On the Donaldson-Uhlenbeck compactification of instanton moduli spaces on class VII surfaces

0.82 MB·

by  Nicholas Buchdahl

ON THE DONALDSON-UHLENBECK COMPACTIFICATION OF
INSTANTON MODULI SPACES ON CLASS VII SURFACES
NICHOLASBUCHDAHL,ANDREITELEMAN,ANDMATEITOMA
7
1
0 Abstract. Westudythefollowingquestion: LetpX,gqbeacompactGaudu-
2 chonsurface,EbeadifferentiablerankrvectorbundleonX andDbeafixed
n holomorphic structure on D :“ detpEq. Does the complex space structure
a onMAaSDpEq˚ inducedbytheKobayashi-Hitchincorrespondenceextendtoa
ASD
J complex space structure on the Donaldson compactification M pEq? Our
a
2 resultsanswerthisquestionindetailforthemodulispacesofSUp2q-instantons
1 withc2“1ongeneral(possiblyunknown)classVIIsurfaces.

]
V
C
.
h
t Contents
a
m
1. Introduction 2
[ 1.1. Themoduliproblemforvectorbundlesoncompactcomplexmanifolds 2

1.2. Extending the complex structure to a compactification of stpEq.
1 MD
The fundamental questions 4
v
9 1.3. The idea of proof of Theorem 1.2 9
3 1.4. The idea of proof of Theorem 1.3 10
3 ASD
2. The topological structure of pEq at the virtual points 11
3 M
2.1. Local models at the reducible virtual points 11
0
ASD
. 2.2. The structure of pEq around the strata of irreducible virtual
1
M
0 points 14
7 3. Singular sheaves with trivial determinant and c “1 15
2
1 3.1. A family of SLp2,Cq-bundles with c2 “0 15
:
v 3.2. Singular torsion-free sheaves with trivial determinant and c2 “1 18
i 4. The map V and its properties 19
X ε
4.1. The embedding V 20
ε
r
a 4.2. Topological properties of the map Vε 24
5. Extending the complex structure 28
5.1. Families of Serre extensions 28
5.2. The proof of Theorem 1.2 30
5.3. The proof of Theorem 1.3 32
6. Appendix 34
6.1. Elementary transformations of sheaves and sheaf deformations 34
6.2. The push-forward of a family under a branched double cover 38
6.3. Extensions and families of ideal sheaves 39
6.4. Compact subsets of the moduli space of simple sheaves 41
6.5. The gluing lemma 43
References 43
1
2 NICHOLASBUCHDAHL,ANDREITELEMAN,ANDMATEITOMA
1. Introduction
1.1. The moduli problem for vector bundles on compact complex man-
ifolds. An old, classical problem in complex geometry concerns the classification
of the holomorphic structures on a fixed differentiable vector bundle on a complex
manifold, up to isomorphism. The moduli problem for holomorphic vector bun-
dles is devoted to studying the corresponding set of isomorphism classes and the
geometricstructures(topologies,complexspacestructures,Hermitianmetrics,etc)
one can naturally put on this set.
Moreprecisely,letX beacompactcomplexmanifoldofdimensionnandE bea
differentiablevectorbundleofrankr onX. Asemi-connectiononE isafirstorder
operatorδ :A0pEqÑA0,1pEqsatisfyingtheB¯-Leibnizrule: δpfsq“B¯fs`fδpsqfor
any f P 8pX,Cq and sPA0pEq. The natural de Rham-type extension A0,ppEqÑ
C
A0,p`1pEq will be denoted by the same symbol. A semi-connection δ is called
integrable if F “ 0, where F P A0,2pEndpEqq is the endomorphism-valued p0,2q-
δ δ
form defined by the composition δ˝δ :A0pEqÑA0,2pEq.
Denote by olpEq the set of holomorphic structures on E, and by 0,1pEq
H A
( 0,1pEqint) the space of (integrable) semi-connections on E. The Newlander-
A
Nirenberg theorem gives a bijection 0,1pEqint Ñ olpEq. We will denote by
A H
the holomorphic structure on E associated with an integrable semi-connection
δ
E
δ. For an open set U Ă X the space pUq of holomorphic sections of co-
Eδ Eδ U
incides with the kernel of the first order operator δ : A0pU,Eq Ñ A0,1pU,Eq.
