy, introduced in [SS1,2], and show that the Limit Holonomy is an obstruction to extensibility of holomorphic Hermitian bundles. Here we also state the result of [SS 1, 2] on extension of Sobolev connections with a trivial Limit Holonomy. In Sect. 2 we apply the H6rmander's L2-methods to construct solutions of the nonlinear R-equation. This gives an extension theorem for holomorphic Hermitian bundles with L"-integrable curvature over n-dimensional manifolds. To obtain the general case, in Sect. 3 we use the slicing extension techniques for holomorphic objects.
1 The limit holonomy and Sobolev extension of connections Singularities of real codimension two (e.g. smooth submanifolds or analytic hypersurfaces) appear also in the classical extension theory for holomorphic function. However, there is an important distinction. In general, holomorphic functions do not extend through analytic hypersurfaces. On the other hand, every holomorphic function extends through a smooth submanifold of real codimension two if at some point its tangent space is not complex. In the Hermifian bundle case the situation is opposite. In [She2] it is shown that: i) if Q c X is an analytic set, then every holomorphic Hermitian line bundle with locally L2-bounded curvature extends through Q; ii) if Q is a smooth two-codimensional submanifold of X such that TqQ is not complex some for q E Q, then there exists a topologically trivial holomorphic Hermitian line bundle E over X\Q with zero curvature which does not extend holomorphically through Q. Let us consider the last statement in more detail. For simplicity we suppose thatX = I132 and Q = R 2. Let T denote the group {z E IE: N = I } and the corresponding constant sheaf, and let ~ be the sheaf Of pluriharmonic functions. One can include T and ~ in the short exact sequence of sheaves (see e.g. [Shif]) 0~T *~C9" , ~ 0 , where ~ is the natural imbedding and 2 ( f ) = In [f]. Consider the corresponding exact cohomology sequence HO(X\Q,d? *) ~*, HO(x\Q,#~)
~, HI(XX~Q,T) - ~ H1(X\Q,d~*).
Every holomorphic bundle over X = 1122 is trivial, so non-triviality of E over X\Q implies its non-extendability. A holomorphic Hermitian line bundle (E,h) over X\Q with a flat connection D is, up to a connection preserving isomorphism, uniquely determined by a homomorphism from nl(X\Q) ~- Z to T. This homomorphism can be regarded as an element of H I ( X \ Q , T )
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v.v. Shevchishin
Hom(Ir~(X\Q),T) ~ T. Denote by [E]T the corresponding cohomology class. Then E is holomorphically trivial iff e. [E ]T = 0. This is equivalent to the equality [E]T = &f for some f E H~ But every pluriharmonic function f on X \ Q extends to X and can be represented as a real part of some holomorphic function g c (9(X\Q). Consequently [E]x = t~2.(eg) = 0. Thus e, is an imbedding and every nontrivial element of H I ( X \ Q , T ) gives an example of a non-extendible holomorphic line bundle with zero curvature. Note that [E]T is essentially the holonomy of D along any smooth closed curve, which generates 1h(XkQ). This encourages us to consider the holonomy
as an obstruction to extension of" holomorphic bundles. More precisely, the holonomy is an obstruction to extension of connections, e.g. Yang-Mills connections. In this physical context it appeared first in [FHP1,2] and then was systematically studied in [SSI,2]. As result one gets the codimension two removable singularity theorem for (coupled) YangMills fields with the condition of triviality of the Limit Holonomy (see [SS1,2], [She4]). Other applications of this approach are understanding global properties of Yang-Mills fields [SSU], [Smi], as also four-dimensional topology [Kro 1,2]. The case of general bundles and unitary connections brings difficulties of a new type. The first of them is related to the fact that for r = rank E > 2 the Lie group U ( r ) is nonabelian. The parallel displacement along any smooth closed curve ? starting and ending at x is a unitary automorphism Jr,x of the fiber Ex ([KoNo]). However, since one has no preferable basis of Ex, it does not determine an element of U(r). One can associate to Jv,x only a conjugacy class in the group U(r). This class is called the holonomy of the connection along ~. In physical terminology, the conjugacy class is a gauge invariant characteristic of Jr,x. Another difficulty arises when the curvature is non-zero. In this case we cannot consider holonomy as a function on rcl (XkQ). For homotopic but different curves the corresponding holonomies can, in general, be different. Moreover, as in the case of Riemannian connections, the curvature reflects the dependence of the holonomy on the choice of such a curve. The description of connections with two-codimensional singularities, given in [SSI, 2], is appropriate to our purpose. Namely, there is shown that the L2-integrability condition on the curvature implies the following phenomenon. If one shrinks a closed curve around a singular set Q to a (general) point q on Q, corresponding holonomies tends to some limit. This limit is independent of the choice of a point q E Q. Following [SS1,2] we call it the Limit Holonomy. The condition on a singular set, imposed in the Main Theorem, provides that the corresponding Limit Holonomy is trivial. Here we collect the final form of results about the Limit Holonomy, which are used in this paper. Let M is a smooth Riemannian manifold, diml~M = m ;> 4, E a smooth complex vector bundle over M with the fiber Ex c~ C', r = rankc E ~ 2, and D a eormection in E with the curvature F.
