Analytic cycles in flip passages and in instanton moduli spaces over non-K\"ahlerian surfaces

0.3 MB·English

by  Andrei Teleman

ANALYTIC CYCLES IN FLIP PASSAGES AND IN INSTANTON
MODULI SPACES OVER NON-KA¨HLERIAN SURFACES
5
1 ANDREITELEMAN
0
2
Abstract. LetMst (Mpst)beamodulispaceofstable(polystable)bundles
n
with fixed determinant on a complex surface with b1 = 1, pg = 0, and let
a Z ⊂ Mst be a pure k-dimensional analytic set. We prove a general formula
J forthehomologicalboundaryδ[Z]BM ∈H2BkM−1(∂Mˆpst,Z)oftheBorel-Moore
7 fundamental class of Z in the boundary of the blow up moduli space Mˆpst.

Theproofisbasedontheholomorphicmodeltheoremof[Te5],whichidentifies
G] aneighborhoodofaboundarycomponentofMˆpstwithaneighborhoodofthe
boundaryofa“blowupflippassage”.
D
We then focus on aparticular instanton modulispace which intervenes in
. ourprogramforprovingtheexistenceofcurvesonclassVIIsurfaces. Usingour
h
result,combinedwithgeneralpropertiesoftheDonaldsoncohomologyclasses,
t
a we prove incidence relations between the Zariski closures (in the considered
m moduli space) of certain families of extensions. These incidence relations are
crucial for understanding the geometry of the moduli space, and cannot be
[
obtained usingclassicalcomplexgeometricdeformationtheory.

1
v
1
5 1. Introduction
6
1 Let (X,g) be a Gauduchon surface [Gau], (E,h) a Hermitian rank-2 bundle
0 over X, a holomorphic structure on the determinant line bundle det(E) and a
1. the CherDn connection of the pair ( ,det(h)). The moduli space pst ( st) of
D M M
0 polystable (stable) holomorphic structures on E with det( )= can be identi-
5 fied with the instanton moduli space ASDE(respectively EASD∗)Dof (irreducible)
1 M M
projectively ASD unitary connections A on E with det(A) = a [DK], [Bu], [LT],
:
v [Te1], [Te3], [Te5]. The stable part st pst is open and has a natural com-
i M ⊂ M
X plex space structure, which, in general, does not extend across the reduction locus
:= pst st (the subspace of reducible instantons).
r R M \M
a The set of topological decompositions of E (as direct sum of line bundles) can
be identified with the set
c H2(X,Z) c(c (E) c)=c (E)
ec(E):={ ∈ | 1 − 2 } ,
D ∼
where is the equivalence relation defined by the involution c(cid:14) c1(E) c. We
∼ 7→ −
assume that c (E) 2H2(X,Z), which implies that this involution has no fixed
1
6∈
points. Moreover,underthis assumption,weshowed[Te3],[Te5]that, forasurface
with b (X)=1, p (X)=0 (in particular for a class VII surface [BHPV], [Na]),
1 g
R
decomposes as a disjoint union of circles
(1) = C ,
λ
R
λ∈D[ec(E)
TheauthorhasbeenpartiallysupportedbytheANRprojectMNGNK,decisionNr. ANR-10-
BLAN-0118.
1
2 ANDREITELEMAN
where C := [ ] pst has a direct summand with c ( ) λ . Choosing
λ 1
E ∈ M | E L L ∈
a representative c λ, and putting d := 1deg ( ), C can be identified with the
(cid:8) ∈ 2 g D λ (cid:9)
circle
C := [ ] Picc(X) deg ( )=d
c {L ∈ | g L }
(see section 2.1). Blowing up pst at a circle C of regular reductions yields a
λ
M
proper map p : ˆpst pst, defined on a space ˆpst which has a natural
λ Mλ → M Mλ
structureofamanifoldwithboundaryaroundtheexceptionallocusPλ :=p−λ1(Cλ)
[Te5]. This exceptional locus is a fiber bundle over C with a complex projective
λ
space as fiber. It’s important to point out that, in our framework, a moduli space
st contains distinguished locally closed complex subspaces c, which correspond
M Pε
to families of stable extensions, and are described in Remark 1.1 below. For [ ]
L ∈
Pic(X) define
P := [ ] st is an extension of by ,
[ ] ∨
L {E ∈M | E K⊗L L}
and for a subset A R, and a class c H2(X,Z), put
⊂ ∈
Picc(X) := [ ] Picc(X) deg ( ) A .
A {L ∈ | g L ∈ }
When A is a singleton (an open interval), Picc(X) is a circle (an annulus).