U
Usingthisbijectionandthenatural 8-topologyonA0,1pEqint weobtainanatural
C
topology on the space olpEq of holomorphic structures on E. The moduli space
H
of holomorphic structures on E is the topological space obtained as the quotient
L L
olpEq 0,1pEqint
H AutpEq “A AutpEq
of olpEq by the group AutpEq :“ ΓpX,GLpEqq of differentiable automorphisms
H
of E, which acts naturally on this space.
Since GLprq (the structure group of a vector bundle) is not a simple group,
one considers a natural refinement of our moduli problem, which has a simple
“symmetry group”. Let be a holomorphic structure on the determinant line
D
bundle D :“ detpEq, and λ P 0,1pDqint be the corresponding integrable semi-
A
connection. A -oriented holomorphic structure on E is a holomorphic structure
D
on E with detp q“ .
E E D
We recall that an SLpr,Cq-vector bundle on X is a rank r vector bundle E
endowed with a trivialization of its determinant line bundle. Therefore, in this
case, D comes with a tautological trivial holomorphic structure Θ; an SLpr,Cq-
holomorphic structure on E is just a Θ-oriented holomorphic structure in E in our
sense. This shows that classifying -oriented holomorphic structures on a fixed
D
differentiable vector bundle gives the natural generalization of the classification
problem for SLpr,Cq-holomorphic structures on an SLpr,Cq-vector bundle.
The set ol pEq of -oriented holomorphic structures on E can be identified
D
withthesubHspace 0,1pEDqint Ă 0,1pEqintdefinedbytheconditiondetpδq“λ. The
gauge group C :“AΓλpX,SLpEqqAacts naturally on this subspace. We will focus on
GE
the moduli space of -oriented holomorphic structures on E, which, by definition,
D
ON THE DONALDSON-UHLENBECK COMPACTIFICATION 3
is the quotient space
L L
ol pEq 0,1pEqint
DpEq:“H D C “Aλ C.
M GE GE
In general this quotient is highly non-Hausdorff and cannot be endowed with a
natural complex space structure. It can be identified with the topological space
associated with a holomorphic stack, but up till now it is not clear if, in our non-
algebraiccomplexgeometricframework,thisapproachleadstoeffectivenewresults.
Ontheotherhandtheclassicalpointofview(studyingmodulispaceswhosepoints
correspond to equivalence classes of holomorphic structures) has been used with
effective results, for instance in making progress towards the classification of class
VII surfaces [Te2], [Te4].
We recall that a holomorphic structure on E is called simple if H0p ndp qq“
E E E
CidE. In contrast with MDpEq, the moduli space MsDipEq of simple D-oriented
holomorphic structures has a natural, in general non-Hausdorff, complex space
structure. Thisstructurecanbeobtainedintwodifferentways, which, aposteriori
turnouttobeequivalent. Thefirstapproach[FK]usesclassicaldeformationtheory.
The second [LO] uses complex gauge theory. The equivalence of the two points of
view has been established by Miyajima [Miy].
Asinalgebraicgeometry,inordertodefineHausdorffmodulispacesofholomor-
phic structures, one needs a stability condition, which depends on the choice of an
additionalstructureonX. Whereasinalgebraicgeometrythisadditionalstructure
isapolarisationonX (i.e. thechoiceofanamplelinebundleonX),inourgeneral
complex geometric framework we need a Gauduchon metric on X, i.e. a Hermitian
metric g on X whose associated p1,1q-form ω satisfies the Gauduchon condition
g
ddcpωn´1q“0 [Gau]. Such a metric defines a Lie group morphism
g ż
degg :PicpXqÑR, deggpLq“ c1pL,hq^ωgn´1,
X
where c p ,hq denotes the Chern form of the Chern connection associated with a
1
L
Hermitianmetrichon . Usingthisdegreemaponecanintroduceaslopestability
L
condition in the same way as in the algebraic framework:
For a coherent sheaf on X we put deg p q :“ deg pdetp qq. A non-zero
F g F g F
torsion-free sheaf on X is called (semi-)stable (with respect to g) if for every
F
subsheaf Ă with 0ărkp qărkp q one has
H F H F
deg p q deg p q deg p q deg p q
g H ă g F respectively g H ď g F .
rkp q rkp q rkp q rkp q
H F H F
is called polystable (with respect to g) if it splits as a direct sum “‘k of
F F i“1Fi
non-zero stable subsheaves such that deggpFiq “ deggpFq for 1ďiďk.
