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Documenta Mathematica

Gegr¨ undet 1996 durch die Deutsche Mathematiker-Vereinigung

K

1

(A

R

) −−→

nrAR

ζ (A

R

)

×

ˆδA,1 R

−−→ Cl(A, R )

ρ

 

y

ρ

 

y

ρ

  y K

1

(B

R

) −−→

nrBR

ζ (B

R

)

×

ˆδB,1 R

−−→ Cl(B, R ).

Diagram from

“Tamagawa Numbers for Motives with Coefficients”

see pages 501–570

Band 6

·

2001

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thematischen Gebieten und wird in traditioneller Weise referiert.

Documenta Mathematicaerscheint am World Wide Web unter:

http://www.mathematik.uni-bielefeld.de/documenta

Artikel k¨onnen als TEX-Dateien per E-Mail bei einem der Herausgeber ein- gereicht werden. Hinweise f¨ur die Vorbereitung der Artikel k¨onnen unter der obigen WWW-Adresse gefunden werden.

Documenta Mathematicapublishes research manuscripts out of all mathe- matical fields and is refereed in the traditional manner.

Documenta Mathematicais published on the World Wide Web under:

http://www.mathematik.uni-bielefeld.de/documenta

Manuscripts should be submitted as TEX -files by e-mail to one of the editors.

Hints for manuscript preparation can be found under the above WWW-address.

Gesch¨aftsf¨uhrende Herausgeber / Managing Editors:

Alfred K. Louis, Saarbr¨ucken louis@num.uni-sb.de

Ulf Rehmann (techn.), Bielefeld rehmann@mathematik.uni-bielefeld.de Peter Schneider, M¨unster pschnei@math.uni-muenster.de Herausgeber / Editors:

Don Blasius, Los Angeles blasius@math.ucla.edu Joachim Cuntz, Heidelberg cuntz@math.uni-muenster.de Bernold Fiedler, Berlin (FU) fiedler@math.fu-berlin.de

Friedrich G¨otze, Bielefeld goetze@mathematik.uni-bielefeld.de Wolfgang Hackbusch, Leipzig (MPI) wh@mis.mpg.de

Ursula Hamenst¨adt, Bonn ursula@math.uni-bonn.de Max Karoubi, Paris karoubi@math.jussieu.fr Rainer Kreß, G¨ottingen kress@math.uni-goettingen.de Stephen Lichtenbaum, Providence Stephen Lichtenbaum@brown.edu Alexander S. Merkurjev, Los Angeles merkurev@math.ucla.edu

Anil Nerode, Ithaca anil@math.cornell.edu

Thomas Peternell, Bayreuth Thomas.Peternell@uni-bayreuth.de Wolfgang Soergel, Freiburg soergel@mathematik.uni-freiburg.de G¨unter M. Ziegler, Berlin (TU) ziegler@math.tu-berlin.de

ISSN 1431-0635 (Print), ISSN 1431-0643 (Internet) SPARC

Leading Edge

Documenta Mathematicais a Leading Edge Partner of SPARC, the Scholarly Publishing and Academic Resource Coalition of the As- sociation of Research Libraries (ARL), Washington DC, USA.

Address of Technical Managing Editor: Ulf Rehmann, Fakult¨at f¨ur Mathematik, Universit¨at Bielefeld, Postfach 100131, D-33501 Bielefeld, Copyright °c 2000 for Layout: Ulf Rehmann.

Typesetting in TEX, Printing: Schury Druck & Verlag, 83064 Raubling, Germany.

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Band 6, 2001

Raymond Brummelhuis, Norbert R¨ohrl, Heinz Siedentop Stability of the Relativistic Electron-Positron Field of Atoms in Hartree-Fock Approximation:

Heavy Elements 1–9

Matei Toma

Compact Moduli Spaces of Stable Sheaves

over Non-Algebraic Surfaces 11–29

J¨org Winkelmann

How Frequent Are Discrete Cyclic Subgroups

of Semisimple Lie Groups? 31–37

Manfred Streit and J¨urgen Wolfart

Cyclic Projective Planes and Wada Dessins 39–68 C. Deninger

On theΓ-Factors of Motives II 69–97

R. Skip Garibaldi, Anne Qu´eguiner-Mathieu, Jean-Pierre Tignol Involutions and Trace Forms on Exterior Powers

of a Central Simple Algebra 99–120

Henning Krause

On Neeman’s Well Generated

Triangulated Categories 121–126

J´erˆome Chabert and Siegfried Echterhoff Permanence Properties

of the Baum-Connes Conjecture 127–183

Jan Stevens

Rolling Factors Deformations

and Extensions of Canonical Curves 185–226

Daniel Huybrechts

Products of Harmonic Forms

and Rational Curves 227–239

Mikael Rørdam

Extensions of StableC-Algebras 241–246

Brooks Roberts

Global L-Packets for GSp(2) and Theta Lifts 247–314 Annette Werner

Compactification

of the Bruhat-Tits Building of PGL

by Lattices of Smaller Rank 315–341

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Partition-Dependent Stochastic Measures

andq-Deformed Cumulants 343–384

Oleg H. Izhboldin and Ina Kersten

Excellent Special Orthogonal Groups 385–412

J¨org Winkelmann

Realizing Countable Groups As Automorphism Groups

of Riemann Surfaces 413–417

Rutger Noot

Lifting Galois Representations,

and a Conjecture of Fontaine and Mazur 419–445 P. Schneider, J. Teitelbaum

p-Adic Fourier Theory 447–481

Amnon Neeman

On the Derived Category

of Sheaves on a Manifold 483–488

Gr´egory Berhuy, David B. Leep

Divisible Subgroups of Brauer Groups

and Trace Forms of Central Simple Algebras 489–500 D. Burns and M. Flach

Tamagawa Numbers for Motives with

(Non-Commutative) Coefficients 501–570

J¨urgen Hausen

A Generalization of Mumford’s

Geometric Invariant Theory 571–592

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Stability of the Relativistic Electron-Positron Field of Atoms in Hartree-Fock Approximation:

Heavy Elements

1

Raymond Brummelhuis, Norbert R¨ohrl, Heinz Siedentop

Received: September 15, 2000 Communicated by Alfred K. Louis

Abstract. We show that the modulus of the Coulomb Dirac oper- ator with a sufficiently small coupling constant bounds the modulus of the free Dirac operator from above up to a multiplicative constant depending on the product of the nuclear charge and the electronic charge. This bound sharpens a result of Bach et al [2] and allows to prove the positivity of the relativistic electron-positron field of an atom in Hartree-Fock approximation for all elements occurring in na- ture.

