GROUP ACTIONS ON DONALDSON INVARIANTS

Kyungpook MathematicaJ Journal

Vo1ume 3'2, Number 3, Octobel", 1992

GROUP ACTIONS ON DONALDSON INVARIANTS

Yong Seung Cho

Dedicated to Professor Younki Chae on his 60th birthday

1.

Introduction

Let M be a smooth

,

closed

,

simply conneded 4-manifold. Let a group G acts on M as isometries. Suppose that E.:'...M is on SU(2) vector bundle and the G-action on M lifts to the bundle E such that π is a G-equivariant map. In [8] Finashin

,

Kreck and Viro studied on an따lαt“，-바-

!t씨u따Jt띠tior이on on the Do이19ach따lev surfaces. They got non-diffeomorphic but home- omorphic knottings in the 4-sphere. In [9] Fintushel and Stern studied on the equivariant moduli space of self-dual connection. They have infinite cobordism group of homology 3-spheres. In

[2]

and [3] we studied on fioite group actioos on the moduli space of self-dual connections. We have some cohomology obstruction classes to get a smooth G-moduli space. ln [6]

Donaldson gave a comment on anti-holomorphic iovolution on I<3-space.

The G-buodle E즈.M induces an SU(2) orbit bundle E,'éJ.N where N is the G-orbit space of M. The moduli space of gauge equivalence classes of anti-self-dual connections of the G-bundle E is also a G-space^‘ Io this paper we would like to study the relations between the G-moduli space and the moduli space for the orbit bundle E_{1 .} Roughly speaking

,

Donaldson invariants are just the cohomology pairings on the moduli spaces. We would like to study the Donaldson invariants on the G-moduli space and the moduli space obtained from the orbit bundle aod their relations

R.eceived April 17, 1992

This research was supported by Korea Science and Engineering Foundation 321

1n section 2 we introduce simple examples of algebraic surfaces. Dol- gachev surface

,

fibersum and logarithmic transformation

1n section 3 we introduce Finashin, Kreck and Viro’s results on anti- holomorphic involution on Dogachev surfaces and anti-holomorphic invo- lution K3-surface. We study the topology on the orbit spaces. [n section 4 we study equivariant geometry on vecter bundles

,

the space of connec- tions and the moduli space of anti-self-dual connections. 1n section 5 we study involution on K3 surface. The involution induces an involution on the muduli space of anti-self-dual connections. We compute the dimen- sion of the invariant moduli space. We show that the moduli space on the orbit bundle can be identified one of the components of the invariant moduli space. 1n section 6 we show that Donaldson μ-map is functorial to the projection from given space onto the orbit space. We introduce the determinant line bundle which induces the μ-map. We compute a curva- ture form of the line bundle as Lie algebra valued 2-form^‘ 1n sect ion 7 we introduce the results of Friedman and Morgan about homeomorphic but not diffeomorphic examples. FinaUy for an anti-holomorphic involution on a !{3 surface

,

we show that the Donaldson invariant on the moduli space obtained from the orbit bundle can be computed by the fixed point set of the G-moduli space

,

and that the anti-holomorphic involution preserves the Donaldson invariants

2. Algebraic Surfaces

The simple examples of a smooth

,

c1osed

,

simply connected algebraic surfaces are Cp2

,

S2 ^X S2 an the K3 surface. They are zero sets in CP3 of polynomials of degree 1

,

2 and 4 respectively. An elliptic surface is a compact complex surface S which is the total space of a holomorphic fibration S.!'...C onto a Riemann surface such that the generic fibres of π are complex tori. As an example we would like to introduce the simplest elliptic surface. We start with a generic pencil of cubic curves on Cp2 Let

f ,

^and

^h

^be a pair of homogeneous cubic polynomials on C3 in gen- eral position. For each point x [xo

,

xd E CP' in the homogeneous coordina.tes

,

let Zx be the zero set of the polynomial xof

, ⁺

^{Xd2 in CP2}

Then Zx is a smoothly embedded torus in CP2 except for finitely many values of x E CP'. Let Z be the intersection of the zero sets of

f ,

^and

h .