Fi rkpFiq rkpFq
We denote by sipEq, stpEq, pst the moduli space of simple, stable, re-
MD MD MD
spectively polystable -oriented holomorphic structures on E. Any stable bundle
is simple and polystaDble, so stpEq Ă sipEq, stpEq Ă pstpEq. Stability
MD MD MD MD
is an open condition with respect to the classical topology [LT], hence stpEq is
open in both sipEq, and pstpEq. In particular stpEq comes withMaDnatural
MD MD MD
complex space space structure induced from sipEq.
MD
Note that in general, in the non-K¨ahlerian framework, semi-stability is not an
open condition (even with respect to the classical topology).
4 NICHOLASBUCHDAHL,ANDREITELEMAN,ANDMATEITOMA
By choosing a Hermitian metric h on E, pstpEq can be identified with the
MD
moduli space of projectively Hermitian-Einstein connections on E, defined as fol-
lows. Denote by pEq the space of Hermitian connections on pE,hq. For a fixed
A
Hermitian connection a on the Hermitian line bundle pD,detphqq we put
pEq:“tAP pEq| detpAq“au.
a
A A
An element of pEq will be called an a-oriented connection on pE,hq. We de-
a
A
fine the space and the moduli space of projectively Hermitian-Einstein a-oriented
connections on pE,hq by
L
AHaEpEq:“tAPAapEq| ΛgFA0 “0, pFA0q0,2 “0u , MHaEpEq:“AHaEpEq E,
G
where F0 stands for the trace-free part of the curvature F of A, and :“
A A GE
ΓpX,SUpEqqistheSUprq-gaugegroupofpE,hq. Ifn“2,theconditionsΛ F0 “0,
g A
pF0q0,2 “0 are equivalent to the anti-selfduality condition pF0q` “0, so HEpEq
A A Ma
is just the moduli space ASDpEq of projectively ASD, a-oriented connections on
Ma
pE,hq.
The Kobayashi-Hitchin correspondence [Bu1], [LT], [LY] generalising Donald-
son’s fundamental work [D1] states that
Theorem (The Kobayashi-Hitchin correspondence). Let be a fixed holomorphic
D
structure on D “detpEq and let a be the Chern connection of the pair p ,detphqq.
D
The map AÞÑEB¯A induces a homeomorphism
HEpEq´»Ñ pstpEq,
Ma MD
which restricts to a homeomorphism HEpEq˚´»Ñ stpEq between the moduli
Ma MD
space of irreducible projectively Hermitian-Einstein, a-oriented connections on E,
and the moduli space of stable -oriented holomorphic structures on E.
D
Remark 1.1. InthespecialcasewhenE isanSLpr,Cq-bundlewecanchoosehsuch
that the distinguished trivialisation of detpEq is unitary. With this choice pE,hq
becomesan SUprq-vector bundle, a coincideswith thetrivial connectionassociated
with this trivialisation, pEq is the space of SUprq-connections on pE,hq, and
a
A
HEpEq is the moduli space of Hermitian-Einstein SUprq-connections on pE,hq.
Ma
InthecaseofanSLpr,Cq-bundleE (orofanSUprq-bundlepE,hq)wewillomitthe
subscripts ,ainournotation,hencewewillwrite pstpEq, stpEqforthemoduli
D M M
spacesofpolystable(stable)SLpr,Cq-structuresonE,and HEpEq, HEpEq˚ for
M M
the moduli spaces of (irreducible) Hermitian-Einstein SUprq-connections on pE,hq.
TheKobayashi-Hitchincorrespondencehasimportantconsequences: usingstan-
dard gauge-theoretical techniques one can prove that the quotient topology of
HEpEq is Hausdorff. The point is that, for this moduli space, the gauge group
Ma
is the group of sections of a locally trivial bundle with compact standard fibre.
E
GTherefore pstpEq, stpEq are Hausdorff spaces, and in particular stpEq is a
MD MD MD
Hausdorffcomplexsubspaceofthe(possiblynon-Hausdorff)modulispace sipEq.