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465
Let G denote the group U(r) or its closed connected subgroup, and let fl be the Lie algebra of G. For simplicity, one can restrict itself only to the case G = U(r). We shall always consider elements of G and g as corresponding complex matrices. Denote by [0]6 := {hgh -1: h E G} the conjugacy class of g in G. Let GlAd := {[g]a: g E G) denote the set of conjugacy classes in G, equipped with the natural distance function
dist([gl]G,[g2]G) := min{distr(gl,hgah-l): h E G},
(1.I)
where dista(gbg2) is the invariant distance function in G. One can easily check that dist([gl]c,[g2]a) is essentially the Hausdorff distance between the sets [gl]a and [gz]a. We suppose that E is a G-bundle and D is a G-connection (see [KoNo]). For G = U(r) this simply means that E is equipped with a Hermitian metric ~", 9)e and D is a Hermitian connection. In any case, a G-invariant Hermitian metric in E is defined and D is a Hermitian connection with respect to this metric. Let a = ( a l , . . . , c r r) be a local frame o r E over U C M. Then there exist the matrices of forms A, = {A~} and F , = {E~} such that
D(6 i) = ~]A~o "j
or, in matrix form,
D ( a ) = A, 9 a,
(1.2')
F(a i) = ~-~F~aJ ]
or, in matrix form,
F(a) = F,~ 9 a.
(1.2")
J
One calls Aa (resp. F~) the matrix of the connection D (resp. of the curvature F ) in the frame a. These matrices are related by
F~ = dAo - A,~ A A,r 9
(1.3)
If a is a G-frame (for G = U ( r ) this simply means unitarity of a), then A, and F~, considered as matrix 1- (resp. 2-) forms, are g-valued (Hermitian, if G = U(r)). If ~ = (T1. . . . . r,) is another frame and g is the transition function from a to v, then Az and F~ are obtained as follows: A~ = g . A~ . g - I + d g . g - I
(1.4')
F~ = 0 9 F~ 9 g - i = Ad(g)F.
(1.4")
Now we introduce a wider class of "singular sets", for which the Main Theorem is valid.
Definition. A subset Q of a smooth maniJbld M is called a Lipschitz stratified subset of codimension two, if there exist a sequence of closed subsets Q(i) c M such that i) Q(O) = Q, Q(i+l) c Q(i), NeQ(i) = ~ ; ii) Q(i)kQ(i+l) is a Lipschitzian submaniJbld in MkQ (i+1) of codimension 2 (resp. > 2) i f i = 0 (resp. i ~_ 1).
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Such a stratification {Q(i)} is not unique. So considering a set of this type we always suppose that some stratification is fixed. Then the set Q\Q(t) is called the main stratum o f Q, whereas the connected components of Q\ Q(1) are called the main components of Q. An obvious example of a Lipschitz stratified subset is given by a locally finite union of submanifolds which intersect each other transversely. Another one is an analytic set in a complex manifold. The first main result of [SS1,2] is the existence of the Limit Holonomy. The considered situation is local, and the following notations are used. Q denotes the m - 2-cube {(xi . . . . ,Xm_2): 0 < X i < 1}, A := {re i~ 6 IE: r < l, 0 < O < 2n} is the unit disk, A := A\{0} is a punctured disc, s := Q x A, and s := Q • z~. Proposition 1.1. (Existence of the Limit Holonomy). Let E be a vector G-bundle over {2, and D a G-connection in E with the curvature F, Let ~ttso [Jq.r]c 6 G/Ad, q 6 Q, 0 < r < 1, denotes the holonomy o f D along the circle o f radius r in the disk {q} • A. Suppose F 6 LZ(f2). Then there exists the uniqueIy defined [JIG e G/Ad such that Jbr almost every q 6 Q the limit limr--,O[Jq.r]C exists and equals
:o [:1o. The introduced conjugacy class [J]6 is called the Limit Holonomy of D. We say that the Limit Holonomy is trivial if [J]6 = [1]G, the conjugacy class o f the identity element of G. Since the Limit Holonomy is an obstacle to extension of holomorphic Hermitian bundles, one is interested in conditions, which suffice its triviality. In [She4] it is shown that one can globalize the Proposition 1.1 for bundles, which are defined outside a Lipschitz stratified set Q of codimension two and have locally L2-bounded curvature, and define the Limit Holonomy as a locally constant G/Ad-valued function on a main stratum of Q. In particular, one obtains the following
Proposition 1.2.