2000 Mathematics Subject Classification: 35Q40, 81Q10

Keywords and Phrases: Dirac operator, stability of matter, QED, generalized Hartree-Fock states

1. Introduction

A complete formulation of quantum electrodynamics has been an elusive topic to this very day. In the absence of a mathematically and physically complete model various approximate models have been studied. A particular model which is of interest in atomic physics and quantum chemistry is the the electron- positron field (see, e.g., Chaix et al [4, 5]). The Hamiltonian of the electron- positron field in the Furry picture is given by

H:=

Z

d3x: Ψ(x)Dg,mΨ(x) : +α 2 Z

d3x Z

d3y: Ψ(x)Ψ(y)Ψ(y)Ψ(x) :

|x−y| ,

1Financial support of the European Union and the Deutsche Forschungsgemein- schaft through the TMR network FMRX-CT 96-0001 and grant SI 348/8-1 is gratefully acknowledged.

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where the normal ordering and the definition of the meaning of electrons and positrons is given by the splitting ofL2(R2)⊗C4into the positive and negative spectral subspaces of the atomic Dirac operator

Dg,m= 1

iα· ∇+mβ− g

|x|.

This model agrees up to the complete normal ordering of the interaction energy and the omission of all magnetic field terms with the standard Hamiltonian as found, e.g., in the textbook of Bjorken and Drell [3, (15.28)]. (Note that we freely use the notation of Thaller [8], Helffer and Siedentop [6], and Bach et al [2].)

From a mathematical point of view the model has been studied in a series of papers [2, 1, 7]. The first paper is of most interest to us. There it is shown that the energyE(ρ) :=ρ(H) is nonnegative, ifρis a generalized Hartree-Fock state provided that the fine structure constant α := e2 is taken to be its physical value 1/137 and the atomic number Z does not exceed 68 (see Bach et al [2, Theorem 2]). This pioneering result is not quite satisfying from a physical point of view, since it does not allow for all occurring elements in nature, in particular not for the heavy elements for which relativistic mechanics ought to be most important. The main result of the present paper is

Theorem 1. The energy E(ρ) is nonnegative in Hartree-Fock statesρ, ifα≤ (4/π)(1−g2)1/2(p

4g2+ 9−4g)/3.

We use g instead of the nuclear number Z = g/α as the parameter for the strength of the Coulomb potential because this is the mathematically more natural choice. For the physical value of α ≈ 1/137 the latter condition is satisfied, if the atomic numberZ does not exceed 117.

Our main technical result to prove Theorem 1 is Lemma 1. Letg∈[0,√

3/2]and

d= (1

3(p

4g2+ 9−4g) m= 0 p1−g2 13(p

4g2+ 9−4g) m >0. Then we have form≥0

|Dg,m| ≥d|D0,0|. (1)

The following graph gives an overview of the dependence of don the coupling constantg

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Our paper is organized as follows: in Section 2 we show how Lemma 1 proves our stability result. Section 3 contains the technical heart of our result. Among other things we will prove Theorem 1 in that section. Eventually, Section 4 contains some additional remarks on the optimality of our result.

2. Positivity of the Energy

As mentioned in the introduction, a first – but non-satisfactory result as far as it concerns heavy elements – is due to Bach et al [2]. Their proof consists basically of three steps:

(i) They show that positivity of the energy E(ρ) in generalized Hartree-Fock states ρis equivalent to showing positivity of the Hartree-Fock functional

EHF :X →R,

EHF(γ) = tr(Dg,mγ) +αD(ργ, ργ)−α 2 Z

dxdy|γ(x, y)|2

|x−y| whereD(f, g) := (1/2)R

R6dxdyf(x)g(y)|x−y|1is the Coulomb scalar prod- uct,X is the set of trace class operatorsγfor which|D0,m|γis also trace class and which fulfills−P≤γ≤P+, andργ(x) :=P4

σ=1γ(x, x). (See [2], Section 3.)

(ii) They show, that the positivity ofEHF follows from the inequality

|Dg,m| ≥d|Dg,0| (Inequality (1)), ifα≤4d/π (see [2], Theorem 2).

(iii) They show this inequality for d= 1−2g implying then the positivity of E(ρ) in Hartree-Fock statesρ, ifα≈1/137 and Z≤68.

From the first two steps, the proof of Theorem 1 follows using Lemma 1. – Step (iii) indicates that it is essential to improve (1) which we shall accomplish in the next section.

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3. Inequality Between Moduli of Dirac Operators

We now start with the main technical task, namely the proof of the key Lemma 1. We will first prove Inequality (1) in the massless case. Then we will roll back the “massive” case to the massless one.

Because there is no easy known way of writing down|Dg,0|explicitly, we prove the stronger inequality

D2g,0≥d2D20,0 (2)

again following Bach et al [2]. However, those authors proceeded just using the triangular inequality. In fact this a severe step. Instead we shall show (2) with the sharp constant d2 = (p

4g2+ 9−4g)2/9 in the massless case. Since the Coulomb Dirac operator is essentially selfadjoint on D:=C0(R3\ {0})⊗C4 forg≤√

3/2, (2) is equivalent to showing

kDg,0fk22−d2kD0,0fk22≥0 for allf ∈ D.

Since the Coulomb Dirac operator – and thus also its square – commutes with the total angular momentum operator, we use a partial wave decomposition.

The Dirac operator Dg,m in channelκequals to hg,m,κ:=

µm−grdrd +κr

d

dr+κr −m−gr

¶ .

It suffices to show (2) for the squares ofhg,0,κandh0,0,κforκ=±1,±2, ....

Notice that hg,0,κ is homogeneous of degree -1 under dilations. Therefore it becomes – up to a shift – a multiplication operator under (unitary) Mellin transform. The unitary Mellin transform M : L2(0,∞) → L2(R), f 7→ f# used here is given by

f#(s) = 1

√2π Z

0

r1/2isf(r)dr.