MD
1.2. Extending the complex structure to a compactification of stpEq.
MD
Thefundamentalquestions. ThefirstnaturalquestionrelatedtotheKobayashi-
Hitchin correspondence is: Does pstpEq have a natural complex space structure
MD
extending the canonical complex space structure of stpEq? Theexplicitexamples
MD
describedin[Te2],[Te4]andthegeneralresultsprovedin[Te5]showthatingeneral
the answer is negative. For instance, for a class VII surface X with b pXq “ 1, a
2
ON THE DONALDSON-UHLENBECK COMPACTIFICATION 5
bundle E with c pEq “ 0, c pEq “ c p q and a suitable Gauduchon metric g on
2 1 1 X
X,themodulispace pstpEqcanbeidKentifiedwithacompactdisk,whoseinterior
MD
corresponds to stpEq. Moreover, on a class VII surface X with b pXq “ 2 one
obtains in a simMilaDr way a moduli space pstpEq which can be ident2ified with S4,
MD
and in this case stpEq corresponds to the complement of the union of two circles
MD
in S4. These examples show that, in the general (possibly non-K¨ahlerian) frame-
work the complex space structure on stpEq does not extend to a complex space
structure on pstpEq. On the otherMhaDnd, in general pstpEq is not compact,
MD MD
and when this is the case, even if the complex structure of stpEq does extend to
pstpEq, the result is not satisfactory. This motivates theMfolDlowing:
MD
Question 1. Let pX,gq be a compact Gauduchon manifold, E be a differentiable
rankr vectorbundleonX and beafixedholomorphicstructureonD :“detpEq.
D
Does the complex space structure of stpEq extend to a complex space structure
on a natural compactification of it, wMhicDh contains the space pstpEq?
MD
Notethat,inthecasen“2,onehasagoodcandidatefor“anaturalcompactifi-
cation of stpEq which contains pstpEq”: we identify stpEq with ASDpEq˚
MD MD MD Ma
viatheKobayashi-Hitchinisomorphism,andweembedthelatterspaceintheDon-
aldson compactification of ASDpEq. Therefore, in the case n “ 2, one can ask a
Ma
more precise version of Question 1:
Question 2. LetpX,gqbeacompactGauduchonsurface,E beadifferentiablerank
rvectorbundleonX and beafixedholomorphicstructureonD :“detpEq. Does
D
the complex space structure on ASDpEq˚ induced by the Kobayashi-Hitchin cor-
Ma
respondenceextendtoacomplexspacestructureontheDonaldsoncompactification
ASD
pEq?
Ma
Astheexamplesaboveshow,inourgeneral(possiblynon-K¨ahlerian)framework
bothquestionshavenegativeanswer. Indeed, inourexamples thespace ASDpEq
Ma
is already compact, and admits no complex space structure at all.
On the other hand we believe that both questions have positive answer when
pX,gq is K¨ahler. In other words
Conjecture 1. LetpX,gqbeacompactK¨ahlermanifold,E beadifferentiablerankr
vectorbundleonX and beafixedholomorphicstructureonD :“detpEq. Then
D
the natural complex space structure of stpEq (induced from sipEq) extends to
a natural compactification of it whichMconDtains pstpEq. ForMnD“ 2 the natural
MD
complex space structure of stpEq extends to the Donaldson compactification
MD
ASDpEq of ASDpEq.
Ma Ma
Thisconjectureisknowntoholdintheprojectivealgebraicframework(whenω
g
is the Chern form of an ample line bundle on X) for SLp2,Cq-bundles. In this
H
ASD
case, forn“2oneproves[Li]that pEqcanbeidentifiedwiththeimageina
Ma
projectivespaceof a regularmap defined ona Zariskiclosedsubset of theGieseker
moduli space associated with the data c “ 0, c “ c pEq, r “ 2. This Gieseker
1 2 2
moduli space is a projective variety. Taking into account Li’s result, and the fact
that any compact K¨ahler surface admits arbitrary small deformations which are
projective ([Kod1], [Bu3], [Bu4]), Conjecture 1 becomes very natural. The recent
results of [GT] concern the higher dimensional projective case, and give further
6 NICHOLASBUCHDAHL,ANDREITELEMAN,ANDMATEITOMA
evidence for this conjecture.