Since

f ,

^and

^h

^are cubic polynomials in general position, the set Z is the set of 9 distinct points. The set Z is a subset of the zero set Zx for all x E CP'. For each point p E CP2 - Z there is a. unique point

Group Actions on Donaldson Invariants 323

x E Cpl such that p E Zx. If we blow up the 9 points in Z, the zero sets Zx are disjoint. We have an elliptic fibration CP²tt9δP깐， Cpl given by π (Zx)

=

x. Let Dl

= CP개9δF2-

Let N C D_l be the inverse image of π of a small open 2-disc in

cp

^l. Let rþ : 8N • 8N be a fiber-preserving

,

orientation reversing self-diffeomorphism where 8N is the boundary of N which is diffeomor- phic to T² X SI. If we glue D_{l \} N and DI \ N along thier boundaries via the map rþ. This is called a fiber sum which is not holomorpbic processes in general. Since DI bas Euler characteristic x(Dd = 12 and signature a(Dd

=

-8

,

we have X(D2 )

=

24 and a(D2 )

=

-16. In fact the fiber sum D2 is diffeomorphic to the f(3-surface. The fiber sum turns out to be independent of all choices.

We introduce another operation on an elliptic surface S • C which is called logarithmic transformation. In the smooth category let N be the inverse image of a small neighbourhood of a generic point in C as before. We delete N and glue T² x D² back in by some diffeomorphism rþ:T² xSl • 8N. The multiplicity is defined to be the absolute value of the wir비 ng number of π 。 rþ: poin t x SI • π (8N) ε SI. In the holomorphic category for a generic point p E C the fiber Sp is a complex torus. Let 6., be a small coordinate disc about p in C. Over 6., our fibration is isomorphic to a fibration X C x 6./ L where L (Z

+

Z，μ이‘

a k E N with (k

, m)

1 and form the smooth quotient X' of C x 6../(Z

+

Zω (sm)) by the cyclic group of order m generated by (z

,

s) • (z

+

k/m, ehi/ms). We define X' • 6., by(z,s) • sm

=

t. We have an elliptic fibration X' • 6., which has only one singular fìber X o' at 0, of multiplicity m. The map given by (z

,

s) •• (z - k/2πi log s

,

sm) induces a bibolomorphic fiber preserving map h : X'\zoI • X\Xo. If we identify X

’\

Xo' with the ^{Slð\{이 •} A\{O} C C via h

,

then the union (S\Sp) U X o' becomes an elliptic fibration S' • C with the two properties S

’\

^Sp’

biholomorphic to S\sp and sp' has multiplic띠 m. Some I<3-surfaces admit elliptic fibrations.

Let p and q be relatively prime integers and X(p

,

q) denote the man ifold obtained from a 1<3-surface X by logarithmic transformations of multiplicities p and q. If p and q are both odd

,

then X(p

,

q) adimits a spin structure. Applying Freedman c1assification theorem for simply con nected

,

c10sed topological 4-manifolds

,

we have

Theorem 2.1. 1f X(p, q) is obtained from a K3-surface X by Jogarithmic

tmr페ormations of multiplicities p and q

(1) X (p

,

q) is homeomorphic 10 the K3-surface X if p and q are odd.

(2) X(p

,

q) is homeomorphic to

3CP재19EF2

otherwise-

3. Involution on AIgebraic Surface

Let c be the standard antiholomorphic involution on CP 2. The fixed point set of c is the standard embedding of Rp2 into 54 with normal Euler number -2 in CP 2

I

c ~ 5" There are antiholomorphic involution con the Dolgachev surface D

,

(2

,

2k

+

1) with fixed point set F(k)

=

~10Rp2

with normal Euler number 16 is obtained by taking the connected sum (54

,

RP2) ~9(-5<， Rp2)

=

⁽⁵

<,

F(k))

Theorem 3.1. (Finashin

,

Kreck

,

Viro) (1) The knotiings (5

<,

F(k

,))

^and

(5

<,

F(k2)) are not diJJeomorphic f07' k

, ^i-

k2.

(2) F(k) = ÜlO(Rp2).

(3) π1 (54\F(k))

=

Z2'

(4) The n01'1nal Eu/e1' numbe1' with local coefficients of F(k) in 54 is 16

(5) The knotlings (5

<,

F(k)) are homeomorphic.

More generally let us start with a complex algebraic surface X defined over the real numbers. So there is an antiholomorphic involution a : X • X with fixed p이 nt set a real form XR of X which is a real algebraic surface. Let Y be the quotient space X \σ Since the fixed point set has real codimension 2

,

Y is a smooth manifold.

Lemma3.2. (l)IfXiss잉m때r

p/y connecled since the nontrη11νVI끼IU띠/loop in Y can be lifted to be a nonlrivial loop in X.

(2) b"(Y)

=

^Pg(X).

(3) 2X(Y)

=

X(X)

+

X(XR )

(4) The manifold X caη be recovered by the double cover ofY

,

branched

oνer the fixed point set XR

Theorem 3.3. (Donaldson) Let (5

,

a) be a /(3 surface wilh anlih%mor- phic involution a. Then

51"

is one of slandard raliona/ surfaces 52 X 52 01' Cp2~nCp2.