Unitarity can be seen by considering the isometry ι: L2(0,∞) −→ L2(−∞,∞)

f :r7→f(r) 7→ h:z7→ez/2f(ez) .

The Mellin transform is just the composition of the Fourier transform andι.

We recall the following two rules forf#=M(f) on smooth functions of com- pact support in (0,∞).

(rαf)#(s) =f#(s+iα) µ d

drf

#

(s) = (is+1

2)f#(s−i) These two rules give

Mhg,0,κ

µf+ f

=

µ −g −is−12+κ +is+12+κ −g

¶ µMf+(s−i) Mf(s−i)

¶ .

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If we denote above matrix byhMg,0,κ, we see that (2) is equivalent to (3) (hMg,0,κ)hMg,0,κ−d2(hM0,0,κ)hM0,0,κ=

µ g2+ (1−d2)(s2+ (κ+12)2) −2(κ−is)g

−2(κ+is)g g2+ (1−d2)(s2+ (κ−12)2)

≥0, whereκ=±1,±2, . . .. This is true if and only if the eigenvalues of the matrix on the left hand side of (3) are nonnegative for alls∈Randκ=±1,±2, . . .. The eigenvalues are the solutions of the quadratic polynomial

λ2−2λ¡

g2+(1−d2)(s22+1 4)¢

g2+(1−d2)(s22+1 4)¢2

−(1−d2)2κ2

−4g2(s22).

Hence the smaller one equals

λ1=g2+ (1−d2)(s22+1 4)−p

(1−d2)2κ2+ 4g2(s22).

Here we can already see that d may not exceed 1, and that d = 1 is only possible forg= 0. It the following we therefore restrictdto the interval [0,1).

At first we look at the necessary condition λ1(s= 0)≥0. Now, λ1(s= 0) =g2+ (1−d2)(κ2+1

4)− |κ|p

(1−d2)2+ 4g2 is positive, if|κ|not in between the two numbers

p(1−d2)2+ 4g2±p

(1−d2)2+ 4g2−4(1−d2)(g2+ (1−d2)/4) 2(1−d2)

=

p(1−d2)2+ 4g2±2gd 2(1−d2) . But since we are only interested in integer |κ| ≥1, we want to get the critical interval below 1 (to get the interval above 1 would requireg >√

3/2), i.e., p(1−d2)2+ 4g2+ 2gd

2(1−d2) ≤1, or – equivalently –

p(1−d2)2+ 4g2≤2(1−d2)−2gd.

Since by definition of dwe have g≤(1−d2)/d, the right hand side of above inequality is non-negative. Hence, the above line is equivalent to

4g2+ 8dg−3(1−d2)≤0.

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Solving (4) fordyields d≤1/6¡

−8g+p

16g2+ 36¢

= 1/3³p

4g2+ 9−4g´ . (5)

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We also need the solution forg:

g≤ 1 2(p

3 +d2−2d) =3 2

1−d2

√3 +d2+ 2d. (6)

We now compute the derivative

∂λ1

∂s = 2s[1−d2−2g2¡

(1−d2)2κ2+ 4g2(s221/2

].

The possible extrema ares= 0 and the zeros of [. . .]. We will show below that under condition (5) only s= 0 is an extremum. It is necessarily a minimum, since λ(s = ±∞) = ∞, which concludes the proof. Now we show [. . .] >0.

The expression obviously reaches the smallest value if we choose κ2 = 1 and s= 0. In this case we get the inequality

4g4−(1−d2)2((1−d2)2+ 4g2)<0, which implies

g2<1 +√ 2

2 (1−d2)2. (7)

By the necessary condition (6) we get a sufficient condition for (7) to hold 3

2

1−d2

√3 +d2+ 2d <

s 1 +√

2

2 (1−d2).

Becaused <1 this is equivalent to 3<√

2 q

1 +√ 2(p

3 +d2+ 2d) and the right hand side is bigger than 3 for alld.

Before we proceed to the massive case, we note that we did not loose anything in the above computation, i.e., our value ofd2 is sharp for Inequality (2).

Next, we reduce the massive inequality to the already proven massless one. We have the following relation between the squares of the massive and massless Dirac operator

Dg,m2 =D2g,0+m2−2mβg/|x|.

The above operator is obviously positive, but we will show in the following that we only need a fraction of the massless Dirac to control the mass terms.

To implement this idea, we show

²Dg,02 +m2−2mβg/|x| ≥0, (8)

if and only if²≥g2.

To show (8), we note that from the known value of the least positive eigenvalue of the Coulomb Dirac operator (see, e.g., Thaller [8]) we haveDg,m2 ≥m2(1− g2). Scaling the mass with 1/²and multiplying the equation by²yields

²m2(1−g2)

²2 ≤²D2g,m/²=²Dg,02 +1

²m2−2mβg/|x|.

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It follows that

²D2g,0+m2−2mβg/|x| ≥ µ

1−1/²+1−g2

²

¶ m2=

µ 1−g2

²

¶ m2, showing (8), if²≥g2. This is also necessary, since all inequalities in the proof are sharp forf equal to the ground state eigenfunction.

With (8) the massive inequality follows in a single line:

Dg,m2 = (1−g2)D2g,0+g2D2g,0+m2−2mβg/|x| ≥(1−g2)d2D0,02 . 4. Supplementary Remarks on the Necessity of the Hypothesis

g <√ 3/2

We wish to shed some additional light, on whygin our lemma does not exceed

√3/2. In this section we will show again that for the “squared” inequality Dg,m2 ≥d2D20,m

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we inevitably get d2 ≤ 0 for g = √

3/2. This is because there are elements of the domain of D3/2,m whose derivatives are not square integrable. One example is the eigenfunction of the lowest eigenvalue.

For generalg∈[0,√

3/2] this function is given in channelκ=−1 as ng

µ −g 1−s

rsegmr, where s=p

1−g2 andng is the normalization constant for theL2-norm. Its derivative is square integrable, if and only ifs >1/2 or equivalentlyg <√

3/2.