In this article we will study in detail Question 2 in an interesting special case:
the moduli space of SLp2,Cq-structures on an SLp2,Cq-bundle E with c2pEq “ 1
on a class VII surface X. One has
` ˘
ASDpEq“ ASDpEqY ˆX
M M M0 ` ˘ ` ˘ (1)
“ ASDpEq˚Y Y ˚ˆX Y ˆX ,
M R M0 R0
where Ă ASDpEq is the subspace of reducible SUp2q-instantons with c “ 1,
2
R M
and ( ˚, )standsforthemodulispaceofflat, respectivelyflatirreducible,
M0 M0 R0
flat reducible, SUp2q-instantons. The last three terms in the decomposition (1) are
compact. Therefore in our case, Question 2 reduces to a set of three more specific
questions:
Does the complex space structure of ASDpEq˚ “ stpEq extend across the com-
M M
pact strata (a) ˆX, (b) ˚ˆX, (c) ?
R0 M0 R
Thefactor ofthethirdsummandin1canbefurtherdecomposedasadisjoint
0
R
unionasfollows. DenotebyCpXqthegroupofcharactersHompH1pX,Zq,S1q. One
has a natural identification
L
CpXq:“CpXq ´»Ñ ,
xjy 0
R
where j is the involution induced by the conjugation S1 Ñ S1. The group CpXq
fits into the following commutative diagram with exact rows:
0 (cid:45) C0pXq»S1 (cid:45) CpXq (cid:45) TorspH2pX,Zqq (cid:45) 0
X X
(cid:63) (cid:63) (cid:107)
0 (cid:45) Pic0pXq»C˚ (cid:45) PicTpXq c(cid:45)1 TorspH2pX,Zqq (cid:45) 0,
where
C0pXq:“HompH1pX,Zq{Tors,S1q»S1,
PicTpXq:“tr sPPicpXq| c1p qPTorspH2pX,Zqqu.
L L
The central vertical monomorphism maps a character χ P CpXq to the isomorphy
class of the associated holomorphic Hermitian line bundle , and identifies CpXq
χ
L
with kerpdeg q, which is independent of the Gauduchon metric g. More
g PicTpXq
preciselythecongruenceclassCcpXqPCpXq{C0pXqassociatedwithatorsionclass
cPTorspH2pX,Zqqisidentifiedwiththevanishingcircleofdegg onthecongruence
class PiccpXqPPicpXq{Pic0pXq.
The involution j : CpXq Ñ CpXq maps CcpXq diffeomorphically onto C´cpXq,
in particular leaves invariant any circle CcpXq associated with a class c belonging
to the Z2-vector space ` ˘
Tors2pH2pX,Zqq:“ker H2pX,Zq´2Ѩ H2pX,Zq .
Let µ2 be the multiplicative group t˘1u. For cPTors2pH2pX,Zqq the set ρcpXq of
fixedpointsoftheinducedinvolutionCcpXq´jÑc CcpXqisaµ -torsor(inparticular
2
has two points), and the quotient
L
CcpXq:“CcpXq
xj y
c
ON THE DONALDSON-UHLENBECK COMPACTIFICATION 7
is a segment. More precisely, the choice of a fixed point l P ρcpXq gives a homeo-
morphism
L
CcpXq´´h´l»Ñ S1z ÞÑz¯»r´1,1s.
Let TpXq be the quotient of TorspH2pX,Zqq by the involution c ÞÑ ´c, and
denote by T pXq, T pXq the subsets of TpXq which correspond respectively to
0 1
Tors2pH2pX,Zqq and TorspH2pX,ZqqzTors2pH2pX,Zqq.