Proof Yau’s solution of the Calabi conjedure gives a a-invariant Kähler metric on 5

,

compatible with a family of complex structures. With respect to one such complex structure

,

^J

,

the map a is a holomorphic involution

Group Actions on Donaldson Invariants 325

S. ln fact this complex structure is orthogonal to the original one present in our explicit complex description of S. Then J induces a complex struc- ture on the quotient space Y = S/a

,

such that the projection map is a holomorphic branched cover. But it is a simple fact from the complex surface theory that a K3 surface is a branched cover of a surface Y than Y is a rational surface.

Remark 3.4. By the argument of Finashin

,

^{Kreck and Viro}

,

^{the quotient}

of again a loganthrMdlffeomorphIC to one of these standard manifolds

、TVe

^{get knotted}

complex curves in this manifold

,

that is

,

embedded surfaces homologous to a complex curve of the same genus

,

but not isotopic to a complex curve

Let a be the standard antiholomorphic involution on a K3-surface X.

Then the fixed point set XR is a real algebraic variety o[ real dimension 2.

Let the Kähler metric be h = ε hI3dz’ 0dε} in the theorems. The Kähler from is ~ ε h펴z'

^

^{dz}. Since π :X • Y}

^,

^Y

=

X/a is a holomorphic branched cover with branch set λ R.

If p E X \ X R, then clearly there is a neighbourhood U of p such that rr lu is holomorphic (b‘). That is π is double cover on the subset X \ X_R

,

π

local diffeomorphism and 1r‘h = h on Y \ X_{R .}

If p E X_R, then there is a neighbourhood U of p in X such that

U " , U

,

^{X U2 where a is trivial on U1}

^,

^{U2 '" D2 and a} is the reflection with

respect to the origin. Thus

U /a '" (U

,

^{x U2)/a ε U}

,

^{x (U2}/a) '" U

,

^{X U2}‘

Here we can choose a(U) U by taking U

=

V

n

a(V) where V is a neighbourhood of p

Let G = {l

, a}

be the group of order 2. The intersection [rom of X is

-2Es EÐ

³ 101

I

^{~ ~ and is posi뼈 on 딱 (X) 은 Z3 and is} negative on H"!.(X) '" Z19. By Lefschétz-fixed point theorem.

Theorem 3.5

ign X\G

= ]」 ε

^{(trglm - trgIH') =}

식(3

^{- 19)}

+

^{(traIH+ - tralH-)].}

|G| aeG + 2

By G-signature theorem for the involution a :

X

•

X.

sign (a

,

X)

=

tral악 (X) ^- trak(x)

=

^sign(X)2

=

^{sign (X}R)2

where X_R is a real 2-dimensional surface in X. X융 is a zero dimensional submanifold of X consisting of finitely many isolated points with sign.

Thus we have

Theorem 3.6. sign (Y)

=

^{상 16}

+

sign(XR)2]. In particular

,

If Y

=

S2 x S2

,

theη sign (Y)

=

^{0 and sign(X}R

J2 =

^{16. IfY}

=

Cp2~nCp2 ， then sign (Y)

=

^1-n

=

^![-16

+

sigη (X_R

J2]

and sign (X_R)2

=

^{2(1 - n)}

+

^16.

4. The Equivariant Geometry on Bundles

Let G be a finite abelian group. Let G act on a smooth

,

closed

,

simply connected 4-manifold Af and lift to an SU(2) vector bundle π :E • M on which π is a G-map.

Choose Riemannian metrics on λ1 and E on which G acts as isometries A Riemannian connection is a linear map \7 !ìO(E) • !ì¹(E) which satisfies

\7(fa) d

<

a

,

T

>

df 0 a+ f\7a

<

\7a

,

T

> + <

a

,

6 T

>

where

f

^E C∞ (M) and a

,

T E !ìO(E) and

<, >

^is the given Riemannian metric on E. There are Riemannian connections on E. Let C be the space of a1l Riemannian connections on E. 1n fact C is an affine space as a model space !ì1(adP) where adP is the associated Lie algebra bundle of E. For

\7 E C let F'" E !ì2(adP) be the curvature of \7. Let

ç

be the gro때 of gauge transformalions of

E

which are the sections of the associated Lie group bundle by lhe adjoinl action of SU(2) on itself. The group

ç

^acts

on by g(\7)

=

^{9 0 \7 0 g-1}

Let B = C /

ç

be the orbit space and π :C • B be the projection. Let

*

be the Hodge star operator on the oriented Riemannian 4-manifold Jν

A connection \7 E C is said anti-self-dual if *F'"

+

F'" =

o.