A
Remark 1.1. Suppose that b (X) = 1, p (X) = 0, c (E) 2H2(X,Z), let λ
1 g 1
6∈ ∈
ec(X) such that C is a circle of regular reductions, and choose c λ. Then
λ
D ∈
1. There exists an open neighborhood of C in pst such that st is a
λ λ λ
U M U ∩M
smooth complex manifold of dimension 4c (E) c2(E),
2 − 1
2. There exists ε>0 such that
i) for any [ ] Picc(X) the subspace P is a complex submanifold of
(d ε,d) [ ]
stL. T∈his submani−fold can be identifiedLwith P(H1( 2 )), where
λ ⊗ ∨
U ∩M L ⊗K
1
dim(H1( 2 )))=r := (2c c (E))(2c c (E)+c (X)) .
⊗ ∨ c 1 1 1
L ⊗K −2 − −
ii) The union P is disjoint, defines a smooth submanifold
c of [Lst],∈Paincdc(Xth)e(d−nεa,dt)ura[Ll]map c Picc(X) is a fiber bundle
wPiεth fiUbeλr∩PMSCrc−1. Pε → (d−ε,d)
The following remark highlights an advantage of the blow up moduli space:
Remark 1.2. With the notations, suppose r >0. For sufficiently small ε>0
c
12.. TThhee cclloossuurreeP˜¯εccooffPεcciinnMˆpλpststisisaoPbtCrac−in1e-dbufnrodmle o˜vcerbePiccocl(lXap)s[idn−gε,dto],points the
fibers over thPeεcirclePεPicc M. Pε
d
{ }
Note that, in general, is very hard to describe the Zariski closure of c in st,
Pε M
even when X is a known surface (for instance a Kato surface). Understanding the
Zariskiclosuresofthesefamilies,andtheincidencerelationsbetweentheseclosures
is very important for understanding the geometry of the moduli space, and plays
animportantroleinourprogramtoproveexistenceofcurvesonclassVII surfaces.
This is why we are interested in the general properties of the analytic subsets of
st. The main result of this article concerns the following
M
Problem 1. Let Z st be a pure k-dimensional analytic set, and let [Z]BM
⊂ M ∈
HBM( st,Z) its fundamental class in Borel-Moore homology. Determine the ho-
2k M
mological boundary δ([Z]BM) H ( ,Z) of [Z]BM.
2k 1 λ
∈ − P
ANALYTIC CYCLES IN FLIP PASSAGES AND INSTANTON MODULI SPACES 3
Why is this problem relevant for understanding the geometry of an instanton
moduli spaces pst and its families of extensions? In order to explain this, recall
M
first [DK] that st = ASD∗ is naturally embedded in the infinite dimensional
M M
moduli space of irreducible unitary connections A on E with det(A) = a. The
∗
B
space isendowedwithtautologicalcohomologyclasses[DK],whichwillbecalled
∗
B
Donaldsonclasses. Ingeneral,aDonaldsonclassν H ( ,Q)cannotbeextended
∗ ∗
∈ B
across a circle of reductions, but it does extend to the exceptional locus
λ
ˆpst associated with a circle C of regular reductions. This a second impoPrtan⊂t
Mλ λ
advantage of the blow up moduli space. With this remark we have
Proposition 4.1. Suppose that b (X)=1, p (X)=0, c (E) 2H2(X,Z), pst
1 g 1
6∈ M
is compact, and all reductions in pst are regular. For any Donaldson cohomology
M
class ν Hk 1( ,Q) and any ξ HBM( st,Q) we have
∈ − B∗ ∈ k M
(2) ν ,δ ξ =0 ,
λ
h Pλ i
λ∈DXec(E)
where δ ξ denotes the homological boundary of ξ in H ( ,Q).
λ k 1 λ
− P
Thepointisthattherestrictionsν havebeencomputedexplicitly[Te2,Corol-
lary2.6]. Therefore,assumingthatPrPoλblem1issolved,formula(2)yieldsa strong
obstructiontotheexistenceofananalyticsetZ st withprescribedtopological
⊂M
behavior around the circles of reductions C . Note that (in the relevant cases) the
λ
particularmodulispace pst,usedinourprogramtoprovetheexistenceofcurves
M
on class VII surfaces, satisfies the assumptions of Proposition 4.1 [Te3].
The mainresult ofthis article gives a solutionto Problem1. The proofis based
on the holomorphic model theorem [Te5], which states that a neighborhood of the
boundary in pst can be identified with a neighborhood of the boundary of a
C
P M
standardmodel, whichwe calledablow up flip passage, andwhoseconstructionwe
recall briefly below.
