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DATE DUE DATE DUE DATE DUE

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

MSU ammo». Action/Equal Opportunity Institution
WM‘

 

SPECIAL METRICS ON SYMPLECTIC
MANIFOLDS

By

Tech 0'. Draghz'cz'

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

1997

ABSTRACT

SPECIAL METRICS ON SYMPLECTIC
MANIFOLDS

By

Tech 0. Draghz'cz'

The central idea of this work is to find geometric and topological consequences of
the existence of special types of Riemannian metrics on compact symplectic manifolds.
The first part of the thesis is devoted to a conjecture of Goldberg about Einstein
metrics on symplectic manifolds and to some related questions coming from a natural
variational problem. The main result of chapter three is the relation we find between
almost Kahler metrics and Hermitian conformal classes. The final chapter deals with
a conjecture about the Seiberg-Witten invariants and the orientations of compact

4-manifolds.

To my parents and my wife,

iii

ACKNOWLEDGMENTS

I am very grateful to my thesis advisor, Professor David Blair, for his constant
help, advice, and encouragement. I would also like to thank Professors Jon Wolfson
and Thomas Parker for many helpful conversations.

iv

TABLE OF CONTENTS

Introduction 1
Preliminaries 2
1.1 Almost Kahler structures ......................... 2
1.2 4-dimensional almost Kahler structures ................. 12
The Goldberg Conjecture and Related problems 16
2.1 The Goldberg Conjecture ......................... 16
2.2 Critical associated metrics ........................ 24

Hermitian conformal classes and almost Kéihler structures on 4-

manifolds 29
3.1 Two Problems ............................... 30
3.2 Main Result ................................ 32
3.3 Yamabe and fundamental constants of Hermitian surfaces ....... 42
3.4 Conformal transformations of almost Kahler metrics on 4-manifolds . 51
3.5 Kahler forms versus symplectic forms .................. 53
Seiberg-Witten Invariants when Reversing Orientation 57
4.1 Statement of the result .......................... 57
4.2 Proof of Theorem 4.1 and Examples ................... 58
BIBLIOGRAPHY 63

CHAPTER 0

Introduction

The main idea of this work is to study geometric and topological consequences of the
existence of special types of Riemannian metrics on compact symplectic manifolds.
After an introductory part, the second chapter of the thesis is centered around a
conjecture of Goldberg from 1969 which states that any Einstein metric compatible
to a symplectic structure on a compact manifold is, in fact, a Kahler Einstein metric.
Some new positive partial results are given. Related to the Goldberg conjecture, we
also study critical metrics coming from a natural variational problem on symplectic
manifolds. The main result of chapter three is the relation we find between almost
Kahler metrics and Hermitian conformal classes. We show that on most compact
complex surfaces which also admit symplectic forms, each Hermitian conformal class
contains almost Kahler metrics. We also give results about the number of symplectic
forms compatible to a given metric. As applications, we obtain alternative proofs for
results of LeBrun on the Yamabe constants of Hermitian conformal classes and give
some answers to a question of Blair about the isometries of almost Kahler metrics.
The final chapter deals with a conjecture from the Seiberg-Witten theory stating that
for any compact, orientable, simply connected 4-manifold, with one of the orientations
all the invariants will vanish. We prove this conjecture for a large class of complex

surfaces. Kahler Einstein metrics play an important role in the proof.

CHAPTER 1

Preliminaries

This chapter sets notations and presents some basic results which we are going to use
throughout the thesis. Central is the notion of almost Kahler structure and the whole
chapter presents definitions and properties related to this notion. Section 1.1 discusses
the case of general dimension, the particular aspects of almost Kiihler structures in

dimension 4 being treated in Section 1.2.

1.1 Almost Kéihler structures

An almost Kiihler structure on a manifold M 2" is a triple (g, J,w) of a Riemannian

metric g, a g-orthogonal almost complex structure J and a symplectic form w given

by

w(X, Y) = g(X, JY). (1.1)

Alternatively, an almost Kahler structure is an almost Hermitian structure (9, J, w)
whose fundamental 2-form w defined by (1.1) is closed. A Riemannian metric which
admits an almost Kahler structure will be called almost K c'ihler metric.

If a symplectic form to is given on M, then there are many almost Kahler structures

with fundamental form w. Let us denote by AM“, the set of associated metrics to
w, that is, all Riemannian metrics g on M for which there exists an almost complex
structure J, such that (9, J, w) is an almost Kahler structure. The following two
propositions describe some of the properties of the set of associated metrics to a

symplectic form.
Proposition 1.1: The space AM“, is a non-empty, contractible Space.

Proof: We will use a well known fact from linear algebra known as the polar decompo-
sition: any m x m matrix A E GL(m, R) can be uniquely written as A = F - G, with
F E 0(m) and G 6 3,. (m), where 0(m) denotes as usual the group of orthogonal
m x m matrices and S+(m) denotes the group of symmetric, positive definite m x m
matrices.

Remark that if A is a skew-symmetric, non-singular matrix to start with, the
matrices F and G from its decomposition satisfy F2 = —Im and FG = GF. Indeed,
from A = -A‘, it follows that FGF = —G, which can also be written as, FZF‘GF =
-—ImG. But now note that F2 6 0(m) and F‘GF E 5+ (m) and by the uniqueness of
the decomposition it follows that F2 = —Im and FG = GF.

Fixing a Riemannian metric k on a manifold M m, the above statements imme-
diately translate to statements about non-singular endomorphisms of the tangent
bundle, where the properties of symmetry, skew-symmetry, positive definiteness are
all understood with respect to the metric k.

Now let (M 2", w) be a symplectic manifold and choose I: an arbitrary Riemannian
metric on M. The symplectic form w and the metric k induce an endomorphism
A of the tangent bundle at every point defined by w(X , Y) = k(X,AY). This is
clearly skew-symmetric and non-singular at every point since w is so. Therefore A
can be uniquely written as A = JG, where J 6 l"(End(TM)) is an orthogonal en-

domorphism, and G E I‘(End(TM)) is a symmetric and positive definite. Moreover

J2 = -ITM and JG = GJ. Define a bilinear form 9 on TM by g(X, Y) = k(X,GY).
This is clearly a Riemannian metric and it is an associated metric to w the corre-
sponding almost Kahler structure being easily checked to be (9, J, w).

This proves that AM“, is not empty. Remark that the above construction gives,
in fact, a map, say pm from the space M of all Riemannian metrics to the space of
associated metrics AMw. But M is contractible. Indeed, choosing go 6 M, the map
Ft(g) = (1 — t)go + tg defines a contraction of M to go. The composition prt defines

a contraction of AM“, to a point. [3

Proposition 1.2 The space of associated metrics AM“, is an infinite dimensional

Frechet manifold.

The proof of this result could be found in [12].

We see that despite the fact that there are many almost Kiihler structures associ-
ated with a given symplectic form, any two such almost Kiihler structures have homo-
topic almost complex structures. In particular, the Chern classes c,-, for z' 6 {1, ..., n},
are invariants of the symplectic structure (in fact, of the homotopy class of the sym-
plectic structure). Finally, let us also remark that the symplectic form determines a
volume element a on M 2", hence an orientation. All associated metrics to w induce

this volume element, namely:

for all g 6 AM“.

For a Riemannian manifold (M, 9) we denote by V, R, p, s the Levi-Civita con-
nection, the Riemannian curvature tensor, the Ricci tensor and the scalar curvature
respectively. Our conventions for the definitions of the curvature and the Ricci tensor

are the following:

R(X1Y) : [VXa VY] _ V[X,Y]a

p(X, Y) = tr(Z ——> R(Z, X)Y).

A Riemann metric g on a manifold M m is said to be Einstein if the Ricci tensor p is

(at each point) a multiple of the metric. Equivalently,

Note that the second Bianchi identity implies for m > 2 that the scalar curvature 3
must be a constant. Einstein metrics will play a central role in this thesis.

The curvature operator acting on 2-forms, denoted by R, is defined by

R(a)ij = Rijaba’ab,

for any a E A2M. We also denote by (, ) and | | the local scalar product and norm

induced by the metric on various types of tensor fields.

Let now (M 2", 9, J, w) be an almost Hermitian manifold. A (2,0) tensor D on M

is called J-invariant (or J-Hermitian) if it satisfies

D(JX, JY) = D(X, Y),

for any tangent vectors X, Y. We say that D is J-anti invariant (or J-anti Hermitian)
if it satisfies

D(JX, JY) = —D(X, Y).

For example, the metric g and the fundamental form w of any almost Hermitian
manifold are, by definition J-invariant tensors. Also, the Ricci tensor of Kahler

metrics is J-invariant. In general, for almost Hermitian metrics this last statement is

no longer true and it will be useful to use the decomposition

inv anti

p=p +p ,
where p‘"” is the J-invariant part of the Ricci tensor defined by
p‘""(X. Y) = goat Y) + pox, M).
and pa"ti is the J-anti invariant part given by
p°"“(X, Y) = goon Y) — pox, M)

The *-Ricci tensor, p* is an analog of the Ricci tensor, but involving also the

almost complex structure:
p*(X, Y) = tr(Z ——> R(X, JZ)JY).
Alternatively, p* is given by:
p*(X, JY) = R(w)(X, Y).