To make the argument precise, we compute theL2-norm ofh3/2,m,−1Ψβ and h0,m,1Ψβ withβ ∈(1,2],g=p

3/2,s= 1/2,m0>0, and Ψβ:=nβ

µ −g

−(s−1)

rβsegm0r

with the normalization constant nβ. We will see that asβ →1, the first one stays finite and the second one tends to infinity. This only leaves d2 ≤0 for g =√

3/2 in (9). The value of m0 is not relevant; it is just necessary to take m6=m0 ifm= 0 to keep Ψβ square integrable. Now,

hg,m,−1Ψβ =nβ

µ−gm+g2/r+ (s−1)drd + (s−1)/r

−gdrd +g/r+ (s−1)m+ (s−1)g/r

rβsegm0r

=nβ

µg2+ (βs+ 1)(s−1) +r(−gm−(s−1)gm0)

−gβs+g+ (s−1)g+r(g2m0+ (s−1)m)

rβs−1e−gm0r. Writing the above function as

nβ

µf1(β) +r·h1

f2(β) +r·h2

rβ/21egm0r,

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we get the following expression for its norm n2β

Z 0

((f1(β) +r·h1)2+ (f2(β) +r·h2)2)rβ2e2gm0rdr.

The potentially unbounded terms are those involving fi2. Now, f1(β) = (1− β)/4, f2(β) = (1−β)√

3/2, and for a ∈ (−1,0), b > 0 we have the straight forward inequality

Z 0

rae−brdr≤ 1

a+ 1 +eb b . Hence

(1−β)2 Z

0

rβ2e2gm0r dr→0 for β→1.

Proceeding as before we get in the free case h0,m,1Ψβ=nβ

µ−gm+ (s−1)drd + (s−1)/r

−gdrd +g/r+ (s−1)m

rβse−gm0r

=nβ

µ(βs+ 1)(s−1) +r(−gm−(s−1)gm0)

−gβs+g+r(g2m0+ (s−1)m)

rβs1egm0r. But now the terms that depend on r like rβs1 do not vanish for β → 1.

Therefore the L2-norm is unbounded.

References

[1] Volker Bach, Jean-Marie Barbaroux, Bernard Helffer, and Heinz Sieden- top. Stability of matter for the Hartree-Fock functional of the relativistic electron-positron field.Doc. Math., 3:353–364 (electronic), 1998.

[2] Volker Bach, Jean-Marie Barbaroux, Bernard Helffer, and Heinz Siedentop.

On the stability of the relativistic electron-positron field. Commun. Math.

Phys., 201:445–460, 1999.

[3] James D. Bjorken and Sidney D. Drell. Relativistic Quantum Fields. In- ternational Series in Pure and Applied Physics. McGraw-Hill, New York, 1 edition, 1965.

[4] P. Chaix and D. Iracane. From quantum electrodynamics to mean-field theory: I. The Bogoliubov-Dirac-Fock formalism.J. Phys. B., 22(23):3791–

3814, December 1989.

[5] P. Chaix, D. Iracane, and P. L. Lions. From quantum electrodynamics to mean-field theory: II. Variational stability of the vacuum of quantum elec- trodynamics in the mean-field approximation. J. Phys. B., 22(23):3815–

3828, December 1989.

[6] Bernard Helffer and Heinz Siedentop. Form perturbations of the second quantized Dirac field. Math. Phys. Electron. J., 4:Paper 4, 16 pp. (elec- tronic), 1998.

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[7] Dirk Hundertmark, Norbert R¨ohrl, and Heinz Siedentop. The sharp bound on the stability of the relativistic electron-positron field in Hartree-Fock approximation.Commun. Math. Phys., 211(3):629–642, May 2000.

[8] Bernd Thaller. The Dirac Equation. Texts and Monographs in Physics.

Springer-Verlag, Berlin, 1 edition, 1992.

Raymond Brummelhuis Department of Mathematics Universit´e de Reims

51687 Reims France

raymond.brummelhuis@univ-reims.fr

Norbert R¨ohrl Mathematik, LMU Theresienstraße 39 80333 M¨unchen Germany

ngr@rz.mathematik.uni-muenchen.de Heinz Siedentop

Mathematik, LMU Theresienstraße 39 80333 M¨unchen Germany

hkh@rz.mathematik.uni-muenchen.de

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Compact Moduli Spaces of Stable Sheaves over Non-Algebraic Surfaces

Matei Toma

Received: October 18, 2000 Communicated by Thomas Peternell

Abstract. We show that under certain conditions on the topolo- gical invariants, the moduli spaces of stable bundles over polarized non-algebraic surfaces may be compactified by allowing at the border isomorphy classes of stable non-necessarily locally-free sheaves. As a consequence, when the base surface is a primary Kodaira surface, we obtain examples of moduli spaces of stable sheaves which are compact holomorphically symplectic manifolds.

2000 Mathematics Subject Classification: 32C13

1 Introduction

Moduli spaces of stable vector bundles over polarized projective complex sur- faces have been intensively studied. They admit projective compactifications which arise naturally as moduli spaces of semi-stable sheaves and a lot is known on their geometry. Apart from their intrinsic interest, these moduli spaces al- so provided a series of applications, the most spectacular of which being to Donaldson theory.

When one looks at non-algebraic complex surfaces, one still has a notion of stability for holomorphic vector bundles with respect to Gauduchon metrics on the surface and one gets the corresponding moduli spaces as open parts in the moduli spaces of simple sheaves. In order to compactify such a moduli space one may use the Kobayashi-Hitchin correspondence and the Uhlenbeck compactification of the moduli space of Hermite-Einstein connections. But the spaces one obtains in this way have a priori only a real-analytic structure. A different compactification method using isomorphy classes of vector bundles on blown-up surfaces is proposed by Buchdahl in [5] in the case of rank two vector

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bundles or for topological invariants such that no properly semi-stable vector bundles exist.

In this paper we prove that under this last condition one may compactify the moduli space of stable vector bundles by considering the set of isomorphy clas- ses of stable sheaves inside the moduli space of simple sheaves. See Theorem 4.3 for the precise formulation. In this way one gets a complex-analytic structure on the compactification. The idea of the proof is to show that the natural map from this set to the Uhlenbeck compactification of the moduli space of anti- self-dual connections is proper. We have restricted ourselves to the situation of anti-self-dual connections, rather than considering the more general Hermite- Einstein connections, since our main objective was to construct compactificati- ons for moduli spaces of stable vector bundles over non-K¨ahlerian surfaces. (In this case one can always reduce oneself to this situation by a suitable twist). In particular, whenXis a primary Kodaira surface our compactness theorem com- bined with the existence results of [23] and [1] gives rise to moduli spaces which are holomorphically symplectic compact manifolds. Two ingredients are needed in the proof: a smoothness criterion for the moduli space of simple sheaves and a non-disconnecting property of the border of the Uhlenbeck compactification which follows from the gluing techniques of Taubes.