Let B be a Riemann surface, p :E B, p :E B holomorphic Hermitian
′ ′ ′′ ′′
→ →
bundles of ranks r , r , and f : B R a smooth function which is a submersion
′ ′′
→
at any vanishing point, and such that C := f 1(0) is a circle. The direct sum
−
E := E E is endowed with the C -action ζ (y ,y )) = (ζy ,ζ 1y ). The zero
′ ′′ ∗ ′ ′′ ′ − ′′
⊕ ·
locus Z(mf) of the map mf :E R defined by
→
1
mf(y ,y )= ( y 2 y 2)+f(b) , (y ,y ) E ,
b ′ ′′ 2 k ′k −k ′′k ∀ ′ ′′ ∈ b
is a smooth, S1-invariant hypersurface. The induced S1-action on Z(mf) is free
awayofC (embeddedinE viathezerosection). PutF :=E ,F :=E . The
′ ′ C ′′ ′′ C
normal bundle of C in Z(mf) can be identified (as an S1-bundle) with F F¯ ,
′ ′′
⊕
hence the spherical blow up Z\(mf) of Z(mf) at C is a manifold with boundary,
C
whose boundary can be identified with the sphere bundle S(F F¯ ) (see section
′ ′′
⊕
2.1 in this article, [Te5]). The blow up flip passage Qˆ associated with the data
f
(p :E B ,p :E B,f) is defined by
′ ′ ′ ′′ ′′
→ →
Qˆf :=Z\(mf)C S1 .
Thisquotientisasmoothmanifoldwhoseboun(cid:14)dary∂Qˆ canbeidentifiedwiththe
f
projective bundle P(F F¯ ). The interior Qˆ ∂Qˆ can be identified with the
′ ′′ f f
⊕ \
quotient Qs :=Es/C , where Es :=C (Z(mf) C) is open in E. Therefore this
f f ∗ f ∗· \
4 ANDREITELEMAN
interior comes with a natural complex structure. Note also that Qˆ comes with a
f
map q : Qˆ B, whose restriction to Qs is a holomorphic submersion. Choosing
f → f
points b B := ( f) 1(0, ) (and supposing r > 0, r > 0), the fibers
− ′ ′′
q 1(b )±are∈smo±oth co±mplex ma∞nifolds related by a flip. This explains the choice
−
±
of the terminology ”flip passage”. More precisely, put P := P(E ), P := P(E )
′ ′ ′′ ′′
andletP (P )betherestrictionofP (P )toB . TheprojectivebundlesP ,P
′ ′′ ′ ′′ ′ +′′
are natur±ally±embedded in Qs. With these notati±ons we see that the fiber q −1(b )
f − +
is obtained fromthe fiber q 1(b ) by “replacing”P with P . We can state now
− − b′− b′+′
Problem 2. Let Z Qs be a pure k-dimensional analytic set, and [Z]BM
⊂ f ∈
HBM(Qs,Z) its fundamental class in Borel-Moore homology. Determine the ho-
2k f
mological boundary δ([Z]BM) H (P(F F¯ ),Z) of [Z]BM.
2k 1 ′ ′′
∈ − ⊕
The two problems Problem 1, Problem 2, are related by the the holomorphic
model theorem proved in [Te5], which we explain briefly below (see section 2.2 for
details). Coming back to our gauge theoretical framework, let λ ec(E) with
∈ D
a circle of regular reductions C , and fix c λ. The holomorphic model theorem
λ
∈
gives a system (p : E B ,p : E B ,f ) as above, a diffeomorphism Ψ
′c c′ → c ′c′ c′′ → c c c
betweena neighborhoodO of ∂Qˆ inQˆ anda neighborhood of in ˆpst,
c fc fc Oλ Pλ Mλ
where Qˆ is the blow up flip passage associated with (p : E B ,p : E
fc ′c c′ → c ′c′ c′′ →
Bc,fc). Moreover, Ψc induces a diffeomorphism ∂Qˆfc = P(Fc′ ⊕ F¯c′′)−≃→ Pλ, a
Pbic′h,′+olo⊂mQorsfpchoinsmto tOhce\ex∂tQeˆnfsci−o≃→n spOaλce\sPPλεc,, Panεc1d(Em)−acps⊂tMhestprreosjpecetcitvieveblyu.ndles Pc′,−,
We explain now our answer to Problem 2, which concerns an arbitrary blow up
flip passage Qˆ (see section 3.3). Let Θ, Θ be the tautological line bundles of
f ′ ′′
the projective bundles P :=P(E ), P :=P(E ), and denote by Q the total space
′ ′ ′′ ′′
of the line bundle p (Θ) p (Θ ) over the fiber product P P . We identify
∗1 ′ ⊗ ∗2 ′′ ′×B ′′
this fiber product with the zero section of Q. One has a natural biholomorphism
j :Q (P P ) Qs (P P ).
1 \ ′×B ′′ → f \ −′ ∪ +′′
Theorem 3.9. With the notations above, suppose r >0, r >0, and let Z Qs
′ ′′ ⊂ f
be an analytic subset of pure dimension k 1 such that dim(Z (P P )) < k.