The trace of p* is called *-scalar curvature and is denoted by 3*. For Kahler met-
rics the *‘RICCI tensor coincides with the Ricci tensor. This is not true for general
almost Hermitian manifolds. In fact, in this case the *-RICCI tensor is not necessarily

symmetric. We may remark though, that it satisfies the following identity:
p*(JX, JY) = p*(Y,X),

for any tangent vectors X, Y. This implies that the symmetric part, p*’3’"‘, is a J-

invariant tensor, whereas the skew-symmetric part, p*"‘e‘”, is a J -anti-invariant tensor.

Using the analogous condition on the *—RlCCi tensor, we can define air-Einstein metrics.
However, in this case it does not follow that the *—scalar curvature is a constant, so
we have two notions of *-Einstein. An almost Hermitian metric is said to be:
weakly *—Einstein if at every point the *-R.ICCI tensor is a multiple of the metric (the
air-scalar curvature need not be constant);
*-EIIISteIII if the *-RICCI tensor is a constant multiple of the metric (the *—scalar
curvature is constant).

For 2-forms on an almost Hermitian manifold the following pointwise, orthogonal

decomposition is useful:
A2M = Rw EB A3"”M EB AantiM,

where the factors denote respectively multiples of the fundamental form w, J—invariant
2—forms of zero trace, and J-anti—invariant 2-forms. For a 2—form 'y, the components

With [ESPECt to this decomposition are:
1 inv anti
IY — —(7?(U)CI,+’YO +7 _

7‘3"“ denotes the J-anti-invariant part of '7. The J-invariant part is

i 1 inv

7M=EWMW+%-

If we complexify the tangent space and consider the usual decomposition of complex
2—forms in (2,0), (1,1) and (0,2) forms, it is easy to see that the J-invariant forms
are real parts of (1,1) forms and J-anti-invariant forms are real parts of (2,0) forms

(equivalently, of (0,2) forms).

By easy computation we have:

11—1 1

’7 A w = gww = (n -1)! (M00; (12)

n— ”—1 inv ani
VA’YAW 2=(n-2)![-n—(“1,w)2-l'ro 2+|7 ‘|2]0= (1-3)

= (n - 2)! Mud)"2 - IY'WI2 + |7“"“|le.

Let us recall now some formulas specific to almost Kahler manifolds. Let
(M 2", g, J, Lo) be an almost Kéihler manifold and let V be the Levi-Civita connection

of the metric g. The fact that w is closed is equivalent to

9((VleY, Z) + 9((VYJ)Z, X) + 9((VzJ)X, Y) = 0, (1-4)

for any vector fields X, Y, Z on M. As a consequence of (1.4) we have
(VJXJ)JY = —(VXJ)Y, (1.5)

which is known as the quasi-Kahler condition. In dimension 4, relations (1.4) and (1.5)
are equivalent, but in higher dimensions there are examples of manifolds satisfying
(1.5), but which are not almost Kiihler (look at 56 for instance, with the standard
metric and the standard almost complex structure).

Taking trace in (1.5), we see that the co-differential of w vanishes, hence on is also
a co-closed form. Therefore for any almost Kahler structure (9, J, w), the symplectic

form w is harmonic with respect to the associated metric g.

In our study of almost Kahler manifolds two symmetric tensor fields appear quite

often, so we give them names. We call B and D the global tensor fields whose local

expressions are
Bij = (Vink)(Vbij)a ng = (ViJsk)(VjJsk)- (1.6)

Here and in many other places throughout this work, for local notations and compu-
tations we are using local J-basis, that is an orthonormal basis of the form
{61,J61,...,8mJe,—,}. We adopt the summation convention of Einstein on repeated
indices, but, as we work with orthonormal base, there is no need to raise and lower
the indices. The tensor D has a nice invariant description, D(X, Y) = (V x J, VyJ),
but we do not have a good invariant form for B. It can be easily seen from (1.5) that
both B and D are J-invariant, symmetric tensor fields, with trace equal to IVJI2.

For any almost Kahler manifold the following relation is due to Koto ([29]):

~ 1
pisym : pint) + 58 (17)

This formula implies the relation between the scalar curvatures:
* 1 2

Therefore, we see that an almost Kahler structure is Kiihler if and only if s* = 3.

Besides the Levi-Civita connection V, on an almost Kahler manifold (M, g, J,w)
it is useful to consider the first canonical connection V0, defined by Lichnerowicz in
[38] to be

var = VXY — %J(VXJ)(Y). (1.9)

Since V0 preserves J, its Ricci form 7 represents 21rc1. Using relation (1.9) it is not

hard to obtain the expression for 7 in terms of the Levi-Civita connection V:

10

7(X, Y) = p*(X, JY) — Ell-D(X, JY). (1.10)

Using (1.2), (1.10) and (1.8), we get the expression of the first Chern number in

terms of curvature computed by Blair in [14]:

21rc1U[w]”"l = (n — 1)! /M(*y,w)o = (n —1)!/M 8*: so (1.11)

 

The right hand-side of this equality which a priori seems to depend on the metric,
turns out to be a symplectic invariant.
Relations (1.8) and (1.11) imply the basic scalar curvature inequality for almost

Kahler metrics:

 

/M sdo g (73.71)!“ U [w]"-l, (1.12)

with equality if and only if the metric is Kahler.
From (1.3), (1.10) and (1.8)

47r2c¥ U [w]"'2 = f 7 A '7 A can—2 = (1.13)
M

_ _ l ____| _ *8 ew _ _ *sym _ _ _ *
(n 2).] [ 1 + 2|p | 2]p | 2|D| + (p ,D)]do.

A short computation making use of Koto’s formula (1.7) gives:

1 *sym 2 1 2 1 * _
= __ mu __ _ M) B _ mu D _ _ _ D .

11

Therefore, replacing in (1.13) we get:

4W2 2 11—2 (8 + 3*)2 1 taken: 2
We U [“1 - til—1‘6— + 5'” ' ‘ (“4)
1 inv 2 1 1 1 2

Next we give an integral formula which holds on any compact almost Kahler man-
ifold. It was derived by Sekigawa in [42] and will play an important role in the next
section. Sekigawa used the connections V, V0 and the Chern-Weil homomorphism
to obtain two different representatives, p(V) and u(V°), for the first Pontrjagin class
p1 (M). The 4-form u(V°) — u(V) is then an exact form and hence by Stokes Theorem,

/M(u(v0) — av» A W = 0.

After an extensive calculation, Sekigawa obtains from the above relation the fol-

lowing:

Proposition 1.3 (Sekigawa, [42]) For any almost Kahler manifold (M 2", g, J,w)

the following integral formula holds:
1
[M [Zfl + 22(Vipbj — Vjpbi)(Vink)ij+ (1.15)

I 1
2 4 *skew2
+ (,0,B)+—4 f2+—2 [VJ] +4|p |]a= 0,

where
f1 = z (7209 A 63' _ Jei /\ Jej),e° A 6’ — J6“ /\ Jeb)2 and f2 = 201i " ’\j)2;

A1 = An“ S S A" = A2,, being the eigenvalues of the tensor B.

12

1.2 4-dimensional almost Kahler structures

There are a few special features of the dimension 4. It is well known that the Hodge
operator of a Riemannian 4-manifold (M, 9) satisfies *2 = id acting on 2-forms.

Therefore we have the splitting of the bundle of 2-forms

A2M = A+M ea A’M,

into self-dual 2-forms and anti-self-dual 2-forms, corresponding to the +1 and -1-
eigenspaces of *. It is well known that a 4-dimensional Riemannian manifold (M, g)

is Einstein if and only if the curvature Operator satisfies:

(12a, o) = 0, Va 6 AiMfi e AEM.

Basic topological invariants of 4-manifolds can be nicely described in terms of various
parts of the curvature of a Riemannian metric g. Thus, if 0(M) is the signature and

x(M) is the Euler class of a compact 4-manifold M, then

 

1 _
ow) = 1,7,, [Mun/+12 — IW ma, (1.16)
1 S2 2 _
2X(M) = H [4(6- — % + |W+|2 + |W |2)o. (1.17)

Now let (M 4, 9, J, w) be a compact, 4-dimensional almost Hermitian manifold. We
have the following equalities due to Hirzebruch and Wu relating topological invariants

of the manifold with the Chern classes induced by the almost complex structure:

Ci(M) = 30(M) + 2MM), 02(M) = X(M), P1(M) = 30(M)- (1-18)

The decomposition into self-dual and anti-self—dual 2-forms is very nicely related

13

with the decomposition induced by J:
AiM .—_ Rw ea A“"“M, A2_M = AgnvM. (1.19)

The behavior of the curvature operator with respect to decomposition (1.19) char-
acterizes some interesting geometrical conditions on almost Hermitian 4-manifolds.
The following can be proven by easy computations:
(i) pan“ = 0 if and only if (R(A°"“M), A3"”M) = 0;
(ii) ,0“ka = 0 if and only if (R(A“"“M), Ra!) = o.