AcknowledgmentsI’d like to thank N. Buchdahl, P. Feehan and H. Spindler for valuable discussions.

2 Preliminaries

LetX be a compact (non-singular) complex surface. By a result of Gauduchon any hermitian metric on X is conformally equivalent to a metric g with ∂∂-¯ closed K¨ahler formω. We call such a metric aGauduchon metricand fix one onX. We shall call the couple (X, g) or (X, ω) apolarized surface andω thepolarization. One has then a notion of stability for torsion-free coherent sheaves.

Definition 2.1 A torsion-free coherent sheafFonX is calledreducibleif it admits a coherent subsheafF0 with 0<rankF0 <rankF, (andirreducible otherwise). A torsion-free sheafFonX is calledstably irreducibleif every torsion-free sheaf F0 with

rank(F0) = rank(F), c1(F0) =c1(F), c2(F0)≤c2(F) is irreducible.

Remark that ifX is algebraic (and thus projective), every torsion-free coherent sheaf F on X is reducible. But by [2] and [22] there exist irreducible rank- two holomorphic vector bundles on any non-algebraic surface. Moreover stably irreducible bundles have been constructed on 2-dimensional tori and on primary Kodaira surfaces in [23], [24] and [1].

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We recall that on a non-algebraic surface thediscriminantof a rankrtorsion- free coherent sheaf which is defined by

∆(F) =1 r

³c2(F)−(r−1)

2r c1(F)2´ is non-negative [2].

LetMst(E, L) denote the moduli space of stable holomorphic structures in a vector bundle E of rank r > 1, determinant L ∈ Pic(X) and second Chern classc∈H4(X,Z)∼=Z. We consider the following condition on (r, c1(L), c):

(*) every semi-stable vector bundleE withrank(E) =r, c1(E) =c1(L) andc2(E)≤cis stable.

Under this condition Buchdahl constructed a compactification ofMst(E, L) in [5]. We shall show that under this same condition one can compactifyMst(E, L) allowing simple coherent sheaves in the border. For simplicity we shall restrict ourselves to the case degωL = 0. When b1(X) is odd we can always reduce ourselves to this case by a suitable twist with a topologically trivial line bundle;

(see the following Remark).

The condition (*) takes a different aspect according to the parity of the first Betti number ofX or equivalently, according to the existence or non-existence of a K¨ahler metric onX.

Remark 2.2 (a) Whenb1(X) is odd (*) is equivalent to: ”every torsion free sheafF on X with rank(F) = r, c1(F) =c1(L) andc2(F) ≤c is irre- ducible”, i.e. (r, c1(L), c) describes the topological invariants of a stably irreducible vector bundle.

(b) Whenb1(X) is even andc1(L) is not a torsion class inH2(X,Zr) one can find a K¨ahler metricg such that (r, c1(L), c) satisfies (*) for allc.

(c) Whenb1(X) is odd or when degL= 0,(*) impliesc <0.

(d) If b2(X) = 0 then there is no torsion-free coherent sheaf on X whose invariants satisfy (*).

ProofIt is clear that the stable irreducibility condition is stronger than (*).

Now if a sheaf F is not irreducible it admits some subsheaf F0 with 0 <

rankF0 <rankF. Whenb1(X) is odd the degree function degω: Pic0(X)−→

R is surjective, so twisting by suitable invertible sheaves L1, L2 ∈ Pic0(X) gives a semi-stable but not stable sheaf (L1⊗ F0)⊕(L2⊗(F/F0)) with the same Chern classes asF. Since by taking double-duals the second Chern class decreases, we get a locally free sheaf

(L1⊗(F0)∨∨)⊕(L2⊗(F/F0)∨∨)

which contradicts (*) for (rank(F), c1(F), c2(F)). This proves (a).

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For (b) it is enough to take a K¨ahler classω such that

ω(r0·c1(L)−r·α)6= 0 for allα∈N S(X)/Tors(N S(X))

and integersr0 with 0< r0 < r. This is possible since the K¨ahler cone is open in H1,1(X).

For (c) just consider (L⊗L1)⊕ O⊗(r−1)X for a suitable L1 ∈Pic0(X) in case b1(X) odd. Finally, suppose b2(X) = 0. Then X admits no K¨ahler structure hence b1(X) is odd. IfF were a coherent sheaf on X whose invariants satisfy (*) we should have

∆(F) =1 r

³c2−(r−1)

2r c1(L)2´

=1 rc2<0 contradicting the non-negativity of the discriminant. ¤ 3 The moduli space of simple sheaves

The existence of a coarse moduli spaceSplX for simple (torsion-free) sheaves over a compact complex space has been proved in [12] ; see also [19]. The resulting complex space is in general non-Hausdorff but points representing stable sheaves with respect to some polarization onX are always separated.

In order to give a better description of the base of the versal deformation of a coherent sheafF we need to compare it to the deformation of its determinant line bundle detF. We first establish

Proposition 3.1 Let X be a nonsingular compact complex surface, (S,0) a complex space germ,F a coherent sheaf onX×S flat overSandq:X×S→X the projection. If the central fiberF0:=F|X×{0}is torsion-free then there exists a locally free resolution ofF overX×S of the form

0−→qG−→E−→ F −→0 whereGis a locally free sheaf on X.

ProofIn [20] it is proven that a resolution ofF0 of the form 0−→G−→E0−→ F0−→0

exists onX withGandE0locally free onX as soon as the rank ofGis large enough and

H2(X,Hom(F0, G)) = 0.

We only have to notice that whenF0 and Gvary in some flat families overS then one can extend the above exact sequence overX×S. We chooseSto be Stein and denote byp:X×S→S the projection.

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From the spectral sequence relating the relative and global Ext-s we deduce the surjectivity of the natural map

Ext1(X×S;F, qG)−→H0(S,Ext1(p;F, qG)).