For a class c“rcsPTpXq denote by CcpXq the quotient of CcpXqYC´cpXq by
the involution induced by j on this union. Note that for c “ tcu P T pXq one has
0
CcpXq“CcpXqwhereas, forcPT pXq, thechoiceofarepresentativecPcgivesan
1
identification CcpXq»CcpXq. Therefore the space of reducible flat instantons
0
R
on X decomposes as a finite disjoint union of segments and circles:
ˆ ˙ ˆ ˙
ď ď ď ď
»CpXq“ CcpXq“ CcpXq CcpXq . (2)
0
R
cPTpXq cPT0pXq cPT1pXq
The set of all vertices appearing in the first term of (2) can be identified with the
set of fixed points of j, which coincides with
ď
ρpXq:“H1pX,µ2q“HompH1pX,Zq,µ2q“ ρcpXq,
cPTors2pH2pX,Zqq
and has an important gauge theoretical interpretation: it corresponds to the mod-
uli space of reducible flat SUp2q-instantons with non-abelian stabiliser SUp2q. All
other points of correspond to reducible flat SUp2q-instantons with abelian sta-
0
R
biliser S1. Therefore, from a gauge theoretical point of view, the first term in
the decomposition (2) (which is always non-empty) is substantially more complex
than the second, because a segment CcpXq contains both abelian and non-abelian
reductions. Our main result states
Theorem 1.2. Let X be a class VII surface, and let E be an SLp2,Cq-bundle on
X with c pEq “ 1. The complex space structure of ASDpEq˚ “ stpEq extends
2
M M
ASD
across ˆX, and pEq is a smooth complex 4-fold at any reducible virtual
0
R M
point prAs,xqP ˆX.
0
R
This result is surprising for two reasons: first, is a union of segments and
0
R
circles, which are not complex geometric objects; second, since the vertices of the
segments CcpXq (c P T pXq) are isolated non-abelian reductions, one expects es-
0
sential singularities at these points.
Oursecondresultconcernstheextensibilityofthecomplexspacestructureacross
˚ˆX:
M0
Theorem 1.3. Let X be a class VII surface, and let E be be an SLp2,Cq-bundle
on X with c pEq “ 1. Then ˚ consists of finitely many simple points, and the
2 M0
complex space structure of ASDpEq˚ “ stpEq extends across ˚ ˆX. For
M M M0
ASD
every rAs P ˚ the surface trAsuˆX has an open neighbourhood in pEq
M0 M
which is a normal complex space whose singular locus is trAsuˆX, and the normal
cone of this singular locus can be identified with the cone bundle of degenerate
elements in _ bS2p q, where is the holomorphic bundle associated with A.
KX E E
Here we denoted by S2p q the second symmetric power of ; an element ηbσ P
E E
pxq_bS2p pxqqisdegenerateiftheassociatedlinearmap pxqÑ pxq_b pxq_
X X
K E E E K
8 NICHOLASBUCHDAHL,ANDREITELEMAN,ANDMATEITOMA
has non-trivial kernel. Since rkp q “ 2, this is equivalent to the condition that σ
E
belongs to the image of the squaring map pxqÑS2p pxqq.
E E
Finally, the extensibility of the complex space structure across the subspace
Ă ASDpEq has been studied in detail in a more general framework in [Te5]. In
R M
our special case the result is the following
Theorem 1.4. [Te5] Let X be a class VII surface endowed with a Gauduchon met-
ric g with deggpKXq ă 0, and let E be an SLp2,Cq-bundle on X with c2pEq “ 1.
Then ASDpEq˚ “ stpEq is a smooth 4-fold, and the reduction locus Ă
M M R
ASDpEq is a union of b2pXq|TorspH2pX,Zqq| circles. Any such circle has a
M
neighbourhood which can be identified with a neighbourhood of the singular circle
in a flip passage; in particular the holomorphic structure of ASDpEq˚ does not
M
extend across any of these circles.
In other words, for class VII surfaces with b ą 0, the holomorphic structure
2
doesnotextendacrossthecirclesofreductionsinthemodulispace,but(supposing
deg p q ă 0) the structure of the moduli space around such a circle is perfectly
g KX
understood. Notethattheconditiondeg p qă0isnotrestrictive. Indeed,using
g KX
the classification of class VII surfaces with b “ 0 [Te1] and the results of [Bu2]
2
(see also [Te4, Lemma 2.3]), it follows that:
Remark 1.5. Any class VII surface whose minimal model is not an Inoue surface
admits Gauduchon metrics g such that deg p qă0.
g KX
A surprising corollary of our results is:
Corollary 1.6. Let pX,gq be a primary Hopf surface endowed with a Gauduchon
metric, and E be an SLp2,Cq-bundle on X with c2pEq “ 1. Then the natural
complex structure on stpEq is smooth and extends to a complex structure on
M
ASD
pEq, which becomes a 4-dimensional compact complex manifold.