Then ≥ ∩ −′ ∪ +′′
(1) TsuhbesectloosfuQrewZ˜itohfdj1i−m1((ZZ˜\((PP−′ ∪P+′P′)))i)n<Qki,s a pure k-dimensional analytic
′ B ′′
∩ ×
(2) Choosing a point x C, the equality
∈
δ[Z]BM =[C]⊗JEEx′x′′ [Z˜]BM ·(Px′ ×Px′′)
holds in H1(C,Z)⊗H2k−2(P(Fx′ ⊕(cid:0) F¯x′′),Z)=H2k−1((cid:1)P(F′⊕F¯′′),Z).
In this theorem JEx′′ :H (P(E ) P(E ),Z) H (P(E E¯ ),Z) denotes
Ex′ 2s x′ × x′′ → 2s+2 x′ ⊕ x′′
the join morphism defined in section 3.2 (see formula (17), Remark 3.7).
Our explicit applications concern the moduli space considered in our program
for proving the existence of curves on class VII surfaces [Te3]. Let (X,g) be a
class VII surface endowed with a Gauduchon metric, the canonical line bundle
K
ofX, K its underlying differentiable line bundle, (E,h)a Hermitianrank 2-bundle
on X with c (E) = 0 and det(E) = K, and st, pst the two moduli spaces
2
M M
associated with the data (X,g,E,h, ). pst is always compact [Te3, Theorem
K M
ANALYTIC CYCLES IN FLIP PASSAGES AND INSTANTON MODULI SPACES 5
1.11]. Moreover, assuming that deg ( ) < 0, X is minimal and is not an Enoki
g K
surface, it follows that st is a smooth manifold of dimension b (X), and all the
2
M
reductions in pst are regular [Te3, Theorem 1.3]. We suppose for simplicity that
M
H (X,Z)= Z, which implies that H2(X,Z) is torsion free. Let (e ) be a
Do1naldson basis of H2(X,Z), i.e. a basis which satisfies the conditiio1n≤si≤b2(X)
b2(X)
e e = δ , e =c ( ) .
i j ij i 1
· − K
i=1
X
[Te3, section 1.1]. Put I:= 1,...,b (X) , and for I I put
2
{ } ⊂
I¯:=I I , e := e .
I i
\
i I
X∈
Since c1(E) = i Iei, we have Dec(E) = {eI,eI¯} I ⊂ I , hence Mpst has
2b2(X)−1 circlesPof∈reductions. Using the nota(cid:8)tion intr(cid:12)oduced(cid:9)in Remark 1.1, we
have r = I¯. Therefore, putting k := 1deg ( ), we(cid:12)obtain, for every I ( I, a
eI | | 2 k K
preoIjeinctivestb.uInndplearPtiεecIul→ar P0iceisI(oXpe)n(k−inε,k) wsti.thThfiibseorpPen|CI¯|s−u1b,saentddeafinneesmabnedenddin(gthoef
P M Pε M
“known end”) of st. As explained before, in the general case, for a given index
M
set I ( I (including for ) is very hard to describe the Zariski closure of eI in
∅ Pε
st, even when X is a known class VII surface. The result below, proved using
M
the methods developed in this article, will be used in [Te6].
Proposition 4.2. Let X be a minimal class VII surface with H (X,Z) Z and
1
≃
b (X)=3, which is not an Enoki surface. Let g be a Gauduchon metric on X with
2
deg ( )<0. Then
g K
1. The component π ( pst) of 0 contains all four circles of reductions,
M0 ∈ 0 M Pε
2. Let i I. The Zariski closure of ei in st has pure dimension 2, does not
∈ Pε M
intersect 0, and contains the curve eI for a subset I I of cardinal 2.
Pε Pε ⊂
If a stable bundle is the central term of an extension of the form
E
0 0
∨
→L→E →K⊗L →
with c ( ) = e , we agree to say that is a stable extension of type I. The first
1 I
L E
statement of Proposition 4.2 shows that, for every I ( I there exists a family of
stable extensions of type which convergesin st to a stable extension of type I.
∅ M
Thesecondstatementshowsthatforanyi IthereexistsI Iofcardinal2,and
i
∈ ⊂
a family of extensions of type i which converges in st to a stable extension of
{ } M
type I . These statements cannot be proved using complex geometric arguments.
i
2. Flip passages and the holomorphic model theorem
2.1. The spherical blow up. Let M be an m-manifold, W M a closed r-
dimensional submanifold. The spherical blow up Mˆ of M at⊂W is a manifold
W
with boundary, which comes with a smooth map p : Mˆ M with the following
W
→
properties:
1. The boundary ∂Mˆ of Mˆ coincides with the sphere bundle NM /R ,
W W { W}∗ >0
where NM denotes the complement of the zero sectionin the normalbundle
{ W}∗
NM of W in M,
W
6 ANDREITELEMAN
2. One has p 1(W) = ∂Mˆ , and the restriction p : ∂Mˆ W coincides
− W ∂MˆW W →
with the bundle projection NM /R W,
{ W}∗ >0 →
3. The restriction p :Mˆ ∂Mˆ M W is a diffeomorphism.
MˆW\∂MˆW W \ W → \
We refer to [AK] for the construction and the functoriality properties of the
spherical blow up. We will give now explicit constructions for the spherical blow
up Mˆ in several situations which are of interest for us.