(Sufficient condition for triviality of the Limit Holonomy.) Let M be a smooth manijbld, Q a Lipschitz stratified subset of codimension two and let E be a G-bundle over M \ Q equipped with a G-connection D o f the curvature F. Suppose that F 6 L~oo(M) and that E and D extend to an open set U C M, having a nonempty intersection with every main component o f Q. Then Jbr every main component of Q the corresponding Limit Holonomy [ ] ] a is equal to [lie. The second main result of [SS1,2] gives the complete description for a wide class of connections in a neighbourhood of a two-codimensional singularity. Namely, under the hypothesis of the Proposition 1.1 the additional assumption ]lFllt,,:(o) < s(m, G) implies that one can represent D in the form D = D ~ + A ~, where D b is a fiat (i.e. with a zero curvature) G-connection in E with the holonomy [JIG, and A ~ admits a Sobolev-type estimate
IIA ItL 4, Q a Lipschitz stratified subset of codimension two and let E be a G-bundle over M \ Q equipped with a G-connection D of the curvature F. m/2 Suppose that F E Lloo (M) and that the Limit Holonomy o l D is trivial. Then ]br every q E Q there exists a neighbourhood U C M and a G-]?ame tr o r E over U\Q, such that the corresponding connection ]brm A~ of D is Sobolev Ll,m/z-bounded in U. For m = 4, G = SU(2), and Q smooth submanifold of codim R Q = 2 this result was obtained in [SSI, 2], the general case was considered in [She4]. If the curvature is only Le-integrable with 2 < p < m/2, one can obtain a slice-type extension for connection, and then apply it for extension of holomorphic bundles. Formulating corresponding result of [She4] we use the following notations, m > k > 4 are integers, 1" is a cube, ~z: 1'n ~ I m-k is the natural projection on the first m - k coordinates, Q c I m is a Lipschitz stratified subset of codimension two in I m, for y E I m-k we set Iyk := l r - l ( y ) and Qy := Q Iq Iyk. Proposition 1.4. (Slice extension of connections with trivial Limit Holonomy). Let E be a vector G-bundle over Ira\Q, and D a G-connection in E with the curvature F. Suppose that F is Lk/~-inteyrable in I n' and that the Limit Holonomy o l D is trivial. Then ]'or almost all y E I "-k there exists an extension o f the restricted bundle EIryk\Qr with the connection Dl~ky\Qy to a
Sobolev L2,k/2-bundle Ey over Iky with a Sobolev Ll,k/2-connection ff)y. The later means (see [Uhll], [SS2], or [She4]) that ]br any such y E I n-k and every x E Qy there exists a nei9hbourhood U C 1~ of x and a frame a of Ellyk\ay over U\Qy such that the corresponding connection form A~ is L l,k/2bounded in U. Considering in Sect. 2 solutions of R-type equations we shall use some elliptic regularity theory. The result we apply is the following Lemma 1.5. Let I2 be a bounded domain in IR'n, m ~ 3, K a compact subset o f O, 9 : C~176 k) --* C~176163 I) an elliptic differential operator of de qree one with smooth coefficients, and A E Lm(O, Mat( l x k, lR )). Suppose u E L2(fJ, IRp:) satisfies the equation ~ u + Au = f
(1.6)
V.V. Shevchishin
468
with f ~ LP(~-2,~.I), 2 < p < m. Then u is L~oP-junction in f2 with the interior estimate
[lUlIL,.p(K) < C(f2, K, ~ , A, p) 9 (l[ullL2ca~ + II/IIL,, C1 = CI(~2,K,~, p) such that if [IA[[L,,(a) < el, then
IlullL,.,,tx> < c, 9 ([lutlLz~a} + [IfllL,,~a)).