Lemma 4.l. For a때 \7 E C and a때 h E G

,

ψe have h(\7) E C.

Proof Let v E TM

,

σ， a1 ， a2 E !ìO(E) and f E C∞ (M). h(\7)(fa) = h [\7h- 1o(fa)oh]

=

^{h[\7 fo(h-1。σ。h)]}

=

^{h(df0h-1oaoh)} 十 h (f\7 h-¹oah) = df 0 σ

+ J[

h( \7)σ]

Compatibility of lhe connection h(\7) with the Riemannian structures

<

h(\7)v^O"], a2 > E"

+ <

이， h(\7)v^0"2 >E"

=<

h( \7h~lvh-] 0 0"0 h)

,

a2 >E"

+ <

^이 ， h( \7 h~lvh-1 0 0"2 0 h) >E"

Group Actions on Donaldson Invariants 327

=< '\1h-;' vh-1

ah,h- 1a2 >Sh-I(.)

+

< h'a, '\1h-;lv(h-1

ah) >Sh-I(r)

byhisisometryonE=< '\1vh-lah,h-lah >Sr

+

< h-1a,h,'\1vh- 1a2h) >Sr

by h is isometry on M = v < h-1a

,

h

,

h-1a2h >s.= v < a" a2

>

by isome tries on M and E

Let

A

be the subspace of C consisting of all anti-self-dual connections on E. The action of ç on C can restrict on A. The orbit space A/Ç 드 M is called the moduli spa.ce of the ga.uge equivalence class of anti-self-dual connections on E. The action of G on the bundle E • M induces on action of G on C. lf a E nO(E) and h E G then we define h(a) = hoaoh- I, where h-¹ is an action on M and h is an action on E as a bundle map. If v E TM, a.nd '\1 E C, then we define h('\1)v(a) = h( '\1h~lvh-l ⁰ a ⁰ h)

Finally we define an action of G on nk(adP) by (h</>)VI"'Vk h。

φ-1 _ ^L-l h .. VI, ...• h.Vk '-' ’‘

Lemma 4.1. Since G act on M as isometπes， the action oJ G on C preserves A. The set oJ invarianl connections oJ E denoles C^G

=

^{{'\1 E}

Clh('\1)

=

'\1} and Ihe invarianl anti-self-dual connections A G

=

CG

n

A.

The G -equivmianl gauge grollp denoles çG

=

^{g E Çlhg

=

gh Jor all h E G}. Then çG acls on C^G and A^G Let B G

=

C^G /çG and λ1^G

=

AG/ÇG and nk(adp)G be Ihe G-invaπanl subspace of n( adP). Conneclion '\1 on E induces a connection '\1 on adP by f07' </> E

n

^{O( adP)} and J07' a E nO(E)

('\1</>)(a) = '\1(</>a) - </>('\1a).

Then we have Ihe following immediate consequences.

Lemma 4.2. lf'\1 is a G-invarianl anli-self-dllal conneclion .

... n ... d^V

(1) 0 •

n

^{O( adp)G ~n' (adp)G}

==+n!

^{(adP) •}

o

^{is an} ellipiic complex

(2) F^{'l E} n 2(adp)G.

(3) ç강 1 acls on

cf

^and

A f

wheπ

q

is the Sobolev-complet'ion {'\1

+

AIA E L

k(

Q'(adp)G} by the Lz-norm.

Rema>'k

4-

3. If the hundle E • M is a.n SO(3)-bundle. Let π :C • B be the projection. Then

B G* CG*jçG = π (C^G)*

M G• AG*jÇG = π (A^G)*.

Proof Suppose \1 is a G-invariant irreducible connection and \1' = g(\1).

For each h E G

,

h(g(\1))

=

h(\1')

=

g(\1)

=

g( h(\1)) and g-l h-¹ gh(\1) =

\1. g-l h-¹gh is an element of the isotropy subgroup

r"

^{of the gauge}

transformation group

ç.

Since the connection \1 is irreducible

,

the gauge transformation 9 is G-invariant

5. Involution on K3 surface

Let an anti-bolomorph.ic involution a on a K3-surface X lift to an SU(2) vector bundle E파 X on whicb π is a-equivariant.