W
1. The case when W =Z(s) for a regular section s Γ(E).
∈
Let p : E M be a real rank r-bundle on M and s Γ(E) a section which
→ ∈
is regular (transversal to the zero section) at any vanishing point. In this case
the zero locus W := Z(s) is a smooth codimension r submanifold of M, and the
intrinsic derivative of s defines an isomorphism NM E . The spherical blow
W → W
up Mˆ can be obtained as follows: endow E with an Euclidean structure, and let
W
π : S(E) M be the corresponding sphere bundle. Then Mˆ can be identified
W
→
with the submanifold
sˆ:= (y,ρ) S(E) [0, ) ρy =s(π(y))
{ ∈ × ∞ | }
ofS(E) [0, ). Notethatthemap(y,ρ) ρy s(π(y))isasectioninthebundle
× ∞ 7→ −
(π p ) (E)overS(E) [0, ),andthissectionistransversaltothezerosectionof
1 ∗
◦ × ∞
this pull back bundle. Via the identification Mˆ =sˆ, the contractionmap sˆ M
W
→
is the restriction π p .
◦ 1 sˆ
2. The case when M is the total space of a vector bundle p : E B, and
→
W =Z(θ ), where θ Γ(E,p (E)) is the tautological section of E.
E E ∗
∈
The tautological section θ Γ(E,p (E)) of E is defined by θ (y) = y, where
E ∗ E
∈
the right hand term is regarded as an element of the fiber p (E) . The zero locus
∗ y
of θ is the zero section B E. This is a special case of 1., hence Eˆ = θˆ .
E B E
On the other hand θˆ can be⊂obviously identified with S(E) [0, ) via the map
E
× ∞
S(E) [0, ) (y,ρ) (ρy,y,ρ) θˆ . Therefore we get an identification
E
× ∞ ∋ 7→ ∈
Eˆ =S(E) [0, ) ,
B
× ∞
and, via this identification, the contractionmap S(E) [0, ) E is (y,ρ) ρy.
× ∞ → 7→
Example 2.1. Consider the inclusion map j : Z(mf) ֒ E of the submanifold
→
intervening in the definition of a blow up flip passage (see section 1, [Te5]). Then
j 1(B) = C and j induces a bundle isomorphism NZ(mf) = NE . This implies
− C B C
thattheblowupZ\(mf) canbeidentifiedwiththepreimageofZ(mf)intheblow
C
up Eˆ of E at the zero locus of its tautologicalsection. Since Eˆ =S(E) [0, )
B B
× ∞
(3) Z\(mf) = ((y ,y¯ ),ρ) S(E E¯ ) [0, ) mf(ρy ,ρy )=0 .
C ′ ′′ ∈ ′⊕ ′′ × ∞ | ′ ′′
(cid:8) (cid:9)
3. The case when M is the total space of a vector bundle p : E B, and
→
W =Z(θ ,p (s)), where s Γ(F) is a section of a bundle q :F B.
E ∗
∈ →
Supposes Γ(F)istransversaltothezerosection,andletW :=Z(s)beitszero
∈
submanifold. Then(θ ,p (s)) Γ(p (E) p (F))istransversaltothezerosection,
E ∗ ∗ ∗
∈ ⊕
Z((θ ,p (s)) canbe identified with W via p, andone has anobvious identification
E ∗
ANALYTIC CYCLES IN FLIP PASSAGES AND INSTANTON MODULI SPACES 7
NE =(E F) . In the same way as in 2. we get an identification
W ⊕ W
Eˆ = ((y,v),ρ) S(E F) [0, ) ρv =s(p(y)) ,
W
{ ∈ ⊕ × ∞ | }
and, via this identification, the contraction map Eˆ E is ((y,v),ρ) ρy.
W
→ 7→
2.2. The holomorphic model theorem. Let X be a surface with b (X) = 1,
1
p = 0. For such a surface the canonical morphism H1(X,C) H1(X, ) is an
g X
→ O
isomorphism, hence Pic0(X)=H1(X, )/H1(X,Z) C .
X ∗
Fixx X,andletL bethePoincaOr´elinebundlen≃ormalizedatx ontheprod-
0 0
∈
uct Pic(X) X. Let p :Pic(X) X Pic(X), p :Pic(X) X X be the two
1 2
× × → × →
projections. The H1(Pic(X),Z) H1(X,Z)-termofthe Ku¨nnethdecompositionof
c (L) can be interpreted as a m⊗orphism δ :H (X,Z) H1(Pic(X),Z).