Tricerri and Vanhecke [47] have also shown the following relations on an almost

Hermitian manifold of dimension 4:

R(JX, JY, JZ, JW) = R(X, Y, Z, W) 4:) (panti = 0, [2”ka = 0); (1.20)

z :11-(3 — 3*)g, (1.21)

Now let us go to almost Kahler 4-manifolds. First of all, the definition can be
given in a different way. If (M 4, g) is a Riemannian 4-manifold, there is a bijective
correspondence between (oriented) orthogonal almost complex structures and self-
dual forms of pointwise constant length J2. Because of this, a harmonic, self—dual
form w, with M 2 J2, induces an almost Kahler structure (9, J, Lu). By virtue of this
equivalent definition in this dimension, when the metric is fixed, we will very often
just refer to the form when thinking of the almost Kahler structure.

Let us recall now a definition due to Gray [26]. For every point p e M, be defines
D, := {X E TpMIVxJ = 0} and calls ’D the Kahler nullity distribution of the almost
Hermitian manifold M. Note that D need not be a distribution in the usual sense

since the dimension might vary with the point.

14

On a 4-dimensional almost Kahler manifold it follows from relation (1.5) that
V xw is a J-anti-invariant form for any vector X. Also, V xw = 0 if and only if
Vwa = 0. Therefore, we conclude that dimDp is an even number.

Now, since M is 4—dimensional, the fibers of AantiM have dimension 2. Therefore,

locally we can write

Vw=a®¢+fl®JQ

where {(1), J<I>} is any (local) orthonormal frame of A°”“M, and a, B are (local) 1-
forms. Hence, DP contains the intersection of kerap and kerfip. This proves that the
Kahler nullity distribution Dr has dimension either 2 or 4. Hence, the J-invariant,

symmetric tensor field D, previously defined in (1.6)
D(X, Y) = (VxJ, VyJ) = 2(wa, Vyw),

has at every point a double eigenvalue A1 = 0, and a double eigenvalue A2 = filVJIZ.

In particular, in dimension 4,

ID)? = fi-IVJI“.

The tensor B, we also defined in (1.6) has an even simpler form in this dimension.
Indeed, if we combine relation (1.21) with the Koto’s formula ( 1.7), we see that for a

4-dimensional almost Kahler manifold we have
1 2

With these facts in mind, the formula (1.14) specializes in dimension 4 to:

1
167r2

 

CHM) = [MW - le‘m’l2 + 2|)0""“"”|2 + (p, D)lU- (1.22)

Also, the formula (1.15) of Proposition 1.3 has a nicer expression in dimension 4.

15

Proposition 1.4 (Sekigawa, [42]) For any 4-dimensional almost Kéihler manifold

(M 4, 9, J, w) the following integral formula holds:
1
-f1 + 2 Z(Vipbj — Vjpb.)(V.J,-,.)J,,.+ (1.23)
M 4

1 1
+§s|VJ|2 + Z|VJ|4 + 4|p*”°ew|2]a = o,

where

f1: 2 (72(6i A ej — Jei A Jej),e“ A e" - Je“ A Je")2.

CHAPTER 2

The Goldberg Conjecture and

Related problems

In this chapter we will analyze some special associated metrics on compact symplectic
manifolds. The first section deals with Einstein associated metrics. It is a conjecture
of Goldberg [25] that any compact almost Kahler Einstein manifold is in fact Kahler
Einstein. We give a few new positive partial results to this conjecture. In section 2.2
we consider critical associated metrics coming from a variational problem studied by
Blair and Ianus [15]. Some parallel results to those mentioned in 2.1 are obtained
here for critical metrics satisfying some additional assumptions. Some of the results

presented in this chapter have been published in [17], [18].

2.1 The Goldberg Conjecture

A long-standing problem on almost Kahler manifolds is the following conjecture for-

mulated by Goldberg in 1969 [25]:

Conjecture (Goldberg, [25]): The almost complex structure of a compact, almost

Kahler Einstein manifold is integrable, hence the manifold is, in fact Kahler Einstein.

16

17

Important progress was made by K. Sekigawa who proved the following partial result

(see [42]):

Theorem 2.1 (Sekigawa): The conjecture is true if we additionally assume that

the scalar curvature is non-negative.

Proof: For an Einstein, almost Kiihler manifold, the integral formula of Proposition

1.3 becomes:

1 3 2 1 1 4 *skew 2 _
/M[4f1+ anJl + 4nf+ 2anJl +4|p |]do _ 0. (2.1)

Since both functions f5 and f are positive, we get the inequality
/ [3|VJI2 + i|‘\'7J|4]da < 0. (2.2)
M n 2n -

If s 2 0, the above inequality implies VJ = 0, hence the metric is Kahler. D

Replacing the Riemannian condition on the scalar curvature with the natural

symplectic condition, the result still holds.

Theorem 2.2: Let (M 2",w) be a compact symplectic manifold.

(a) If the first Chern number c1(M) U [c12]"'1 is non-negative, then any associated
Einstein metric is a Kahler Einstein metric.

(b) If c1(M) U [a2]"’l < 0, then the scalar curvature of any Einstein associated
metric must satisfy

n—l (n — 1)! n-l
261(M) U [LU] S TSVOKM) S 61(M) U [w] . (2.3)

Proof: (a) First, by eventually scaling the symplectic form, we can assume that the

total volume of M is 1. For an Einstein associated metric we have the inequality

(2.2). With our assumption on the total volume and since 3 is a constant, we can

18

rewrite this as:

[M lVJ|4o 3 —2s lM |VJ|20, (2.4)

so by Cauchy-Schwarz inequality we get:

(fM |VJ|20)(2s + [M |VJ|2o) _<_ 0. (2.5)

Our assumption on the sign of the symplectic invariant c1 (M) U [w]"‘1 is expressed

in terms of the scalar curvature by (1.11):
0 g 41rcl(M) U [co]"'1 = (n — 1)! /M(s +-£1i|VJ|2)o.

This and (2.5) imply
/ [Vleo = 0,
M

hence VJ = 0 so the metric is Kahler .

(b) For any associated metric, the inequality

(n — 1)!

47f [M so 3 c1(M) U [w]"‘1,

was proved in (1.12), with equality if and only if the metric is Kahler. If the metric

is not Kahler, from (2.5) we obtain
23 +/ |VJ|20 S 0,
M

and using (1.11), we see that this is exactly the first inequality of (2.3). D

We see that the basic scalar curvature inequality (1.12) plays an important role.
Here is a lemma giving an estimate for the square of the scalar curvature of an Einstein

almost Kahler metric in terms of another symplectic invariant involving the square

19

of the first Chern class cf.

Lemma 2.1: Suppose g is an Einstein associated metric on a compact symplectic

manifold (M 2”,w). Then the following inequality holds:

 

2 11—2 < (n— 1)!] 2
c1 U [w] __ 16n7r2 M 3 do. (2.6)

Equality holds if and only if the metric is Kahler Einstein.

Proof: If the scalar curvature is non-negative, by Theorem 2.1 of Sekigawa the metric
is in fact Kiihler Einstein. It is easy to see from (1.14) that in this case we have
equality in (2.6). So it is enough to assume that we have an Einstein associated

metric of negative scalar curvature. In this case, from (1.14) we obtain

4oz ,,_

(n—2)!C¥U[“’] 2:
__ (3+S*)2 1 *skew2 1 2 1 2 1 2
_/M[ 16 +2Ip | 4118 SnslVJl 32|2B D| [010.

From the integral formula (2.1) we get the inequality:

_. *88 d </ __ .
/ 2Ip w] o 8 slVJ] do

Hence, also making use of (1.8) we get:

(714:2)!ch MM 3 (2'7)

 

n—l 1 1 1 1
_/M[ 4” s +83|VJ| +64|VJ| 4ns|VJ| 32|ZB D| ]do

20
We now distinguish two cases. If n = 2 then

IVJI4

VJ2
I2B-D|2=|—l 2|g-DI2=IDI2= 2

and we see that the inequality (2.7) is exactly (2.6) for n = 2. If n 2 3, we see that

the integral formula (2.1) also gives the inequality:
/ [istl2 + —1—|VJ|4]do < o.
M 32 64 "

Since 3 < 0, using this last relation it is not hard to see that we obtain the inequality
claimed in (2.6). Equality holds if and only if VJ = 0. Therefore the lemma is

proved. E]

Lemma 2.1 has interesting consequences in dimension 4. In this case, it says that

an Einstein associated metric on a symplectic manifold satisfies

ci(M) s

 

2
327r2 [Ms do, (2.8)

with equality if and only if the metric is Kahler Einstein.

Theorem 2.3: Let (M,w) be a compact symplectic 4-manifold which admits an

Einstein associated metric. Then the Miyaoka - Yau inequality

Ci(M) S 3020”)

is satisfied.

Proof: Using the inequality (2.8) combined with relations (1.16), (1.17) and (1.18),
we get right away

c¥<M) s 3(2X(M) — 3T<M)).

21

From the Hirzebruch-Wu equalities (1.18) we see that the above relation gives us the

result. [I
The proof of the Theorem 2.3 immediately implies the following

Corollary 2.1: For compact symplectic 4-manifolds (M, w) which satisfy c§(M) =

3c2(M) the Goldberg conjecture is true.