(1.7) 0 and
(1.8)
Proof Consider the second order differential operator ~ * ~ : Cm(I2, IRk) --+ C~176 F,k), where 9*: C~176 F,.t) ~ C~ IRk) denotes the adjoint to operator. The standard theory of elliptic PDE (see e.g. [Mo] or [GiTr]) can be applied to show that @*~: Llo'q(o) ---+L-l,q(o) is an isomorphism for any q, 1 < q < oo. On the other hand, for v E Llo'q and 1 < q < m, by the Sobolev and H61der inequalities we have
II~.*(Av)llL-,,qr
2n, then g E L~oc(B) and E Lion(B), m particular, both ~ and g -I are continuous;
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V.V. Shevchishin
iv) a vector(matrix)-valued Jknction f E L2o~(B) is a solution of the equation (2.2) iff f = h . g Jot some holomorphic vector(matrix)-valued Junction h inB. Suppose additionally that G C Gl(r, C) is a closed complex subgroup with the Lie algebra g C Mat(r, r and T is a g-valued (0, 1)-form Then such a Jimction g can be chosen G-valued provided either T E L~oc(B) for some p > 2n or G is reductive Lie group.
Remark. As it was remarked to the author by A. Huckleberry, the last statement of the theorem uses only an existence of a holomorphic imbedding of the quotient manifold GI(r,~)/G in some C N. Therefore it is valid for all the closed subgroup G C GI(r,C) for which such an imbedding exists. Closed subgroups with this property are called observable (in GI(r, IE)). This class includes, in particular, all the unipotent groups. The proof is based on the HSrmander's L2-estimates for the R-equation [H8]. For convenience we state the results we need. Let U be a bounded domain in C ~ with C2-smooth boundary, qJ and ~b smooth real-valued functions in U. Define the weighted LZ-spaces of matrixvalued (0,k)-forms, 0 < k < n, ~k := {~ E L~oc(U,A(~
| Mat(r, lE)):
flr ~ . e - * - ~
< oo}
u
i
J
As in [H6], we define linear, unbounded, closed, densely defined operators Sk: E/c__+Ek+l,
k = 0 ..... n - l ,
an element ~ E ~k is in the domain of definition Dom(Sk), iff ~r is defined in the weak sense and belongs to ~k+l, and then we set Sk~ = ~ . The following properties of Sk are proved in [H6]. Proposition 2.1. The operators Sk have the following properties: i) Sk+l o Sk = 0, or equivalently ImSk C KerSk+t; the kernel of So consists of holomorphic Mat(r, C)-valued functions; ii) let p ( x ) = dist(x, aU). Suppose Ile-~p-211L~ < co. Then the space of smooth, .finite in U, Mat(r,r (O,k )-forms is dense in Dom(S~_l)n Dom(Sk) for the graph norm
r ~ (llr
+ IIs~*-1r
+ IlSkr189
;
iii) suppose that k > 1 and that
~,i=t
d2(~+(k+
~zi3~:
l)O )w,r 9 j. > 2 ( l ~ l Z + e ~ ) ~ l w ; I 2, w ~ C ~. -
i=l
(2.3)
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471
Then for every smooth, .finite in U, Mat(r, r
IIS *a-1r
2
(O,k)-jbrm ~ holds
+ IIS,~ll~+, > I1~11~ + f e-'~-(k+l)~' ~ 2
"
=
"
U
i=1
(2.4)
OZi
iv) let Hi and 1t2 be Hilbert spaces, and let S: H1 ~ 1-12 be a linear, closed, densely defined operator. Then Ira(S) = Hz iff jbr some constant C
II~ll~= ~ clls*~lln,, .['or every ~ E Dom(S*), and in this case Ira(S*) = Ker(S) •
[]
Furthermore, we shall use the following small perturbation theorem. Lemma 2.2. Let 1-11, 1-12 be Hilbert spaces and let S: H~ ~ 1-12 be a linear bounded surjective operator. Then there exists a constant ~ which depends on S and has the Jbllowing property: i) every linear bounded operator S: Fit ~ H2 with IIs- sII = 3 and b > O. Set q~ := b(1 - Iz12)l-~' and ~ := slog 1 _ ]zt2. 12
Then Jbr any f E Lion(B) one has the inequality n--1
= 3,
2
1
on- 1
1
- + - = 1, (1 - 0) + , and O(s + 2) = s. Let the functions ~p and p q n q t/J be as above. Then by the HSlder inequality one has
(f(e-~-(k+l)r
~ 1
: (f (e-q~-kr
(1-O)q •
(e-~-kff(1
_
,z[2)'~'+2,~[2)Oq) ~
1--0 __
for all q < oo and p < 1 . Consequently ~9 = - g " T belongs to L~oP(B) ~ C~ for some 2n < / 3 < p. This implies the inequality (2.10). To show that det g is a holomorphic function in B(0, 89 we consider T as matrix of (0, l)-forms. Let T[ denote the corresponding entries and let gi denote the ith r o w of g. Then one can see that f
Odetg =~(gl A...