1 1
→
Fix a Gauduchon metric g on X. For any c H2(X,Z), t R the level
sHet(XPi,cZc()X){Ht}1:(=Pic{cd(eXgg) Pic,cZ(X).)}W−1e(wt)illisalawaciyrscleen,dao∈nwdPδicicn(dXu)ces awni∈thisothmeobrpohuinsdm-
1 t t
ary orient→ation of ∂Picc({X}) . With this convention we g{et} an isomorphism
[t,+ )
H (X,Z) Z given by h ∞δ(h),[Picc(X) ] , which is independent of c
1 t
H2(X,Z),→t R and the Gau7→duhchon metric g.{T}hiis isomorphism defines a distin∈-
∈
guished generator γ H1(X,Z). Therefore, by definition, we have
X
∈
(4) γ ,h = δ(h),[Picc(X) ] h H (X,Z) .
X t 1
h i h { } i ∀ ∈
Let E be a Hermitian rank 2-bundle with c (E) 2H2(X,Z) on X, a holo-
1
6∈ D
morphicstructureondet(E), λ ec(E), andc λ. We define the harmonicmap
f : Picc(X) R by f ([ ]) :=∈πDdeg ( ) d),∈where d := 1deg ( ). The van-
c → c L g L − 2 g D
ishing circle C :=f 1(0)=Picc(X) can be identified with the reduction circle
C pst(E)cviathc−eisomorphism(cid:0)k{d:}C C givenbyk ([ ]):=[ ( )].
λ c c λ c ∨
Usin⊂gMtheDRiemann-RochandGrauert’sthe→oremswe obtain[TLe5, ProLp⊕osiDtio⊗nL3.3]:
Remark 2.1. Suppose C is a circle of regular reductions. There exists a Zariski
λ
open neighborhood U of C in Picc(X) such that the restrictions of the coherent
c
sheaves R1p L 2 p ( ) , R1p L 2 p ( ) to U are locally free of
1∗ ⊗ ⊗ ∗2 D∨ 1∗ ⊗− ⊗ ∗2 D
ranks
(cid:0) (cid:1) (cid:0) (cid:1)
1 1
r = (2c c (E))(2c c (E)+c (X)), r = ( 2c+c (E))( 2c+c (E)+c (X))
c′ −2 − 1 − 1 1 c′′ −2 − 1 − 1 1
respectively, and the fibers of the corresponding bundles at a point y U are iden-
tified with H1 L 2 p ( ) , H1 L 2 p ( ) . ∈
y⊗ ⊗ ∗2 D∨ y⊗− ⊗ ∗2 D
Therefore,f(cid:0)or sufficiently sm(cid:1)all ε(cid:0)>0, we obtain(cid:1)holomorphicbundles E , E of
c′ c′′
ranks r , r on the annulus B :=Picc(X) =f 1( πε,πε).
c′ c′′ c (d−ε,d+ε) c− −
The restrictions E , E can be endowed with natural Hermitian metrics,
c′ Cc c′′ Cc
by identifying them with suitable harmonic spaces [Te5]. Using [Te5, Remark
2.8] we obtain a well defined blow up flip passage, which will be denoted by Qˆ .
fc
Correspondingly, we put F := E , F := E , P := P(E ), P := P(E ),
c′ c′ Cc c′′ c′′ Cc c′ c′ c′′ c′′
P :=P , P :=P .
c′,± c′ Bc,± c′,′± c′′ Bc,±
Theorem 2.2. Under the assumptions and with the notations above there exists
an open neighborhood O of ∂Qˆ in Qˆ and a diffeomorphism Ψ : O onto
c fc c c → Oλ
a smooth open neighborhood of in ˆpst such that
Oλ Pλ Mλ
8 ANDREITELEMAN
1. Ψ induces a smooth bundle isomorphism ∂Ψ which fits in the commutative
c c
diagram
∂Ψ
P(Fc′⊕F¯c′′)=∂Qˆfc c✲ Pλ
(5) p
❄ k ❄λ Pλ
c✲
C C ,
c λ
2. Ψ induces a biholomorphism O ∂Qˆ ,
c c\ fc →Oλ\Pλ
34.. ΨDec(nPoct′−in)g=byPεcν(∩hO),λν,(Ψu)c(Pthc′e+′ )D=onPaεlcd1s(Eon)−ccl∩asOseλs,associated with h H (X,Q),
1
∈
u H (X,Q), we have
2
∈
1
(∂Ψ ) (ν(h))=δ(h) W , (∂Ψ ) (ν(u))= 2c c (E),u W ,
c ∗ c c ∗ 1 c
⊗ 2h − i
where W is the positive generator of H (P(F F¯ ),Z).