Remark: Using Seiberg—Witten invariants, C. LeBrun obtained stronger statements
than Theorem 2.3 and inequality (2.8) in dimension 4. We will indicate in Chapter
4, how these stronger results are proved.

Proposition 2.1: Let M be a compact, symplectic, 4-dimensional manifold with
c1 = 51;).[w], for A E R. Suppose there exists a weakly *-Einstein associated metric
with the property that there exist three real numbers a, b,c (a 2 0,b Z 0, but
a2 + b2 # 0) such that

as + bs" = c.

Then this must be a Kahler metric.

Proof: Without loss of generality we can assume that the total volume of M is 1. Let
7 the representative of 21rc1 given in (1.10). From c1 = 51;).[w], it follows that there
exists a 1-form a such that

87r7 = Aw + da.

Since w is harmonic, it is orthogonal on exact forms, so from the previous relation

1 *
A—5/M(S+S )0’.

Therefore we get

 

1 s+s* 2
2 _
c1(M)—3—2?(/M 2 0)'

22
On the other hand, from (1.22)

1 inn *3 ew
16W, [MW—2|); [2+2Ip k |2+(p,D)]o.

 

ci(M) =

Comparing the two expressions for cf(M) we have

 

l 3*+3 inn *3 cw
,(fM 2 0)2=fM[32—2|p |2+2|p . I2+(p.D)]o- (2.9)

The weak air-Einstein condition is equivalent to p*3"e"’ = O and p*‘y'" = %s*g. From

(1.21) we deduce also that p‘"” = fisg. With these relations, (2.9) becomes

s*+s 2_/ *
(/M 2 o) _ M880,

 

which we can rewrite as

(/M 8*2— so)2 = [M ss*o — (lM so)(/M s*o). (2.10)

Suppose that as + b3* 2 c, with a, b, c constants, a, b 2 0, but not both 0. Without

 

loss of generality we can assume b aé 0. Then

[M ss*o — (fM so)(/M s*o) = %[(/M so)2 — [M szo].

By Cauchy-Schwarz inequality we see that this last expression is non-positive. Hence
the right-hand side of (2.10) is non-positive, whereas the left-hand side is non-negative.

This implies that s* = s and by (1.8), the metric is Kahler. D
Without any effort, we have the following corollaries:

Corollary 2.2: Let M be a compact, symplectic, 4-dimensional manifold with c1 =

§%A[w], for A E R. Any *-Einstein associated metric is a Kahler metric.

23

Corollary 2.3: Let M be a compact, symplectic, 4-dimensional manifold with c1 :2
%A[w], for A E R. Any Einstein, weakly *-Einstein associated metric is a Kahler

metric.

We would like to end this section with some more remarks related to the Goldberg
conjecture in dimension 4. We will refer to results which are included in the later
sections of this thesis since they have more significance there, but we would like to
point out here their consequences to the Goldberg conjecture. Given a compact,
oriented 4-manifold M with an Einstein metric g, a counter-example to the Goldberg
conjecture would be provided by a self-dual, harmonic 2-form L.) which has constant
length J2, but it is not parallel. From Proposition 3.5, we see that we cannot find
such a form if the metric g is a Kahler Einstein metric on M. Moreover, using
the Seiberg—Witten equations, LeBrun [34] showed that any Riemannian metric on a

compact symplectic 4-manifold M satisfies the inequality (2.8):

 

l
cflM) S 321r2 [M SZdU’

with equality if and only if the metric is Kahler Einstein. This implies the following

Proposition 2.2: Let M be a compact, oriented 4-manifold admitting Einstein
metrics. If a connected component of the moduli space of Einstein metrics contains
a Kahler Einstein metric, then all the metrics from that connected component are

Ka'hler Einstein.

Proof: Note first that if gt is a path of Einstein metrics of equal volume on a compact
manifold, then the scalar curvature does not change 3, = so. ([9], [10]). The result
then follows applying the above mentioned result of LeBrun. [3

Thus, a counter-example to the Goldberg conjecture will not be obtained in di-

mension 4 by deforming a little bit a given Kiihler Einstein metric. Another attempt

24

could be to consider both orientations of 4-manifolds with Einstein metrics. For in-
stance, let (M, g) be a compact Kahler surface and consider (M, g). As a consequence
of Theorem 4.1 we see that this also fails to give counter-examples to the Goldberg
conjecture if the signature of M satisfies o(M) S 0. We do not know yet what hap-
pens in the case of positive signature, but in the proof of Theorem 4.2 we give some

potentially interesting examples for this problem.

2.2 Critical associated metrics

A more recent, but parallel problem to the Goldberg conjecture arose from the work
of D. Blair and S. Ianus, [15]. They studied variational problems in the set AM“, of
all associated Riemannian metrics to a symplectic manifold, hoping to characterize
Kahler metrics by a variational method. For the integral of the scalar curvature
functional restricted to AMw, Blair and Ianus showed that the critical metrics are
those for which the Ricci tensor is J—invariant. Kahler metrics are among the critical
metrics and, in fact, from (1.12), we see that they are absolute maxima for this
functional. It is therefore interesting to see whether almost Kiihler metrics with J-
invariant Ricci tensor have to be in fact Kahler. We will refer to this question as
the “Blair-Ianus question” as it originates from [15], and by critical metrics we will
understand for the rest of this section almost Kahler metrics with J-invariant Ricci
tensor. If true, the question of Blair and Ianus would be a stronger statement than
the Goldberg conjecture. However, a series of examples of strictly almost Kahler
manifolds with J-invariant Ricci tensor were given in dimensions 4k + 2 (see [16],
[2]), showing that not all critical metrics are Kahler in general. These examples are
obtained on twistor spaces of quaternionic Kahler manifolds. In spite of this, it is still
interesting to see under what additional assumptions the question of Blair and Ianus

has a positive answer, in view of possible analogies with the Goldberg conjecture. For

25

the rest of this section we will consider critical metrics trying to prove parallel results

to some of those from the previous section.

Theorem 2.4: Let (M2",g,J,w) be a compact almost Kahler manifold with J-
invariant Ricci tensor.

A. If 2n 2 6 and there exist a function A _>_ 0 on M such that
Ag(X.X) S p(X.X) S 2A9(X,X)

for any X 6 TM, then the almost complex structure is integrable, that is, M is a
Kahler manifold.
B. If 2n = 4 and the Ricci tensor is non—negative definite when restricted to the

Kiihler nullity distribution D := {X 6 TM |V xJ = 0} then M is a Kéihler manifold.

Proof: We will use identity (1.15) of Proposition 1.3. Integration by parts and equa-
tion (1.5) give
[MKVinj — Vjpbi)(Vink)ij]0 =

= [Mlpbivj(Vink)ij - ijVi((Vink)ij)]0 = 2 [M Pbivj(Vink)ij0'.
To get the second equality we used the fact that

(Vinkljjk = -Jik(Vbij),

and therefore

Vi((Vink)ij) = - ikvi(Vbij)-

On any almost Kahler manifold we have the following identity which can be easily

obtained after a straightforward computation making use of (1.4) and ( 1.5):

VjVBJik = —VijJik - (Vijs)(VsJilc) — (VbJsk)(VjJis)-

26

Since p is J-invariant, using the formula above and, repeatedly, the quasi-Kahler

condition (1.5) we obtain :
2PM(VijJik)ij = pa(Vka.)(V.Ja) - pbi(Vkas)(VkJis) = (10,13) - (P, D)-

Replacing in (1.15) and neglecting a few terms, we get the inequality:

0 g —2 /M[2(p, B) — (p, D)]o — [M 572%.; (2.11)

Now, if the eigenvalues of p are 0 5 A1 3 3 An then, using the hypothesis 2A1 2 An
we get:

2(1). B) Z 2A1IVJ|2 2 AanJl2 2 00,0)- (212)

This and inequality (2.11) imply IVJ I = 0, thus J is parallel so we proved A.

To prove B, note that for almost Kiihler 4-manifolds it was shown that B =
i—IVleg and that D has a double eigenvalue 0 and another double eigenvalue 11%|:
Assume there exists a point p E M such that VJ (p) aé 0. Then in a neighborhood of p,
dimD = 2. In this neighborhood choose {e1, J e1, e2, J 62} a local J-orthonormal basis
which diagonalizes D. We can assume that {e1, Jel} correspond to the eigenvalue 0

of D so {81,J81} generate D in this neighborhood. Then
1
2(p, B) - (p, D) = §S|VJ|2 - p(62,62)|VJ|2 = A61, e1)|VJ|2 Z 0,

where the last equality is obtained from the hypothesis that p is non-negative definite

on D. Using again (2.11) we get VJ = 0, hence the conclusion for B. E]
Next we prove a result analogous to Corollary 2.3.