M
This study of moduli spaces of SUp2q-instantons on class VII surfaces has sev-
eral motivations. First, in recent articles the second author showed that PUp2q-
instanton moduli spaces can be used to make progress on the classification of class
VII surfaces, more precisely to prove the existence of curves on such surfaces [Te2],
[Te4]. Anaturalquestionis: canoneobtainsimilar(orevenstronger)resultsusing
moduli spaces of SUp2q-instantons? In order to follow this strategy one needs a
thorough understanding of compactified such moduli spaces.
A second motivation is related to Corollary 1.6: according to this result, the
ASD
assignment pX,gq ÞÑ pEq defines a functor from the category of Gaudu-
M
chon primary Hopf surfaces to the class of 4-dimensional smooth compact complex
manifolds. Moreover, it is known that ASDpEq˚ “ stpEq is endowed with a
canonical Hermitian metric g which is sMtrongly KT, i.Me., satisfies BB¯ω “ 0 [LT].
g
TheclassofcompactstronglyKTHermitianmanifoldshasbeenintensivelystudied
in recent years. This class of manifolds intervenes in modern physical theories (II
stringtheory,2-dimensionalsupersymmetricσ-models)andalsoinHitchin’stheory
of generalised K¨ahler geometry. Therefore, it is natural to ask
Question 3. IntheconditionsofCorollary1.6doesthecanonicalstronglyKTmetric
on ASDpEq˚ “ stpEq extend to a smooth Hermitian metric on the complex
M M
ASD
4-fold pEq?
M
ON THE DONALDSON-UHLENBECK COMPACTIFICATION 9
ASD
If this question has a positive answer, the resulting metric on pEq will be
M
stronglyKT,givinganinterestingfunctorfromthecategoryofGauduchonprimary
Hopf surfaces to the category of compact strongly KT 4-dimensional manifolds.
This functor would yield a large class of examples of 4-dimensional strongly KT
compact manifolds. We will come back to Question 3 in a future article.
Athirdmotivation: thenoveltyofthemethodsusedintheproofs,whichempha-
sise surprising difficulties which occur in the non-K¨ahlerian framework. We shall
explain the ideas of proofs and these difficulties in the next subsections, and in the
course of the subsequent proofs themselves, it will be apparent that our methods
will be applicable in many other situations.
1.3. The idea of proof of Theorem 1.2. In the first part of the proof we study
ASD
pEq from the topological point of view. We show that any reducible flat
M
SUp2q-instanton A on a class VII surface is regular, i.e. one has H2A “0. Using the
local model theorem for virtual instantons [DK, Theorem 8.2.4], we will prove that
ASD
pEq is a topological 8-manifold around any virtual point prAs,xqP ˆX.
0
M R
Denoting by ASDpEq˚ Ă ASDpEq the open subspace of regular irreducible
M reg M
instantons, we see that the subspace
` ˘
M:“ ASDpEq˚ Y ˆX Ă ASDpEq
M reg R0 M
is a topological 8-manifold.
Inasecondstepwewillconstructacomplexmanifoldstructureonthetopological
manifoldMusingagluingconstructionbasedonthefollowingsimpleresultproved
in the appendix:
Lemma 6.16. Let be a topological 2n-dimensional manifold, and Ă be an
X Y X
open subset endowed with a complex manifold structure. Let be an n-dimensional
U
complexmanifold,andf : Ñ beacontinuous,injectivemapwiththeproperties:
U X
‚ z Ăimpfq,
X Y
‚ The restriction f : f´1p q Ñ is holomorphic with respect to the
f´1pYq Y Y
holomorphic structure induced by the open embedding f´1p qĂ .
Y U
Then
(1) impfq is an open neighbourhood of z in ,
X Y X
(2) f induces a homeomorphism Ñimpfq,
U
(3) f inducesabiholomorhismf´1p qÑimpfqX withrespecttotheholomor-
Y Y
phicstructuresinducedbytheopenembeddingsf´1p qĂ ,impfqX Ă ,
Y U Y Y
(4) There exists a unique complex manifold structure on which extends the
X
fixed complex structure on , and such that f becomes biholomorphic on its
Y
image.