We can apply the base change theorem for the relative Ext1 sheaf if we know that Ext2(X;F0, G) = 0 (cf. [3] Korollar 1). But in the spectral sequence

Hp(X,Extq(F0, G)) =⇒Extp+q(X;F0, G)

relating the local Ext−sto the global ones, all degree two terms vanish since H2(X;Hom(F0, G)) = 0 by assumption. Thus by base change

Ext1(X;F0, G)∼=Ext1(p;F, qG)0

.mS,0· Ext1(p;F, qG)

and the natural map

Ext1(X×S;F, qG)−→Ext1(X;F0, G) given by restriction is surjective. ¤

LetX, S andF be as above. One can use Proposition 3.1 to define a morphism det : (S,0)−→(Pic(X),detF0)

by associating toF itsdeterminant line bundledetF.

The tangent space at the isomorphy class [F] ∈ SplX of a simple sheaf F is Ext1(X;F,F) since SplX is locally around [F] isomorphic to the base of the versal deformation of F. The space of obstructions to the extension of a deformation of F is Ext2(X;F,F).

In order to state the next theorem which compares the deformations ofF and detF, we have to recall the definition of thetracemaps

trq : Extq(X;F,F)−→Hq(X,OX).

When F is locally free one defines trF : End(F) −→ OX in the usual way by taking local trivializations of F. Suppose now that F has a locally free resolutionF. (See [21] and [10] for more general situations.) Then one defines

trF :Hom(F, F)−→ OX by

trF |Hom(Fi,Fj)=

½ (−1)itrFi , fori=j 0 , fori6=j.

Here we denoted by Hom(F, F) the complex having Homn(F, F) = L

i Hom(Fi, Fi+n) and differential

d(ϕ) =dF◦ϕ−(−1)degϕ·ϕ◦dF

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for local sectionsϕ∈ Homn(F, F). trF becomes a morphism of complexes if we see OX as a complex concentrated in degree zero. Thus trF induces morphisms at hypercohomology level. Since the hypercohomology groups of Hom(F, F) and ofOX are Extq(X;F,F) and Hq(X,OX) respectively, we get our desired maps

trq : Extq(X;F,F)−→Hq(X,OX).

Using tr0 over open sets of X we get a sheaf homomorphism tr : End(F) −→ OX. Let End0(F) be its kernel. If one denotes the kernel of trq : Extq(X;F,F)−→Hq(X,OX) by Extq(X,F,F)0 one gets natural maps Hq(X,End0(F)) −→ Extq(X,F,F)0, which are isomorphisms for F locally free.

This construction generalizes immediately to give trace maps trq : Extq(X;F,F ⊗N)−→Hq(X, N)

for locally free sheavesNonX or for sheavesNsuch thatTorOi X(N,F) vanish fori >0.

The following Lemma is easy.

Lemma 3.2 IfF andGare sheaves onX allowing finite locally free resolutions andu∈Extp(X;F,G), v∈Extq(X;G,F) then

trp+q(u·v) = (−1)p·qtrp+q(v·u).

Theorem 3.3 LetX be a compact complex surface,(S,0)be a germ of a com- plex space andF a coherent sheaf onX×Sflat overSsuch thatF0:=F¯¯¯X

×{0}

is torsion-free. The following hold.

(a) The tangent map ofdet :S→Pic(X)in 0 factorizes as

T0S−→KS Ext1(X;F,F)−→tr1 H1(X,OX) =T[detF0](Pic(X)).

(b) IfT is a zero-dimensional complex space such thatOS,0=OT,0/I for an ideal I of OT,0 with I·mT,0 = 0, then the obstruction ob(F, T) to the extension ofF toX×T is mapped by

tr2CidI : Ext2(X;F0,F0CI)∼= Ext2(X;F0,F0)⊗CI−→

−→H2(X,OX)⊗CI∼= Ext2(X; detF0,(detF0)⊗CI) to the obstruction to the extension ofdetF toX×T which is zero.

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Proof(a) We may suppose thatS is the double point (0,C[²]). We define the Kodaira-Spencer map by means of the Atiyah class (cf. [9]).

For a complex spaceY letp1, p2:Y×Y →Y be the projections and ∆⊂Y×Y the diagonal. Tensoring the exact sequence

0−→ I/I2 −→ OY×Y/I2 −→ O−→0

byp2F forF locally free onY and applyingp1, gives an exact sequence onY 0−→ F ⊗ΩY −→p1,∗(p2F ⊗(OY×Y/I2))−→ F −→0.

The class A(F)∈ Ext1(Y;F,F ⊗ΩY) of this extension is called the Atiyah classofF. WhenFis not locally free but admits a finite locally free resolution Fone gets again a classA(F) in Ext1(Y;F,F ⊗ΩY) seen as first cohomology group of Hom(F, F⊗ΩY).

Consider nowY =X×S withX andS as before,p:Y →S, q:Y →X the projections andF as in the statement of the theorem.

he decomposition ΩX×S =qX⊕pS induces Ext1(X×S;F,F ⊗ΩS×X)∼=

Ext1(X×S;F,F ⊗qX)⊕Ext1(X×S;F,F ⊗pS).

The componentAS(F) ofA(F) lying in Ext1(X×S;F,F ⊗pS) induces the

”tangent vector” at 0 to the deformationF through the isomorphisms Ext1(X×S;F,F ⊗pS)∼= Ext1(X×S;F,F ⊗pmS,0)∼= Ext1(X×S;F,F0)∼= Ext1(X;F0,F0).

Applying nowtr1: Ext1(Y;F,F ⊗ΩY)→H1(Y; ΩY) to the Atiyah classA(F) gives the first Chern class ofF, c1(F) :=tr1(A(F)), (cf. [10], [21]).

It is known that

c1(F) =c1(detF), i.e.tr1(A(F)) =tr1(A(detF)).

Now detF is invertible so

tr1: Ext1(Y,detF,(detF)⊗ΩY))−→H1(Y,ΩY)

is just the canonical isomorphism. Sincetr1 is compatible with the decompo- sition ΩX×S = qX ⊕pS we get tr1(AS(F)) = AS(detF) which proves (a).

(b) In order to simplify notation we drop the index 0 from OS,0,mS,0, OT,0, mT,0 and we use the same symbols OS,mS,OT,mT for the respective pulled- back sheaves through the projectionsX×S→S, X×T →T.

There are two exact sequences of OS-modules:

(1) 0−→mS −→ OS −→C−→0, (2) 0−→I−→mT −→mS −→0.