Theorem 2.5: Let (M 4,w) be a compact, 4-dimensional symplectic manifold with

H 2(M ; R) = R. A critical associated metric g with its *-Ricci tensor symmetric, is a

27

Kahler metric.
We need first the following technical result:

Lemma 2.2: On an almost Kahler manifold (M, g, J, w) let B be a symmetric, J-
invariant tensor field of type (2,0), satisfying (SB = 0 . Then the 2-form B(X, Y) =
B(X, JY) is co-closed (i.e. 66 = 0).

Proof of Lemma 2.2: We work in a local J-basis {€1,J€1,...,6n,Jen}. Locally, fl is

given by 6,,- = B,,J,-_,. Since VgBi, = 0, we have

Vifltj = BisVist-

Using the J-invariance of B and the quasi-Kahler condition ( 1.5) we see that

BisVist : Bisvi-Ijs : BisViJj§ = —BisVist-

Hence Viflij = B,,V,~J,-, = O which concludes the proof. El

Proof of Theorem 2.5: From the fact that p is J-invariant it follows that a defined
by a(X, Y) = p(X,JY) is a J-invariant 2-form on M. We show first that 01 is
closed. The tensor field B = p — %sg is symmetric, J-invariant by hypothesis and,
from the second Bianchi identity, also satisfies (SB = 0 . Applying the Lemma, the
corresponding 2-form, fl = a — %sw is therefore co-closed. Therefore *6 is closed.
Using the decomposition (1.19),

*6 = *(a — st —— 13w): —a + st — st = —a.

Thus a is closed and induces the cohomology class [(1].
Since H2(M ; R) = R, there exist real numbers <, )1 such that c1(M) = 8%A[w]

and [a] = u[w]. Hence 87r7 — Aw and a — W are exact, so orthogonal to w with

28

respect to the global inner product defined on forms. From this we get

—l/(s+s*)o —1/ so
—2M ’“_4M '

Under the assumptions, the 2-form )6 = a- §sw is co—closed, and hence orthogonal

to 87r7 — Aw and a -— pw. Expressing this, we get
1
2 2 _ _ * .
/ [_4|p| +23 + (p, D)]o — 2(/ so)(/ (3 + s)o), (2.13)

[M |p|2o = é/M szo — Ell-([114 so)2. (2.14)

Relations (2.13) and (2.14) combined give

 

[Mm D)0 = ( [M sa)( [M 3*2— 30). (2.15)

Using now (2.14) and (2.15) in (2.9), which holds under our assumptions, we obtain

/.4lpwwva=</.S*. _. mos/”fl”- a.)

 

 

By assumption the *-Ricci tensor is symmetric, so (2.16) gives VJ = 0 completing

the prooffl

CHAPTER 3

Hermitian conformal classes and
almost Kahler structures on

4-manifolds

In this chapter we show that for most compact complex surfaces which also admit
symplectic forms, each Hermitian conformal class contains almost Kiihler metrics.
We also give results about the number of almost Kahler structures a given metric
can be part of. As applications, we obtain alternative proofs for results of LeBrun
on the Yamabe constants of Hermitian conformal classes and give some answers to a
question of Blair about the isometries of almost Kiihler metrics. We end with a section
discussing the difference between the set of Kahler forms and the set of symplectic
forms on a given manifold. Sections 3.1 to 3.4 of this chapter are part of a joint work

with Vestislav Apostolov. With slight modifications the reader could find them in [3].

29

30

3. 1 Two Problems

Let us consider a 4-manifold M which admits symplectic structures. Given a symplec-
tic form, w, the space of associated metrics to w can be also defined in this dimension
by

AM... = {9| (2') w 6 MM, (2) m, = J2}.

Note that there can be no two elements of AM“, in the same conformal class of
metrics. Giving up condition (ii), define the space of conformal associated metrics to
w by,

CAM... = {gl w E A+M}.

Indeed, it is easily seen that

CAM“; = 0:0(M) ° AM“),

where C3?(M) denotes the space of smooth, positive functions on M.
If 5 denotes the set of all symplectic forms on M, then the space of all almost Kahler

metrics and the space of all conformal almost Kahler metrics are, respectively:

AIC = UwesAMw, CAIC = UwesCAMw.

The following easy proposition motivates the questions we are addressing in this

chapter.

Proposition 3.1: Let M be a closed, oriented 4-manifold, admitting symplectic
structures.

(a) If w and w’ are distinct, but cohomologus symplectic forms on M, then CAM“, fl
CAM“), : 0°

31

(b) Let g be a Riemannian metric on M. There exists a finite dimensional vector
subspace V ofC°°(M), such that for any f E Cf(M) with f2 9! V, the metric g’ = fg

is not an almost Kahler metric.

Proof: (a) Assume there exists a metric g E CAM“, fl CAM,» Then w and w’ are
both harmonic with respect to g. But by Hodge decomposition theorem there is a
unique harmonic representative in a given cohomology class. This contradicts the
assumption w # w’.

(b) Let us recall that in dimension 4, harmonic 2-forms are invariant to conformal
changes of metric, as also invariant is the splitting into self-dual and anti self-dual
forms. Given the metric g, assume that the metric g’ = f g is an almost Kiihler metric.

This is equivalent to the existence of a self-dual, harmonic 2-form, w’, with

1
75

|w’|: = |w’|§. = 2.

Let 011, ..., a), form an orthogonal basis for the space of self-dual, harmonic 2-forms
with respect to the global inner product induced by the metric g. As w’ is self-dual,

harmonic with respect to g as well, it must be a linear combination of the a,’s:

I
w = alal + + akak,

for some constants a1, ..., ak.

It follows that

2f2 = Z aiajfija

where f,, are the smooth functions given by the pointwise g-scalar product of a,- and

(1,, fij = ((1,, aj)g. Taking V to be the space generated by the fij’s, the conclusion

32

follows. C]

A short way of rephrasing part (b) of the Proposition 3.1 is that in a given con-
formal class most of the metrics are not almost Kahler. As for (a), it leads to some
questions. First, one may ask under what conditions two symplectic forms share a
same associated metric. As we saw, this is not possible if the forms are cohomologus.

From a metric point of view, this can be restated as follows:

Problem 1: When does a Riemannian 4-manifold (M, g) carry two almost Kahler

structures (g, J1,w1), (g, J2,w2) with w1 74 :l:w2 ?

From a symplectic form w, many others can be obtained by deforming the given
one with “small” closed 2-forms. As symplectic forms in the same cohomology class
have all disjoint sets of conformal associated metrics, it looks that “many” conformal

classes contain almost Kahler metrics. It is natural to ask the following problem.

Problem 2: Find conformal classes which do not admit almost Kiihler metrics.

3.2 Main Result

We give some answers to Problems 1 and 2 for compact complex surfaces where we
consider the space ’H of all Hermitian metrics. First of all, it makes sense to consider
only compact, complex surfaces which also admit symplectic structures. Note that
any closed complex surface with b1 even admits Kahler structures, hence, in particular,
symplectic structures. In the case b1 odd, the situation is more delicate and has been

settled only recently (see [11, 24]).

Proposition 3.2: (O. Biquard [11]) The only complex surfaces with b1 odd that

also admit symplectic structures are primary Kodaira surfaces and blow-ups of these.

Here is the main result of this chapter.

33

Theorem 3.1: Let (M, J) be a compact complex surface which also admits sym-
plectic structures.
(a) If b1 is even then H C CAIC. Moreover:

(a1) Assume that g is a Kahler, non-hyper-Kéihler metric on M, with Kahler form
w. Then w and —w are the only almost Kahler structures compatible to g;

(a2) Assume that g is a non-Kahler, conformally-Kahler metric on M.
If c1 aé 0, one of the following two situations occurs: g has exactly two Sl families of
associated almost Kahler structures, or g is not an almost Kahler metric.
If c1 = 0, one of the following three situations occurs: g has exactly two S1 families of
associated almost Kéihler structures, 9 has exactly one .5'1 family of associated almost

Kahler structures, or g is not an almost Kahler metric.

(b) If b1 is odd, there are two cases:
(b1) If (M, J) is minimal then ’H C CAIC. In this case, each metric g E ’H n AIC
has exactly one S 1 family of almost Kéihler structures associated.

(b2) If (M, J) is not minimal then 71 fl CAIC = (0.

Regarding Problem 1, we see that Kahler, non-hyper—Kiihler metrics have an es-
sentially unique compatible almost Kahler structure. However, as we see in (a2),
there are examples of Hermitian metrics having a whole family of compatible almost
Kahler structures. As c1 aé 0, these examples are not hyper-Kahler.

As an immediate consequence of (b2), here is an answer to Problem 2.

Corollary 3.1: On blow-ups of primary Kodaira surfaces, Hermitian conformal

classes do not contain almost Ka'hler metrics.

The proof of the Theorem 3.1 relies on a series of propositions which we give
below.
Recall that the Lee form (9 of an almost Hermitian 4-manifold (M, 9, J) is defined

by dF = 0 A F, or equivalently 0 = J6F, where F denotes the Kahler form of (g, J),

34

6 is the co-differential operator defined by g, and J acts by duality on 1-forms. It
easily seen that d9 is a conformal invariant, that is, it depends on the conformal class
of g and not on the metric itself. It is also known that Hermitian metrics with d0 = 0
correspond to locally conformal Kahler metrics and the Hermitian metrics with 0 = 0
are, in fact, Kahler metrics.