The hard part of the proof of Theorem 1.2 is the construction of a pair p ,f :
U
Ñ Mq such that, taking :“ ASDpEq˚ with the complex structure induced
U Y M reg
from stpEq , the hypothesis of Lemma 6.16 is fulfilled.
reg
M
Note first that, surprisingly, the Gieseker (semi)stability condition for torsion-
free sheaves, can be naturally extended to arbitrary Gauduchon surfaces. The
point is that, for surfaces, in these conditions only the degree of the sheaf and its
Euler-Poincar´echaracteristicintervene[Fr,p. 97]. Weagreetocallthetorsion-free
sheaves on X, which are not locally free, singular sheaves.
10 NICHOLASBUCHDAHL,ANDREITELEMAN,ANDMATEITOMA
Inoursituationweobtainanaturalhomeomorphismϕ: ´»Ñ ˆX between
0
S R
the moduli space of singular rank 2 Gieseker stable sheaves on X with the
S F
properties
‚ detp q» , c p q“1,
X 2
F O F
‚ the double dual __ is properly semi-stable.
F
Taking into account this identification, a natural choice would be to take for an
U
open neighbourhood of in the moduli space of all (locally free and singular) rank
S
2 Gieseker stable sheaves (with trivial determinant and c “ 1). Unfortunately
2
in our non-K¨ahlerian framework an unexpected difficulty arises: Gieseker stability
is not an open condition. More precisely, in our case, any neighbourhood (in the
moduli space of simple rank 2 sheaves with trivial determinant) of a point r sP
F S
contains points corresponding to sheaves which are not even slope semi-stable.
Taking into account this difficulty, we will take to be a sufficiently small open
U
neighbourhood of in the moduli space si of simple rank 2 sheaves [KO] with
S M
trivial determinant. We will define a map
f : ÑM
U
whose restriction to coincides with ϕ and whose restriction to X stpEq
reg
S U M
coincides with the identity of this set. For a point r s P with not semi-
F U F
stable we put fpr sq:“r s, where is a slope stable locally free sheaf which is
F F
F E E
determined up to isomorphy by the conditions
H0p omp , qq‰0 , H0p omp , qq‰0.
F F
H F E H E F
The proof will be completed by showing that these properties determine a well-
defined map f : ÑM satisfying the assumptions of Lemma 6.16.
U
1.4. The idea of proof of Theorem 1.3. According to the main result of [Pl]
the moduli space ˚ consists of finitely many simple points. Let rAs P ˚ be
M0 M0
a flat irreducible SUp2q-instanton, be the associated stable holomorphic bundle,
E
and π :Pp qÑX its projectivisation. For a point xPX and a line y PPp pxqq we
E E
denote by η : pxqÑq :“ pxq{y the corresponding epimorphism, and we put
y y
E E ` ˘
:“ker Ñ ´´ηÑy q b .
Fy E Etxu y Otxu
We will construct a torsion-free sheaf F on Pp qˆX, flat over Pp q, such that
for any point y P PpEq the restriction F tyuˆX,Eregarded as a sheafEon X, is iso-
morphic with y. The sheaf F defines an embedding Pp q Ñ si in the moduli
F E M
space of simple, torsion-free sheaves with trivial determinant and c “1. The nor-
2
mal line bundle of the image of this embedding can be computed explicitly. We
P
will prove that has an open neighbourhood such that z Ă stpEq . On
reg
P U U P M
the other hand, using Fujiki’s contractibility criterion [Fuj], it follows that, there
exists a modification Ñ on a singular complex space , which contracts the
U V V
projective fibres of . Using the continuity theorem [BTT] we obtain a continuous
P
ASD
map Ñ pEq which induces a homeomorphism between the complex space
U M
ASD
and an open neighbourhood of trAsuˆX in pEq.
V M
The article is organised as follows: Section 2 contains results on the topology
ASD
pEqaroundthevirtuallocus. Theseresultsareobtainedusinggaugetheoret-
M
ical methods. Using the constructions given in Section 3, we construct in Section 4
a holomorphic embedding V :A ˆX Ñ si, where A is an open neighbourhood
ε ε ε
M