Remark 5.1‘ Consider the following diagram E

、

--• T ^o’E

/

---->

X 주→ X

τ’ E^， = E --•

/

"•

J*

E2=Et --• E2

Y=X

\o

^de-g-f-=-

,

^k S4

The involution a X • X can be lifted to Et • E

,

by chooseing an isomorpbism T E • 。*E. Since f bas degree k

, <

^c2(E

,) ,

Y

>=<

C2U* E2)

,

Y

>=<

j"c2(E2)

,

Y

>= <

_c2(E2)

, J*

Y

> = <

C2E2

,

kS'

>=

k

,

and since π is a branched double cover with codimension 2 branch set

<

c2(E)

,

X

>=<

π 'C2E" X

> = <

C2E" π.X

>=<

^C2E

,,

2Y

>=

^2k.

The involution a on the [(3 surface X is an orientation preserving isometry δ acts on tbe set C of connections for E • X and acts on the group

ç

of gauge transformations

,

via

,

^ô'(\1)

=

^{δ?δ-1 and δ (g)}

=

^{δgδ- ，}

for \1

E

C and 9

E ç.

Since ô' is an isometry δ acts on the set of anti-self- dual connections.

Lemma 5.2. (1) δ acts B

,

and M

(2) Let Mk,x be the moduli space of equivalence classes of anti-self- dual connections ψv띠’ u

ofaα‘ Then

dim kt 2k,x

=

2 dlIn kik,Y

Proof of (2)

dimM 2k^,x = 8(2k) - 3(1

+

b+(X)) since b+(X) = 3

= 2(8k - 3(4)) since b+(Y) = 1

= 2[8k - 3(1

+

^{b+ (Y))]}

2dimM^k.Y

Group Actions on Donaldson Invariants 329

Theorem 5.3. The modu/i space M k.y on the orbit space

Y

can be iden- tified one oJ the components oJ the a-invariant modu/i space M~k> X on X.

Proof Consider the fundamental elliptic complex

'l""7 d?

0 • f!

U(

adpr ":::"f!μdP) 。그 f!~(adP)。 • 0

where V E λ1~K.X' By Lefschetz fixed p이 nt theorem

dim

쩌K.X =

ind( 8^'V

+ 앵) = A ε

^L(a

^,

^D)

‘ ueG

where D : T(V+0\ι 0adPc^{)。 •} T(V+0 V+)

’

is the twisted Dirac operator and ad깐 = adP 0R C. Let V : T(V+ 0 ι) • T(V+ 0 ι)u.

L

(1 ,

D)

=

P

,

(adP 0 C)[X]

+

3indV

2k8 - g[2(X(Y) o(Y)) - (dI

냉 )]

2 따 한 ^(Y)

^{o(Y)) -}

3(dz ^따 )]

2dimMkY - 3(dx - 따)

Since a acts trivilly on adPIXR .

Thus dimM값，X

= W ^{(l ,}

^D)

+

L(δ， D)]

=

^{dim Mk}

^,

^y

Remark. When we compute indu(D) iI X is non orientable, twisted coef- ficients are used.

Let π X • Y ⁼ X/a be the projection and let A(E,) be the space of connections with c2(E) ⁼ 2k

,

and let A(Ed be the space of the connec야tμ，on n

‘w，씨l“tμh c2(E，샤)

=

k and E ⁼ π 'E，

As before choose a a-invariant metric 9 on X

,

then π induces a singular metric 9' on Y such that 9 = 7r'(9d. We would like to pull back the connections; ^7r' A(Ed • A(E) by setting for V E A(Ed and 'P E f!O(E)

,

and v E TX

,

π’ (V) v4i = V ~.'ψ( 7r) since E ^π. E

,

ø(πIS a section of E

,

^{• Y}

For any function

J

E C∞(X)， we extend the definition of π‘ (V) to be a connection on E • X

π'(V)v(J ø) = v(J)φ +Jπ 'Vvø

where π·’ (\1v^)tþ = \1~'v(tþ1f). By construction π* (1* = π* and '1r.(J>t< = π.，

0"(π -(\1))vtþ = 0"(π'( \1)U_VO"tþ(1 -1) = (1(\1 ~_Vtþ.(π)= π’ (\1 )vtþ is contained in the inva따nt subspace A(Et. Thus we complete the proof of Theorem 5.3

6. Donaldson Invariant on K3-surface

For any a E Hz(X : Z), we choose an embedded oriented surface ε

representing a. For π

*

(a) =

ß

E H₂(Y : Z)

,

we choose an embedded oriented surface ε1 representing β Let N and N1 be small tubular neigh- borhoods of ε and ε1 respectively. We may assume that π(ε) = εl

Let ^{T); :} B(X) • B(N) be the restriction map and ^T);

, :

B(Y) • B(N1).