A Hermitian metric such that the Lie form is co-closed, i.e. (50 = 0, is called by
Gauduchon a standard Hermitian metric. He proves in [21] the existence of standard
metrics in each Hermitian conformal class (in any dimension) and its uniqueness
modulo a homothety. In some sense, the standard Hermitian metric is the “closest”
to a Kahler metric in its conformal class.

The first result we need is due to Gauduchon. For completeness, we give a proof,

slightly different than the original argument in [23].

Proposition 3.3: (Gauduchon [23]) On a compact complex surface M, endowed
with a standard Hermitian metric g, the trace of a harmonic, self-dual form is a

constant.

Proof: Let (M, g, J, F) be the standard Hermitian structure on M. Any self-dual

form a E A+M can be uniquely written as:

a=aF+fl+R (3.1)

with a E C°°(M) and fl 6 Az'oM. We have to prove that if a is also (co)closed then
a is a constant.

Taking the divergence of both sides of (3.1), it follows

0 = Jda+aJ9+6(fl+B).

Applying J to the above relation, we get:

35

do = —a6 + J6(fl + 3).

Taking inner product of both sides with do and integrating over the manifold implies

[M Idalzd/t = - /M%(9,d(02))d/1 + /M(Ja(p + '3), dam,” =

= — [M $050. a2)du — [Mao + 3). (“MW = 0,

since (50 = 0 and dea e Al’lM. Therefore do = 0, so a is a constant. D

Corollary 3.2: Let (M, g, J, F) be a Hermitian surface. Then any harmonic, self-
dual form w is either the real part of a holomorphic ( 2,0) form or is non-degenerate

everywhere on M.

This result already gives the relations between the spaces 71 and CAIC stated in
the Theorem 3.1 at (a), (b1) and (b2). The next propositions deal with the number

of compatible almost Kahler structures that various Hermitian metrics can have.

Lemma 3.1: Let (M, J) be a complex manifold with c1 319 0, equipped with a
standard Hermitian metric g (which may be Kahler), and let F be the fundamental
form. Suppose al, 012 are two harmonic self-dual 2—forms which satisfy a? = 0%. Then

the traces of these forms (which are necessarily constants) are equal up to sign:

((11, F) = :l:((12, F).

Proof: Using decomposition (1.19), the 2-forms (11,02 can be written uniquely as:

36

0’1 = 01F+51 +31,

012 = 02F+fl2 +32,

where 61, )62 are (2,0) forms and a1, a2 are constants (by Proposition 3).

2 2

Now C21 2 a2 is equivalent to

(“l - a§)F2 = 2(fi2 A B2 — 161 A 31) = Re((fl2 — 51) A (52 + 31))

By the assumption c1 5:5 0, it follows that the form ,82 — 61 must vanish at some
point on M. From the above equality, as F2 is a volume form on M and a1, a2 are

2

constants, it follows al - a3 = 0. El

Proposition 3.4: Let g be a standard Hermitian metric on a complex surface (M, J)
with Cl 75 0, and let F be the fundamental form. Denote by w the unique self—
dual, harmonic form which has trace equal to 1 and is orthogonal to the space of
holomorphic (2,0) forms with respect to the cup product. Suppose a is a harmonic,

self-dual form such that a2 = w2 everywhere on M. Then a = :l:w.

Proof: By Lemma 3.1, tracea = itracew = :l:1. Assume tracea = tracew = 1. In

this case a can be written as

a = w + Rem),

where 6 is a holomorphic (2,0) form. From a2 = w2, it follows the relation

2w A Rem) + Re(fi)2 = 0,

37

everywhere on M. Integrating this relation on M, the first term vanishes because of
the choice of w. Therefore we get Rem) = 0, but as B is a (2,0) form this implies
(3 = 0. Therefore we proved a = w. Similarly, if traced = —tracew = —1, it follows

that a = -—w. E]

Proposition 3.5: Let g be a Kahler metric on M with Kahler form w. Then either
g is a hyper-Kahler metric, or iw are the only almost Ka'hler structures compatible
to g. Moreover, for a hyper-Kahler metric g, all self-dual, harmonic forms of constant

length are parallel.

Proof: The last claim, about self-dual, harmonic forms of constant length on a com-
pact hyper-Kéihler manifold, is probably well-known and could be obtain in many
different ways. For example, it follows from Theorem 2.1 of Sekigawa, since a hyper-
Kahler metric is Ricci flat. It remains to show the first part of the proposition. If
c1 76 0 the conclusion follows immediately from the Proposition 3.4. Our argument
below covers all cases.

Assume there exists another harmonic, self-dual form w’ 7’: :lzw, inducing same

volume form as w. Then w’ is uniquely written as

w’=W+n,

where a is a constant and n is a smooth section of the canonical bundle. From w’2 = w2
we deduce
2
2 I’ll
a + — = 1,
2

hence |a| S 1. If |a| = 1, then 17 = 0, therefore w’ = iw. If |a| < 1, we show that the
metric g is in fact hyper-Kahler. Indeed, wl = 71%;?) is a self-dual harmonic 2-form
of length J2, pointwise orthogonal to w, so it induces another almost Kahler structure

on M, (g, J1,w1), with J and J1 anti-commuting. Since J is parallel with respect to

38

the Levi-Civita connection of 9, it follows that (g, J2 = J 0 J1) is another almost Kahler
structure, with J2 anti-commuting with both J and J1. Now, using an observation of
Hitchin ([27], Lemma 6.8) that any triple of anti-commuting almost Kiihler structures

(g, J, J1, J2) defines a hyper-Kahler structure, we complete the proof. El

Proposition 3.6: Assume that g is a non-Kahler, conformally-Kéihler metric on a
compact complex surface (M, J).

If c1 75 0, one of the following two situations occurs: 9 has exactly two S1 families of
associated almost Kahler structures, or g is not an almost Kahler metric.

If c1 = 0, one of the following three situations occurs: 9 has exactly two S 1 families of
associated almost Kéihler structures, g has exactly one Sl family of associated almost

Ka'hler structures, or g is not an almost Kahler metric.

Proof: First we will consider the case c1 79 0. Let g = f g’ , where f 6 CS? and g’
is a Kahler metric on (M, J) with Kahler form F. Let us assume also that (g, J,w)
is an almost Kiihler structure. Then w is a g-harmonic, self-dual form, of g-length
J2 at every point on M. As g’ is a conformal metric to g, the form w is harmonic
and self-dual with respect to g’ as well. Hence there exists a constant a 75 0 and a

holomorphic (2,0) form ,6 such that

w = aF + Rem).

But in this case, note that the forms

w:r = aF + cos(27rt)Re(fl) + sin(27rt)Im(fi),

w,“ = -aF + cos(27rt)Re(fl) + sin(27rt)Im(fl),

are also harmonic, self-dual and of length J2 with respect to the metric g, for any t E

39

[0, 1]. Therefore 9 has at least two S 1-families of almost Kahler structures compatible
to g.

Suppose now that (g, J’, w’) is some almost Kahler structure compatible to g and
we would like to show that it must be one of the almost Kahler structures described

by the the two Sl-families above. With the same reasoning as above

w’ = a’F + Re(fl’),

2

2_ I
_w,

where a’ is a non-zero constant and [3’ is a holomorphic (2,0) form. Since w
by Lemma 1 we get a’ = :l:a. Let us assume a’ = a, the argument being similar in

the other case. Now w2 = w’2 implies 126(6)2 = Re(fl’)2, which is equivalent to

Real? — fi’) A Rem + (3') = 0-

This means that at every point on M, the form Rem + 3’) is collinear to I m()6 — 6’).
As both Re(fl + 6’) and I m(fl — 6’) are closed, we must have

Rem + fl’) = Almm — 2’).

for A a constant on M. The above relation implies

 

 

 

A2 -— 1 2A
I _ _ .
fl“,\2+15 A2+1’fl’
or, further,
, A2 — 1 2A
Rem) —- ,, +1Re(fl) + ,2 + 11mm).

It is easy to see now that w’ is in fact one of the forms in the family wt+ .
Next, let us consider the case c1 = 0. By Kodaira’s classification Theorem we

distinguish two sub-cases.

40

(i) (M, J) is a hyperelliptic surface or an Enriques surface. For these b+ = 1, hence,
by Proposition 3.1, there are no non-Kahler, globally conformal Kiihler almost Kiihler
metrics.

(ii) (M, J) is a complex torus or a K3 surface. For these b+ = 3 and they have hyper-
Kiihler metrics. Let us first remark that if g’ is such a metric, then all self-dual,
harmonic forms with respect to g’ have constant length. Therefore there is no non-
Kiihler, almost Kahler metric which is conformal to a hyper-Kahler metric. However,
a complex torus or a K3 surface do have Kahler metrics other then the hyper-Kahler
ones. Choose one such metric and denote it again by g’, the corresponding Kahler
form being w’. Suppose that g = f g’ is a non-Kahler, conformally Kahler metric

which has an almost Kiihler structure w. Then we have

w = aw’ + Rem),

where a is a real constant and )6 is a holomorphic (2,0) form. In fact, 6 is everywhere
nondegenerate, so it is a holomorphic symplectic form on M.
Now we have two possibilities: if a = 0, then the metric g has one S 1 family of almost

Kahler structures given by

w, = cos(27rt)Re(fl) + sin(27rt)1m(fl);

if a 9.4 0, then the metric g has two S1 families of almost Kahler structures given by

w,i = iaw’ + cos(21rt)Re(fl) + sin(27rt)Im(fi).