c 2 c′⊕ c′′
The first two statements are proved in [Te5], and the third can be provedeasily
using the construction of Ψ . The fourth follows from [Te2, Corollary 2.6].
c
3. The homological boundary of an analytic set Z Qst
⊂ f
3.1. The real blow up ofaBorel-Moorehomologyclass and itsboundary.
WestartwithabriefaccountonthewellknownintersectiontheoryinBorel-Moore
theory(seeforinstance[BH,section1.12])inaveryparticularframework. Sincein
our results the signs play a crucial role, and in the literature one can find different
conventions for the relevant objects intervening in our formulae (the cap product,
the orientation of the normal bundle of an oriented submanifold of an oriented
manifold, the Thom isomorphism), we will write down carefully the formulae we
need.
In general, if M is a differentiable manifold, and Y M a submanifold of M,
⊂
we will always use the direct sum decomposition T = T NM (defined by
M Y Y ⊕ Y
a Riemannian metric on M) to orient anyone of the three objects M, Y, NM us-
Y
ing orientations of the other two. In particular, the total space E of an oriented
Euclideanbundle overanorientedbaseY isorientedsuchthattheobviousisomor-
phism T = T E is orientation preserving. Note that [BT] uses a different
E Y Y ⊕
convention.
For an oriented r-dimensional Euclideanvector space F we denote by B(F) the
open unit ball of F, by [F,F B(F)] the generator of H (F,F B(F),Z) defined
r
\ \
by the fixed orientation, and by F,F B(F) the corresponding generator of
{ \ }
Hr(F,F B(F),Z).
\
Letp:E Y beanorientedEuclideanrankr-bundleoveraconnected,oriented,
→
closed manifold Y. We denote by Φ the section in the local coefficient system
E
x H (E ,E B(F ),Z)
r x x x
7→ \
given by x [E ,E B(F )], by B(E) the unit ball bundle of E, and by φ
x x x E
7→ \ ∈
Hr(E,E B(E),Z) its Thom class. For any k N the morphism
\ ∈
(6) τE :H (E,E B(E),Z) H (Y,Z) , τE(u):=( 1)krp (φ u)
k r+k \ → k k − ∗ E ∩
is an isomorphism[Sp, Theorem 5.7.10]. In this formula, on the right, we used the
signconventionof [Do, Section VII.12] for the definition of the cap product, which
ANALYTIC CYCLES IN FLIP PASSAGES AND INSTANTON MODULI SPACES 9
doesnotagreewiththeconventionsusedin[Sp], [BH]. Thefactor( 1)kr hasbeen
−
inserted in order to recover the standard isomorphism
h [F,F B(F)] h
⊗ \ 7→
in the case of the trivial bundle E =Y F p1 Y. Having this case in mind, put
× −−→
h Φ := τE 1(h) .
⊗ E { k }−
We will identify H (F,F B(F),Z), H (E,E B(F),Z) with the Borel-Moore
homologyHBM(B(F∗),Z),\HBM(B(E),Z∗)ofF an\dErespectivelyviathecanonical
isomorphism∗s ∗
H (F,F B(F),Z) H (B¯(F),S(F),Z) HBM(B(F),Z) ,
∗ \ → ∗ → ∗
H (E,E B(E),Z) H (B¯(E),S(E),Z) HBM(B(E),Z)
∗ \ → ∗ → ∗
[BH, Section 1.2]. Via the first identification, [F,F B(F)] corresponds to the
\
fundamental class [B(F)] of the oriented manifold F in Borel-Moore homology,
hence Φ corresponds to the section B of the local coefficient system
E E
x HBM(B(E ),Z)
7→ r x
given by x [B(E )]. Therefore the isomorphism τE induces an isomorphism
7→ x k
τB(E) :HBM(E,Z) H (Y,Z), and we can put
k r+k → k
h B := τB(E) 1(h) .
⊗ E { k }−
By our orientation convention for E (and implicitly B(E)), the fundamental
class ofB(E) in Borel-Moorehomologywill be givenby [B(E)]=[Y] B . With
E
⊗
these conventions,the morphismp :H (Y,Z) H (B(E),Z) commutes withthe
∗ ∗ ∗
→
Poincar´eduality isomorphisms on Y and B(E), i.e. one has the identity
(7) p∗(b) [B(E)]=(b [Y]) BE b H∗(Y,Z) .
∩ ∩ ⊗ ∀ ∈
Definition 3.1. Let M be a connected, oriented m-dimensional manifold. For a
class c HBM(M,Z) and a closed, oriented l-dimensional submanifold W M
∈ k ⊂
we define the homological intersection of c with W by
c W :=j (b) [W] H (W,Z) ,
· W∗ ∩ ∈ l+k−m
where j : W ֒ M is the inclusion map, and b Hm k(M,Z) is the Poincar´e
W −
→ ∈
dual of c, i.e. one has c=b [M].