For A E M k.Y

,

consider the twisted Dirac operators over N and N1 :

r(V);- 129E)

r(V;ε; 129 E1 ) ð ••

--+

--ðA •

r( \Ií강 129 E)

r( v;강

k

129E,)

For any ^(0" 129 s) E r(V;딩 129 E

d.

δπ'^ 7r-( ^0" 129 s) ^{8^(π‘(1 129 π'8) = ~π' (1 129 π'8}

+

π ‘(1 129 π‘ (A)(π‘ 8) π-(~ ，(1 129 8

+

(1 129 A(S))

π '8(A)((1 129 s), since πË，(1) = 8E(π’ 0").

Thus the inclusion map 1f' Mk,Y • M 2k,x from Mk,Y into the in- variant anti-self-dual connections o[ λIÍzk^，x induces a bundle map on the determinant line bundles

L);

, _--’

L);

M k,Y --”-.-- M 2k,y.

There is a universal bundle E over X x

c' /9

with its Cbern class c2(E) E H⁴(X X

c* /9).

For any class ε in H2(X)

,

we have a map μ : H2(X) •

H²(C‘

/9)

which is defìned by the slant prodl따 μ(ε) = c2(E)/ ε Lemma 6.1. Given ^OUT tω o fold branched cover π :X • Y

,

ψe have the following commutalive diagl^‘'am

Group Actions on Donaldson Invariants

H2(X)

H

2

(Y)

즈즈• H2(M2k,X)

~ H²(Mk

k

^,y).

331

Proof For any a E H2(X)

,

let ε be an embedded oriented surface repre- senting a. Since the inclusion map π*: Mk^,y • M 2k^{,X induces a bundle} map and μx(a) = c

,

^{(Ld and π‘ (E) = ε1 ， we have}

π*μx(a)

=

π*(c， (LE^{)) = Cl(π‘ (L}E^{)) = c，(Lι)}

= cl(LnE

J

^{= μ y(π}

* (a)).

Given homology classes a[ ... ad E H₂(X Z)

,

we present them by embedded surfaces ^{εI εd} in X in general position such that any triple intersections N‘ ,

n

N_j

n

Nk of smalJ t u bu ular neighborhoods of ε ， εj and

εk respectively are empty. There is a determinant line bundle LE over Bx with a section whose zero set V_E is a codimension 2 submanifold of Bx and meets alJ of the moduli spaces λι for i

<

k transversalJy. If 4k> 3(1

+

bt)

,

then the intersection VE

, n ... n

V_Ed n.ιtk is compact Definition 6.2. For any homology classes a". ^‘ ad E H₂(X

,

Z)

,

tbe Donaldson invariant is deJìned to be

qk서ll't， '"

,

Qd) =

<

μ(atlU"'U μ (ad)，[Mk，xl >

= Ü(\숫，

n ... n

^VEd

n

^Jvt kλ)

where tt denotes a count witb signs

Rema'rk 6.3. Suppose that 91

,

g2 are two different metrics on the base 4- manifold X. The moduli space M (gtl and M(g2) of gauage equivalence classes of anti-self-dual connections with respects to the metrics 9

,

^{and 92}

respectively differ by a boundary in B*. The pairing is independent of the choice of metric. Thus Donaldson invariants are smooth invariants

Let ω be the Kähler class in H²(X : C). By the Hodge index H~= ωH²•^O and bt

=

1

+

2Pg.

We Jìx the orientation of H; by ω^ (complex orientation of H²,O(X C) which speciJì얹 the orientation of the moduli space. Let H be the hyperplane class in H2(X Z) which is the Poincaré dual to ω. As a real manifold H is a compact Riemann surface. Over a small tubular

manifold H in 5

,

we have

,

for each connection A

,

a coupled Dirac operator ÔH : r(V- 0 E) • r(V+ 0 E). We may construct the determinant line bundle LH by using the index of tbe Dirac operators ÔH over Br..

LH = Am'''(kerÔH)* 0 A'뼈‘ (kerôÎf ).

The determinant line bundle LH descends to a bundle over λ1H^. In fact

ι1H is a complex manifold and eacb connection A defìnes tbe associated ô-operator : ÔA ^‘ Ç10(E) • Ç10"(E) and defìnes a holomorphic structure on E. Thus we have the holomorphic line bundle LH • λ1H^.

Theorem 6.4 (Donaldson) (1) The deteπninant line bundle L^/1 • },,1H is an ample line bundle

(2) μ(0') = c

,

^(Lr.) E H²(M.)