In either case, if 9 had other almost Kahler structures, it would follow that 6 has
constant length with respect to g’, which is a contradiction to the fact that g’ is not

hyper-Kahler. El

41

We finally put together the above results to prove Theorem 3.1.

Proof of Theorem 3.1: Let us denote by pg the geometric genus of the complex surface
(M, J), i.e. the complex dimension of the space of holomorphic (2,0) forms. It is well
known that b+ = 2pg when b1 is odd and b+ = 2179 + 1 when b1 is even.

Let us consider first the case b1 odd. By Proposition 3.2 of O. Biquard, the only
compact complex surfaces that also admit symplectic structures are primary Kodaira
surfaces (case of (b1)) and blow-ups of these (case of (b2)). For the primary Kodaira
surfaces it is also known that they do admit holomorphic symplectic structures, that
this, there exists a nowhere vanishing holomorphic (2,0) form. Denote such a form ,8
and consider now a Hermitian metric g. The real form w = Rem) is the real part of
a holomorphic (2,0) form on M hence it is a harmonic, self-dual form for the metric
g. As w is also non-degenerate, there is a conformal metric to 9 such that w and
the new metric define an almost Kiihler structure. We hence proved ’H C CAIC for
primary Kodaira surfaces. Note also that if g is Hermitian, then any almost Kahler
structure, say w, has to be the real part of a holomorphic (2, 0) form since b+ = 2pg.

Hence w = Rem), but then

w, = cos(21rt)Re(fl) + sin(27rt)Im(,B)

is a whole 5'1 family of almost Kahler structures compatible to the metric 9. Finally,
since for a primary Kodaira surface b+ = 2, from Proposition 3.1 it follows that each
Hermitian, almost Kahler metric has exactly one Sl family of compatible almost
Kalhler forms.

To prove (b2) note first that if (M, J) is a blow-up of a primary Kodaira surface,
then c1 ¢ 0 in this case. Let g be a Hermitian metric and let w be a real, self-dual,
harmonic form with respect to g. As above, since b+ = 2pg, it follows that w = Rem),

where [3 is a holomorphic (2,0) form. Since c1 75 0, 6 must vanish at some point on M

42

and so does w. Therefore, for any Hermitian metric there are no harmonic, self-dual,
everywhere non-degenerate forms. This proves (b2).

Let us now consider the case b1 even. In this case b, 2 2p, + 1, so for any
Hermitian metric g, the space of real parts of holomorphic (2,0) forms is strictly
contained in the space of all self-dual, harmonic forms. Let w denote the (unique)
self-dual, harmonic form which has trace equal to 1 and is orthogonal, with respect
to the cup product, to the space of real parts of holomorphic (2,0) forms. This form
is non-degenerate everywhere on M and hence for a conformal metric to 9 this form
will define an almost Kiihler structure. The statements from (al) and (a2) follow

form Propositions 3.5 and 3.6, respectively. El

Remark 3.1: It would be nice to complete (a) in the Theorem 3.1 with a statement
about the possible number of almost Kahler structure compatible to an arbitrary
Hermitian metric (non-Kiihler and not conformally Kiihler). Proposition 3.4 shows
that there are some Hermitian, non-Kahler metrics with a unique, up to sign, almost

Kahler structure. However, we do not know a complete answer to this problem yet.

3.3 Yamabe and fundamental constants of Hermi-

tian surfaces

The Yamabe constant, Y(c), of the conformal class c on a compact 4-manifold M is

defined to be
(M sng9 9]

Y(Z'C) znngc [— m

where 39 is the scalar curvature of the Riemannian metric g and dug denotes it volume
form. It was proved by R. Shoen [43] that each conformal class c contains metrics of

constant scalar curvature. These realize the infimum in the above definition and, for

43

this reason, scalar curvature metrics are also referred as Yamabe metrics. We shall
say that (M, c) is of positive (resp. zero or negative) type if Y(c) is positive (resp.
zero or negative).

It is a remarkable fact that the existence of metrics with positive scalar curvature
on a compact 4-manifold leads to important informations about the differentiable
structure of the manifold. In particular, all Seiberg—Witten invariants must vanish.
This was successfully used by C. LeBrun to prove that on a compact complex surface
(M, J) with even first Betti number the existence of conformal classes (not necessarily
compatible with J) of positive type forces (M, J) to have negative Kodaira dimension,
i.e to be either a rational surface, or a blow up of a ruled surface [33]. Considering only
the conformal classes of Hermitian metrics, LeBrun’s result was previously observed
by several other authors [50, 48, 4]. The main idea dealing with Hermitian conformal
classes is to use the Gauduchon’s vanishing theorem, as it is explained below.

Let (M, J) be a compact complex surface and let c be a conformal class of Her-
mitian metrics on M. For any metric g 6 c we denote by ug the Hermitian scalar
curvature of (g, J), which is defined to be the trace of the Ricci form of the Chem

connection V“, i.e. we have
u, = 2 < R°(F),F >9

where Rc is the curvature of Vc and F, as usually, is the Kiihler form of (g, J). Using
the relation between the Chern connection V“ and the Riemannian connection V,

given by (c.f. [23, 48])

V§Y = VXY — %0(Y)X — %0(JX)JY + %g(X, Y)0,

44

on can easily see (c.f.[23]) that u, and 39 are related by
_ 1 2
ug — sg —. (50 + 2’6I9' (3.2)

The eccentricity function fo(g) of a metric g in c is the positive function determined
by the property 9 = Elmgo, where go is the standard metric of Gauduchon on 0 giving
M a total volume 1 (different normalization than [7]). Note that a metric g is standard
if and only if the corresponding function f0 is a positive constant.

The fundamental constant C (M, J, g) of a compact Hermitian surface we will

define to be (compare with [7]):

C(M, J, g) = [M fo(g)u.du..

Note that C(M, J, g) does not depend on the choice of g E c and is a conformal
invariant of c equal to C (M, J, go) = fM ugodpgo, so we can denote it just as C (M, J, c).

It follows from (3.2) that f M sycdpg0 S C (M, J, c) which gives the estimate

Y(c) S C(M, J, c), (3.3)

with equality in (3.3) if and only if go is a Yamabe-Kahler metric:

The fundamental constant C (M, J, c) is closely related to the complex geometry
of (M, J) in view of the following vanishing theorems of Gauduchon [22]. Denote by
Pm (resp.Qm) the dimension of the space of holomorphic sections of K 3’" (resp. of
K ‘®"‘). Then we have:

(a) C(M,J,c) > 0 => Pm = 0, Vm > 0;

(b) C(M,J,c) < 0 => Q... = 0, Vm > 0;

(c) C(M,J,c) = 0 => Pm = Q... and Pm 6 1,0,Vm > 0.

In particular, for any positive conformal class c, the estimate (3.3) gives G (M, J, c) >

45

0, hence such a surface has to be of negative Kodaira dimension.

It is clear that except for the case when Pm = Qm = 0, Vm > 0 (some surfaces of
negative Kodaira dimension), the sign of G (M, J, c) is independent of c (see [7]). We
also note that the existence of a Hermitian conformal class c with G (M, J, c) = 0 does
imply the existence of a metric g E c of vanishing Hermitian scalar curvature 21,, (see
[7], Corollary 1.9), hence the Ricci form R"(F) (which represents up to multiplication
with—— the first real Chern class of (M, J )) is anti-self-dual. In particular, we have
01 S 0 with equality if and only if c1 = 0. So, on any complex surface (M, J) satisfying
2x(M) + 3o(M) > 0 (or 2x(M) + 3o(M) = 0 and c1 75 0), the sign of C(M, J, c) is
also independent on the Hermitian conformal class c.

On the other hand, for a compact almost Kiihler manifold (M, g, J,w) we have
another estimate for the Yamabe constant, coming from the basic scalar curvature

inequality proved in (1.12).

[M sgdug S 47rc1 - [w],

with equality if and only if the structure is Kahler. It follows that

Y(c ) < 4J27r———— [”1 (3.4)
[w] M
with equality if and only if g is a Yamabe-Kahler metric.
Now we shall use Theorem 3.1 to compare (3.3) and (3.4) on some Hermitian
surfaces. We start with the following proposition, due to LeBrun in a more general

setting [35]:

Proposition 3.7: Let (M, g, J, F) be a Hermitian surface with b1 even and let w be

a harmonic, self-dual form on M of non-negative trace. Then the following inequality

46

holds:
[3%du g 47rc1 - [w],

where s is the scalar curvature, dp is the volume form and | - | is the pointwise norm

determined by the metric g.

Proof: According to Corollary 3.2, we have two cases to consider.

Case 1: The form w is non-degenerate everywhere on M.