∩
Remark 3.2. In the conditions of Definition 3.1 it holds
1. If c is the fundamental class of a closed, oriented submanifold Z M which
⊂
intersects W transversally, then one has
c W =( 1)(m l)(m k)[W Z] ,
− −
· − ∩
whereT =W Z isregardedasasubmanifoldofW endowedwiththeorientation
∩
induced by the orientation of W, and the orientation of NW given by the
W Z
natural isomorphism NW =NM . ∩
W Z Z W Z
2. If M is a complex manifo∩ld, W a com∩plex submanifold, and c the fundamental
class of an k-dimensional analytic subset Z M such that W Z has pure
⊂ ∩
dimension l+k m, then c W is the fundamental class of the analytic cycle
− ·
W Z of W.
∩
10 ANDREITELEMAN
The unit sphere S(F) of an oriented Euclidean space F is given the boundary
orientation of the closed unit disk B¯(F), i.e. the orientation for which the obvious
isomorphism Rη T T is orientation preserving, where η is the outer
⊕ S(F) ≃ F S(F)
normal field of S(F). The sphere bundle S(E) of an oriented Euclidean bundle
p:E Y is oriented using the same rule fiberwise. We denote by S the section
E
→
of the local coefficient system x H (S(E ),Z) given by x [S(E )]. Using
r 1 x x
7→ − 7→
the Leray-Hirsch theorem again, and putting n := dim(Y), we get as above an
isomorphism
H (S(E)) H (Y) H (Sr 1,Z)
n+r 1 n r 1 −
− ≃ ⊗ −
andanidentification[S(E)]=[Y] S . UsingtheBorel-Moorelongexactsequence
E
⊗
HBM(S(E),Z) HBM(B¯(E),Z) HBM(B(E),Z) ∂ HBM(S(E),Z)
···→ j → j → j −→ j−1
associated with the open embedding B(E) ֒ B¯(E) [BH, Section 1.6], and com-
paring it with the long exact sequence of the→pair (B¯(E),S(E)), we get
(8) ∂(h B )=( 1)sh S h H (Y,Z) .
E E s
⊗ − ⊗ ∀ ∈
In particular ∂[B(E)]=∂([Y] B )=( 1)n[Y] S =( 1)n[S(E)].
E E
⊗ − ⊗ −
LetM beanorientedRiemannianm-dimensionalmanifoldandW M aclosed,
oriented l-dimensional submanifold. Let Mˆ be the spherical blow u⊂p of M with
W
center W. If we omit orientations, the boundary ∂M can be identified with the
W
sphere bundle S(N ) of the normal bundle N of W in M.
W W
Letc HBM(M,Z)beak-dimensionalBorel-Moorehomologyclass. Theimage
∈ k
cW := c of c under the canonical morphism HBM(M,Z) HBM(M W,Z)
M\W k → k \
can be regarded as a k-dimensional Borel-Moore homology class of the interior
Mˆ ∂Mˆ of the manifold with boundary Mˆ . Our problem is to compute
W W W
\
explicitly the boundary δ(cW) HBM(∂Mˆ ,Z) in terms of topological invariants
of the triple (M,W,c). ∈ k−1 W
LetK beacompacttubularneighborhoodofW,denotebyU itsinteriorandby
c , c the images of c in HBM(U,Z), HBM(M K,Z) via the canonical mor-
U M\K k k \
phisms. TheBorel-Moorelongexactsequenceassociatedwiththeopenembedding
U (M K)֒ M contains the segment
∪ \ →
HBM(M,Z) HBM(U (M K),Z) δ H (∂K,Z) ... ,
···→ k → k ∪ \ −→ k−1 →
which shows that
δ(c )+δ(c )=0 .
U M K
\
Using the obvious identification HBM(∂Mˆ ,Z)=HBM(∂K,Z), we obtain
W
∗ ∗
δ(cW)= δ(c ) .
U
−
Write c = b [M], where b Hm k(M,Z) is the inverse image of v via the
−
∩ ∈
Poincar´e duality isomorphism. Denoting by j : U ֒ M, j : W ֒ M the
U W
→ →
embedding maps, by pU : U W the projection map, and using formula (7), we
W →
get
cU =jU∗(b)∩[U]=(pUW)∗(jW∗ (b))∩[U]=(jW∗ (b)∩[W])⊗BNW =(c·W)⊗BNW .
Therefore, by (8) we get δ(c )=( 1)l+k m(c W) S . This proves
U − − · ⊗ NW
Proposition 3.3. Under the assumptions and with the notations above one has
(9) δ(cW)=(−1)l+k−m+1(c·W)⊗SNW .