,

^{ψheπ Lr; is} the pull back of LH on Mx and ε is a representative oJ[O']

E

H₂(X : Z).

In the 5U(2) universal bundle E • X x C!Ç}, the orthogonal com plement to orb따 of 5U(2) gives the co이onnectμ，이onon A. The curvature FA of A is a borizontal 2-form with values in tbe Lie algebra su(2). The tangent vectors are of type (2ρ) ， (1

,

1) and (0

,

2) in the tangent space 와(X) x TA(AfÇ). Let p E HJeR(X : R) be the Poincaré dual to a homol- ogy class

I:

ε H₂(X ^‘ Z). The slant product c2(E)! ε E H²(B* : Z) and μ(ε) = c2(E)! ε

=

^{fx c2(E)}

^

^P

=

^{앓 fxTr(댐 ) ^p} is the de-Rham rep resentation of μ(I:). If a

,

b E TAB" ^::e keróA C Ç1' (ÇE)

,

then μ(ε )(a^b) = 뚫 fx tr(a

^

^b)

^

^{p. Let X} be a complex Kähler surface with Kähler form ω Let H be tbe Poincaré dual to ω. μ(H)(a

^

^{b) =} 밟 fxTr(a^b)^ ω

is the K 값11er form on the moduli space M for the usual L2-metric on λ4

For the generic metric on X, the tangent bundle of λ1 is the index bundle for DA : r(V-0 V+0 adP) • r(V+ 0 V+0 adP). If A

,

is a connection on Lr. • Mk

,

then cl(Lε) = 갈 FAI and FAI(a

,

b) = 굉 fztr(a^b)^(PD ε) wbere a

,

b E Ç1'(adP).

Theorem 6.5. 5uppose that Lr; • λ↑ k is the complex line bundle induced by a homology class ε E H2(X : Z) and A is a connection of this bundle.

Then the curvature is given by FA(a

,

b) = 4~; fx tr(a

^

^b)

^

⁰

,

ψhere a

,

b E Ç1'(adP) and 0= Poincaré dual of ε

7. Involution on Donaldson Invariant

Recall that a I<3-surface is a compact

,

simply connected complex surface with trivial canonical bundle. All I<3-surfaces are diffeomorphic and Kãhlerian

,

but not necessarilly biholomorphically equivalent. Some

Group Actions on Donaldson Invariants 333

](3-surface are elliptic surfaces

,

that is they admit a holomorphic map π :X •

CP)

whose generic fiber is an elliptic curve. 양 (X) = +3, the stable range 4k

>

3(1 + b+)

,

i.e. k :::: 4 the invariant qk" is a multilinear function of degree d

=

^{~ dimMk .X}

=

^{4k - 6}

Let Q be the quadratic form of the intersection form on X

,

and let ]( be the linear function]( H2(X) • Z defined by the pairing ]((0')

=<

c)(X)

,O' >

for any 0' in H₂(X). The Donaldson invariants can be expressed as polynouùals in Q and]( : H2(D2(p, q)) • Z and have the form

(7.1) qk.Ð

,

^(p

,

^{q) =} p，qQ[IJ+εf=l a

,

QIl-

,

lK2l where l =

3

⁼

i

^{dlrn Mkι}

and Q[lJ

=

쇄 1 and QI(εl ^{‘ . .} ε21)

=

펴 ε。 Q(ε。，

2::.,)'"

Q(ε。21-1'

L

^0'21)

、!Ve discussed the algebraic surface in section 2. Let D) (2

,

2q + 1) be obtained from D

,

^by logarithuùc transformation of multiplicities 2 and 2q + 1. We introduce the results of Friedman and morgan.

Theorem 7.2. (Friedman

,

Morgan)

(1) No tψo of the manifolds D) (2

,

2q

+

1)( q = 0

,

I ... ) are diffeomor- phic

,

but all homeomorphic

(2) Let D₂(p

,

q) be oblained fmm K3-surface by logarithmic transfor mations of muItiplicilies p and q. Then the pmducl pq is a smooth invari- ant. ]n particular

,

no two of D2(1

,

^2k

+

1) are diffeomorphic

Example 7.3. Let X be a ](3-surface. Let E • X be an SU(2) vector bundle with 야 (E) = 4. The moduli space M.^,x of anti-self-dual connec- tions has dimension 20. Let Q be the quadratic form of the intersection on X.

By (7.1) we have, for ZI," ' , ZIO E H2(X : Z) such that Z = zi(i

=

1 ，" ' ， 1 이，Q(z， z) = 2

q.

,

X(z)

, ...