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(UseI·mT = 0 in order to makemT anOS-module.)

Letj:C→ OSbe theC-vector space injection given by theC-algebra structure ofOS.j induces a splitting of (1). SinceF is flat overSwe get exact sequences overX×S

0−→ F ⊗OSmS −→ F −→ F0−→0

0−→ F ⊗OSI−→ F ⊗OS mT −→ F ⊗OSmS −→0 which remain exact as sequences overOX. Thus we get elements in

Ext1(X;F0,F ⊗OSmS) and Ext1(X;F ⊗OSmS,F ⊗CI) whose Yoneda compo- siteob(F, T) in Ext2(X;F0,F ⊗CI) is represented by the 2-fold exact sequence

0−→ F ⊗OS I−→ F ⊗OS mT −→ F −→ F0−→0

and is the obstruction to extendingF fromX×S toX×T, as is well-known.

Consider now a resolution

0−→qG−→E−→ F −→0

ofF as provided by Proposition 3.1, i.e. withGlocally free onX andElocally free on X×S. Our point is to compareob(F, T) toob(E, T).

Since F is flat overS we get the following commutative diagrams with exact rows and columns by tensoring this resolution with the exact sequences (1) and (2):

0

0

0

0 qG⊗CmS

qG

G0

0

0 E⊗OSmS

E

E0

0

0 F ⊗OSmS

F

F0

0

0 0 0

(1’)

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0

0

0

0 qG⊗CI

qG⊗CmT

qG⊗CmS

0

0 E⊗OS I

E⊗OS mT

E⊗OS mS

0

0 F ⊗OS I

F ⊗OS mT

F ⊗OS mS

0

0 0 0

(2’)

Using the sectionj:C→ OS we get an injective morphism ofOX sheaves G0

idq∗G⊗j

−−−−−→qG⊗CmT −→E⊗OS mT which we calljG.

From (10) we get a short exact sequence overX in the obvious way 0−→(E⊗OS mS)⊕jG(G0)−→E−→ F0−→0 Combining this with the middle row of (20) we get a 2-fold extension

0−→(E⊗OSI)⊕G0−→(E⊗OS mT)⊕G0−→E−→ F0−→0 whose class in Ext2(X;F0,(E⊗OSI)⊗G0) we denote byu.

Letv be the surjectionE→ F and v0:=

µ v⊗idI

0

: (E⊗OS I)⊕G0−→ F ⊗OS I, v00=

µ v0

0

:E0⊕G0−→ F0, theOX-morphisms induced byv.

The commutative diagrams 0 (E⊗OSI)⊕G0

v0

(E⊗OSmT)⊕G0

(v⊗idm0 T)

E

F0

0

0 F ⊗OS I F ⊗OS mT F F0 0

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and

0 (E⊗OSI)⊕G0

(E⊗OS mT)⊕G0 id

E⊕G0

(jGid)

E0⊕G0 v00

0

0 (E⊗OSI)⊕G0 (E⊗mT)⊕G0 E F0 0 show that ob(F, T) =v0·uand

(ob(E, T),0) =u·v00∈Ext2(X;E0⊕G0,(E⊗OS I)⊕G0).

We may restrict ourselves to the situation whenIis generated by one element.

Then we have canonical isomorphisms of OX-modules E0 ∼= E ⊗OS I and F0 ∼=F ⊗OS I. By these one may identifyv0 and v00. Now the Lemma 3.2 on the graded symmetry of the trace map with respect to the Yoneda pairing gives tr2(ob(F, T)) =tr2(ob(E, T)).

ButEis locally free and the assertion (b) of the theorem may be proved for it as in the projective case by a cocycle computation.

Thus tr2(ob(E, T)) = ob(detE) and since det(E) = (detF)⊗q(detG) and q(detG) is trivially extendable, the assertion (b) is true forF as well. ¤ The theorem should be true in a more general context. In fact the proof of (a) is valid for any compact complex manifoldX and flat sheafF overX×S. Our proof of (b) is in a way symmetric to the proof of Mukai in [17] who uses a resolution forF of a special form in the projective case.

NotationFor a compact complex surfaceX and an element Lin Pic(X) we denote bySplX(L) the fiber of the morphism det :SplX →Pic(X) overL.

Corollary 3.4 For a compact complex surfaceXandL∈Pic(X)the tangent space to SplX(L) at an isomorphy class [F] of a simple torsion-free sheaf F with [detF] =L is Ext1(X;F, F)0. When Ext2(X;F, F)0 = 0, SplX(L) and SplX are smooth of dimensions

dim Ext1(X;F, F)0= 2 rank(F)2∆(F)−(rank(F)2−1)χ(OX) and

dim Ext1(X;F, F) = dim Ext1(X;F, F)0+h1(OX) respectively.

We end this paragraph by a remark on the symplectic structure of the moduli spaceSplX whenX is symplectic.

Recall that a complex manifoldM is calledholomorphically symplecticif it admits a global nondegenerate closed holomorphic two-formω. For a surface X, being holomorphically symplectic thus means that the canonical line bundle KX is trivial. For such anX, SplX is smooth and holomorphically symplectic

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as well. The smoothness follows immediately from the above Corollary and a two-form ω is defined at [F] onSplX as the composition:

T[F]SplX×T[F]SplX∼= Ext1(X;F, F)×Ext1(X;F, F)−→

−→Ext2(X;F, F)−−→tr2 H2(X,OX)∼=H2(X, KX)∼=C.

It can be shown exactly as in the algebraic case that ω is closed and nonde- generate onSplX (cf. [17], [9]). Moreover, it is easy to see that the restriction ofω to the fibersSplX(L) of det :SplX →Pic(X) remains nondegenerate, in other words thatSplX(L) are holomorphically symplectic subvarieties ofSplX. 4 The moduli space of ASD connections and the comparison map 4.1 The moduli space of anti-self-dual connections

In this subsection we recall some results about the moduli spaces of anti-self- dual connections in the context we shall need. The reader is referred to [6], [8]

and [14] for a thorough treatment of these questions.

We start with a compact complex surfaceXequipped with a Gauduchon metric gand a differential (complex) vector bundleEwith a hermitian metrichin its fibers. The space of allCunitary connections onEis an affine space modeled onA1(X, End(E, h)) and theC unitary automorphism groupG, also called gauge-group, operates on it. Here End(E, h) is the bundle of skew-hermitian endomorphisms of (E, h). The subset of anti-self-dual connections is invariant under the action of the gauge-group and we denote the corresponding quotient by

MASD=MASD(E).