Agr) = ~ , g l A . " A-~gi A ' " Agr i=1
i,j=l F
= - - ~ g l A--" Agi2"[ A " " Agr i=l
= -tr(T)
9 g = 0.
(2.16)
Now we consider the case when T is only LZ~-bounded. Fix real numbers "'" > /96 = 3 and denote B i : = B(O, pi). Let s = 2 and let be a smooth real function with supp(ct) C BI, such that c~[B2 =- 1. As above, define the unbounded linear operators ,~k : S t' ~ Zt,+l, $1,(~) := ~ + ~ A c~T with the domains of definition 1 > P l > P2 >
Dom(Sk) := {r C ~k: ~ + ~Ac~T E ~ k + l } . Let S~ be the adjoint operators and let S : Y - - - E ~ be the bounded linear operator such that
g(r162
9
+ Si ~*~2,$2r ^
+ $3 ^* ~4,...) 9
Using the Hrlder inequality and boundedness of the functions cp and ff in Bl we obtain
II(S-s)~]l~o~ ~ c , 11131" ITIIIL=(B,)~ C" 11TIIL~"(B)'II~IIL~(~,>
v.v. Shevchishin
478
Due to the ellipticity of S and the Sobolev inequality we have
and consequently
U(~,- S)~II~od ~
c. 1t TIIL=~ k such that g is locally trivial outside Qo. Remark. The statement of the theorem, concerning the codimension of Q0, is sharp. This is shown in the following example. Proposition 3.3. Let B be a unit ball in IF.", n > 3, and let Q = {0}. Define E to be the co-kernel of the injective sheaf' homomorphism r #) ~ 9 n,
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V.V. Shevchishin
r = ( z l f , . . . , Z n f ) . Then E is a bundle over X \ Q , which extends ,as a sheaf (pn/q~(~)) through Q, but does not extend as a bundle. Furthermore, there exists a Hermitian metric h on E such that the curvature F is LP-bounded for any p < n. Proof Since any bundle over B is trivial, E would also be a trivial bundle of rank n - 1. The group Extl(B\{0}; 0 n-l, d~l ) is trivial too [GrHa], so there would exist a splitting homomorphism ~b: tPn --~ ~91 such that ~b o cp = 1. But, due to the choice of tp, this is impossible. For 0 < t < 1 let us consider the map ft: B ~ B with ft(z) := tz, and its lift Ft to the trivial holomorphic vector bundle of rank n, Ft(z,w):= (tz, tw) for z E B and w E C n. It is obvious that Ft defines a holomorphic bundle isomorphism ft*E ~ E and that there exists a Herrnitian metric h in E, such that ft*h = h. Hence ft*F = F for ,the corresponding curvature, which implies IF[ -< Ct -2. Therefore the .curvature F is LP-bounded for any p < n. [] Proof of the Theorem 3.1 a) First we consider the special case when X is the standard polydisc '4n and Q = An- 1 • {0}. Then E is holomorphically trivial. Fix a holomorphic frame of the bundle E over An-l • ~. Let I be an
n- k
-index. Then for almost all w E An-k the restriction of
the b u n d l e E o n ,I:,w k-I x z] has locally Lk-bounded curvature and trivial Limit Holonomy. For such w E A n-k let the holomorphic vector bundle EI,~ be the extension of El~k,;~.,xg defined by Corollary 2.5, and let rlI,w be a holomorphic frame ofEt, w over Ak-I r,w • A. Then tlt, w is locally LP-bounded in Ak-1 t,~ x A for all p < cx~ and for almost all z E '4k-~ r,w one has the following properties: i) the restriction ql, wl{z}xA is locally L r bounded in {z} • '4 for all p