ZIO) = Q(5)(Z)

, ... ,

ZIO)

1

• ζ

^Q(z.

,

z.

,),'"

Q(Z.9Z.

,

O)

J:r

_u

r

_t:JI

’

_O

=

^1.

This result also proved by Fintushel and Stern for homology ](3- surface

Su ppose that 0',

… O'

^{d E} H²(X : Z). We use the Lemma 6.1 to com pute Donaldson invariant

<μx(O'd" .μ X( O'd) ， π• Mk

,

y

>

< 1T*(μX( O'))

.

π·μX( O'd)) ， μ k，Y>

Thus we have a theorem.

<

π*μx( (1) ... π·μ x(od)，A1k^，y>

=

<

μyπ"( ( 1) .. μ y 1r "(Od) ，서 k.Y >

= qk，y(π·gl，

- -,

^T.Qd)

Theorem 7.4. Let a be an anti-h%morphic invo/ution on a K3-surface X. If Y = X/a is the orbit space

,

theπ the Doπ a/dson invarianl ^qk,Y

can be compuled by the pairiη9 lhe in variant modu/i space ，"-'f값 ， x and the

cohomologν c1asses in λ12^{< ，x.}

For a K3-surface X

,

let OQ be the isometry group of the intersection form Q on the integral homology H2(X) and the homomorphism

h: Dif f(X) • OQ.

The isometry group OQ contai 따 an index 2 subgrollP Oð c。찌 sting of transformations which preserve the orientations of the positive part Hi(x ) Z3 of H2 • Since -1 does not lie in Oð

,

there is a splitting OQ

=

Oð EIl OQ.

Theorem 7.5. (Donaldson and Matumoto) The image of h : Dif f(X) • OQ is the subgro1lp Oð

Theorem 7.6. Let a be an anti-holomorphic involution on a [(3 surface X

,

then

。"q2k，X

=

^{q2k.X :} S2d(H2(X

,

Z)) • Z

Proof If a is an anti-holomorphic involution on a K3-surface X

,

then a is an orientation preserving diffeomorphism. By Donaldson and Matllmoto Theorem

,

a" is an isorr빠ry on H²(X : Z) and preserves the orientation of the positive part of the second cohomology group JJ2 (X : Z)+ = ZEIlZEIlZ ThllS a" preserves the Donaldson invariants on S2d(H2(X : Z))^‘

References

(1) w. Barlh, C. Pelers and A. Van de Ven, Compact complex su ψaces ， Springer, Berlin, 1984

[2) Y.S. Cho, Finite 91"0'

‘

p actions On the moduli space 01 selJ-dual connection 1, 까ans­

action AMS, Vol.323, No. 1(1991). 233-26l.

Group Actions on Donaldson Invariants 335

[3] Y.S. Cho, Finite group actions on the moduli space of self-dual connections JJ,

Michigan Math.Jour. 5. 37(1990)

[4] S.K. Donaldson, Connections, cohomology and the intersection froms of 4- manifolds, Jour. Diff. Geom. 24(1986), 275-341

[5] S.K. Donaldson, Polynomial invariants for smooth 4-manifolds, Topology

[야 S.K. Donaldson, Yang-M

‘

^{IIs Jnvaπants} of Four-manifolds, in London Math. Soc.,

Leclure Note Series 150(1989), 6-40

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[10] R. Fintushel and R. Stern, Homotopy I<3-surfaces containing ε(2 ， 3， 7) ， Jour Diff. Geom. 34(1991), 255-265

[1l] R. Friedman and J .W. Morgan, On the diffeomorphism type of cer1ain algebraic surfaces J, Jour. Diff. Geom. 27(1988), 297-398

[12] R. Friedman and J.W. Morgan, Algebraic surfaces and 4-manifolds; some conjec- tures and speculations, Bull. AMS 18 (1) (1988), 1-19

[13] D.S. Freed and K. Uhlenbeck, Jnstantons and Four-Manifolds, Splinger, New York,

1984

[14] R. Gompf, On the topology of algebraic surfaces, in London Math. Soc. Leclure Note Series 150(1989)

[15] D. Kotschick, On manifolds homeomorphic to CP개 8CP2 ， Invent. Math. 95(1989),

591-600

[16] R. Mandelbaum, Four-dimensional topology; an introduclion, Bull. Amer. Math 50c., 2(1980), 1-159

[17] C.T.C. Wail, Diffeomorphisms of 4-manifolds, Jour. London Math. Soc., 39(1964), 131-140

[18] R.O. Wel1s, Jr.,D

‘

^1ferential analysis on complex manifolds, Springer, New York,

1980

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