A unitary connectionA onE is called reducibleifE admits a splitting in two parallel sub-bundles.

We use as in the previous section the determinant map det :MASD(E)−→ MASD(detE)

which associates to A the connection detA in detE. This is a fiber bund- le over MASD(detE) with fibers MASD(E,[a]) where [a] denotes the gau- ge equivalence class of the unitary connection a in detE. We denote by Mst(E) = Mstg(E) the moduli space of stable holomorphic structures in E and byMst(E, L) the fiber of the determinant map det :Mst(E)−→Pic(X) over an element L of Pic(X). Then one has the following formulation of the Kobayashi-Hitchin correspondence.

Theorem 4.1 LetX be a compact complex surface,g a Gauduchon metric on X,E a differentiable vector bundle overX, aan anti-self-dual connection on detE (with respect to g) and L the element in Pic(X) given by ∂¯a on detE.

Then Mst(E, L) is an open part of SplX(L) and the mapping A 7→∂¯A gives

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rise to a real-analytic isomorphism between the moduli spaceMASD,(E,[a])of irreducible anti-self-dual connections which induce[a]ondetEandMst(E, L).

We may also look atMASD(E,[a]) in the following way. We consider all anti- self-dual connections inducing a fixed connectionaon detEand factor by those gauge transformations inGwhich preservea. This is the same as taking gauge transformations of (E, h) which induce a constant multiple of the identity on detE. Since constant multiples of the identity leave each connection invariant, whether on detE or onE, we may as well consider the action of the subgroup ofG inducing the identity on detE. We denote this group bySG, the quotient space byMASD(E, a) and byMASD,∗(E, a) the part consisting of irreducible connections. There is a natural injective map

MASD(E, a)−→ MASD(E,[a])

which associates to anSG-equivalence class of a connectionAitsG-equivalence class. The surjectivity of this map depends on the possibility to lift any unitary gauge transformation of detE to a gauge transformation of E. This possi- bility exists if E has a rank-one differential sub-bundle, in particular when r := rankE >2, since thenE has a trivial sub-bundle of rank r−2. In this case one constructs a lifting by putting in this rank-one component the given automorphism of detE and the identity on the orthogonal complement. A lif- ting also exists for all gauge transformations of (detE,deth) admitting anr-th root. More precisely, denoting the gauge group of (detE,deth) by U(1), it is easy to see that the elements of the subgroupU(1)r:={ur |u∈ U(1)} can be lifted to elements of G. Since the obstruction to takingr-th roots in U(1) lies inH1(X,Zr), as one deduces from the corresponding short exact sequence, we see thatU(1)r has finite index inU(1). From this it is not difficult to infer that MASD(E,[a]) is isomorphic to a topologically disjoint union of finitely many parts of the form MASD(E, ak) with [ak] = [a] for allk.

4.2 The Uhlenbeck compactification

We continue by stating some results we need on the Uhlenbeck compactification of the moduli space of anti-self-dual connections. References for this material are [6] and [8].

Let (X, g) and (E, h) be as in 4.1. For each non-negative integerkwe consider hermitian bundles (Ek, hk) on X with rankEk = rankE =: r,(detEk, dethk)∼= (detE,deth),c2(Ek) =c2(E)−k. Set

U(E) := [

k∈N

(MASD(Ek)×SkX) M¯U(E,[a]) := [

k∈N

(MASD(E−k,[a])×SkX) M¯U(E, a) := [

k∈N

(MASD(E−k, a)×SkX)

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whereSkXis thek-th symmetric power ofX. The elements of these spaces are calledideal connections. The unions are finite since the second Chern class of a hermitian vector bundle admitting an anti-self-dual connection is bounded below (by 12c21).

To an element ([A], Z)∈M¯U(E) one associates a Borel measure µ([A], Z) :=|FA|2+ 8π2δZ

where δZ is the Dirac measure whose mass at a point xofX equals the mul- tiplicity mx(Z) of xin Z. We denote by m(Z) the total multiplicity of Z. A topology for ¯MU(E) is determined by the following neighborhood basis for ([A], Z):

VU,N,²([A], Z) ={([A0], Z0)∈M¯U(E)| µ([A0], Z0)∈U and there is an L23 -isomorphismψ:E−m(Z)|X\N−→E−m(Z0)|X\N

such thatkA−ψ(A0)kL22(X\N)< ²} where ² > 0 and U and N are neighborhoods of µ([A], Z) and supp (δZ) respectively. This topology is first-countable and Hausdorff and induces the usual topology on each MASD(Ek)×SkX. Most importantly, by the weak compactness theorem of Uhlenbeck ¯MU(E) is compact when endowed with this topology,MASD(E) is an open part of ¯MU(E) and its closure ¯MASD(E) inside ¯MU(E) is called the Uhlenbeck compactification of MASD(E).

Analogous statements are valid forMASD(E,[a]) andMASD(E, a).

Using a technique due to Taubes, one can obtain a neighborhood of an ir- reducible ideal connection ([A], Z) in the border of MASD(E, a) by gluing to A ”concentrated” SU(r) anti-self-dual connections over S4. One obtains

”cone bundle neighborhoods” for each such ideal connection ([A], Z) when H2(X,End0(E¯A)) = 0. For the precise statements and the proofs we refer the reader to [6] chapters 7 and 8 and to [8] 3.4. As a consequence of this de- scription and of the connectivity of the moduli spaces of SU(r) anti-self-dual connections overS4(see [15]) we have the following weaker property which will suffice to our needs.

Proposition 4.2 Around an irreducible ideal connection([A], Z)with H2(X,End0(E¯A)) = 0 the border of the Uhlenbeck compactification M¯ASD(E, a) is locally non-disconnecting in M¯ASD(E, a), i.e. the- re exist arbitrarily small neighborhoods V of ([A], Z) in M¯ASD(E, a) with V T

MASD(E, a)connected.

Note that for SU(2) connections a lot more has been proved, [7], [18]. In this case the Uhlenbeck compactification is the completion of the space of anti-self- dual connections with respect to a natural Riemannian metric.

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