Denote by u the (strictly) positive function given by w2 = u4F2, or, equivalently
J2u2 = |w|. The metric g’ = uzg is an associated metric for the symplectic form w.
The almost complex structure induced by g’ and w is homotopic to J, hence it has

the same real first Chern class as J. Using (19), we get:

/ Sgldugr g 47rc1 - [w] (3.5)

Standard formulas for a conformal change of metric g’ 2 (Pg give
39: = (it"sg + 6u‘3Agu,

dug: = u4dpg.

From these we obtain

[Sgidygl =/s g-’-—2\/|_gdpg +6/Idulgdpg_ > [39wa lzgdpg, (3.6)
and the proof is finished for the Case 1.

Case 2: The form w is the real part of a holomorphic (2,0) form.
In this case we have c1 - [w] = 0, since on a complex surface c1 can be represented

by a (1,1) form (the Ricci form of a Hermitian connection). Consider wo a harmonic,

47

self-dual form, nowhere degenerate on M and denote
wt = wo + tw,

for t > 0. Then wt are non-degenerate, harmonic self-dual forms for any t, so we can

apply Case 1 to them. It follows

[31%du S 47rc1 - [wt].

Taking into account that Cl - [w] = 0, this becomes

ta)
[SELL-Jan S 47rc1 - [wo],

J2
and, after dividing by t,

s Iwol2 2 < w,wo > 1 47?
f—J_2( t2 + t + |w|2)2dpg ‘<‘ 761 ‘ M”

 

Taking the limit t —) 00, we obtain the conclusion in this case too. 1:]

Remark 3.2: A more careful application of relation (3.6) implies the inequality
/ sglwtdu. + 6 / urinal/213d)». 3 4mm M,

for any Hermitian metric g and any harmonic, self-dual form w of non-negative trace.
As a consequence, we see that on a scalar-flat Hermitian surface with b1 even, all

holomorphic (2,0) forms have constant length.

Corollary 3.3: Under the same assumptions as Proposition 3.7, we also have the
inequality:

[s2du Z 327r2(cf)2,

48

where cf denotes the harmonic, self—dual part of Cl.

Proof: Apply Proposition 3.6 to the harmonic, self-dual form w which satisfies w =

—cf. We get

much? 5. / -8|w|du s / Isllwldu.

Schwarz inequality implies

Minor)? 3 (/ Sam] may)?

Since w is the harmonic representative of the class cf“, we have

/ lwlzdu = (ctr,

and the conclusion follows. C]

As already mentioned, on a rational surface (M, J) with cf _>_ 0, the sign of
C (M, J, c) does not depend on the Hermitian conformal class c. Therefore it is always
positive, since any rational surface admits a Kiihler metric of positive total scalar
curvature (of. [50, 20]). With this observation and Proposition 3.7 in hand, we prove

the following

Proposition 3.8: Let (M, J) be a rational surface with of Z 0. Then for any

Hermitian conformal class c on M we have

Y(c) S 47r‘/2(cf’)2 S C(M, J, c), (3.7)

where of denotes the harmonic self-dual part of Cl. Moreover, equality in the right-

hand side holds if and only if c contains a K c'ihler metric, while equality in the left-hand

49

side holds if and only if c contains a Yamabe-thler metric.

Proof: Let g E c be an almost Kahler metric, with fundamental 2-form w given by
w = F + Re(a), where F denotes the fundamental 2-form of the standard metric go
and oz is a (2,0) form. The almost complex structure given by g and w is homotopic
to the complex structure J and hence they induce the same first Chern class, c1.

Denoting by 7 = R°(F) the (1,1)-Ricci form of (J, go), we have

cl.[w] = i fM7Aw : (3.8)
[w] - [w] 2” «wa A w
___ 1 IM ”god/1'90 < lC(M J C)
4 1 a

 

4V5” JIM dug. + %fM |Re(a)|2dugo _

with equality if and only if Re(a) vanishes, i.e. if and only if go is a Kahler metric. On
the other hand, since b+(M) = 1 and c1 - [w] > 0 ([45, 32]), we have that (c1)+ = Aw,

for some positive real constant A. Hence

%1— [led]. = A(/[w].[w] = V (Cf?)

which after a substitution in (3.8) completes the proof of the right-hand side inequality
of (3.7). The other inequality is a consequence of Proposition 3.7 and the above

observation. [I]

Corollary 3.4: Let (M, J) be as in Proposition 3.8. Then for any Hermitian con-

formal class c, the fundamental constant C (M, J, c) satisfies

C(M, J, c) 2 47n/2c’f’

with equality if and only if c contains a Kc'ihler metric and the first Chem class has a

self-dual representative with respect to c.

50

Corollary 3.5: For any Hermitian conformal class c on CP2 the Yamabe constant

Y(c) and the fundamental constant C (M, J, c) satisfy
Y(c) S 12J27r S C(M, J, g),

with equality in the right-hand side if and only if c contains a K (ihler metric and with
equality in the left-hand side if and only if c is conformally equivalent to the class of

the Fubini-Study metric.

Proof: Since b‘ (CP2) = 0 we have that 47r(/2(cf)2 = 47r‘/2c¥ = 12J21r. The case of
equality in the left hand side of the inequality follows from the observation that the
only Kahler metric of constant scalar curvature on CP2 is the thini-Study metric.

Cl

Remark 3.3: The inequality Y(g) S 12J27r was proved by LeBrun in [35] for an
arbitrary conformal class on CP2, investigating the “size” of the zero set of a self-
dual form. As was noted there ([35], Corollary 3), this estimate can be used to give a
simple proof the Poon’s result of the uniqueness of the self-dual structure of positive
type on CP2. Our Corollary 3.5, the fact that any Hermitian self-dual structure on
CP2 is of positive type (see [4]) and LeBrun’s arguments give a simple proof in the

framework of Hermitian geometry of the following:

Corollary 3.6 [4] Any self-dual Hermitian conformal structure on CP2 is equivalent

to the standard one.

51
3.4 Conformal transformations of almost Kahler

metrics on 4-manifolds

D. Blair asked in [13] the following question: given a compact almost Ka'hler mani-
fold (M 2", g, J, w) and ()3 an isometry of the almost thler metric, is ()5 necessarily a

symplectomorphism ( or anti-symplectomorphism}?

This is a particular case of our Problem 1 and we use the results proven so far
to give some answers in dimension 4. In fact, in our results as will be a conformal
transformation of the almost Kiihler metric, i.e. the pull-back metric ¢*g is conformal
to g. We first remark that Blair’s question has an affirmative answer for compact 4-
manifolds with b+ = 1, as an easy consequence of Proposition 3.1. From the same

Proposition 3.1, our next partial positive result also follows easily.

Proposition 3.9: Let (M 4, g, J,w) be a compact almost Kahler manifold and let
(p be a conformal transformation of g, homotopic to the identity inside the group of

diffeomorphisms of M. Then ()3 is an automorphism of the almost Kiihler structure
(9. J, w)-

Proof: By assumptions, ¢*w is cohomologus to w and ¢*g is conformal to 9. Since
(V9 is an almost Kahler metric for the symplectic form ¢*w, it follows that g E
CAM“, fl CAM¢.,,. By Proposition 1 (a), this may hold only if ¢*w = w, so (i)
is a symplectomorphism. To conclude that (t is also an isometry just note that a

symplectic form cannot have two distinct, conformal associated metrics.

Remark 3.4: Note that the above result is true in any dimensions if we assume ((5
to be an isometry in the identity component of the diffeomorphisms group. It can
be considered as a slight generalization (in complex dimension 2) of the well-known
results of Lichnerovicz [37] about the connected group of isometries of a compact

Kiihler manifold.

52

Hence Blair’s question has an affirmative answer in this case.
The next result appears as a consequence of Theorem 3.1.

Theorem 3.2: Let (M 4, g, J,w) be a compact Kahler , non-hyper-Kiihler surface. If
(t is a conformal transformation of the Kiihler metric then (15 is a symplectomorphism

or an anti-symplectomorphism.

Proof: Let (t be a positive conformal isometry. Suppose that (b is not an isometry.
Then (15" would be an almost Kahler structure in the conformal class of g. Now,
according to Theorem 3.1,(a2), we have that there is a whole S1 family of almost
Kahler structures with respect to the metric (t‘g. Using (13“, we can induce a 5‘-
family of almost Kiihler structures with respect to g, which contradicts with Theorem
3.1, (a1). So, (I) must be an isometry. We use now Theorem 3.1,(a1) one more time

to complete the proof. El
Remark 3.5: The above result is closely related to Theorem 5.3 in [41].

Now we will give examples when Blair’s question has a negative answer. However,
all such examples that we know so far are very special (all have c1 = 0, for instance).
It might be possible that in most instances isometries of almost Kahler metrics do

indeed preserve (up to sign) the symplectic form.

Remark 3.6: The conclusion of Theorem 3.2 is no longer true for T4 = (S 1)4. Take
the standard metric and consider the Kfihler form w = d01 A d92 + d03 A d04. Let
(f) be the diffeomorphism which acts as identity on the first and third components
and switches the second and the fourth. This is an isometry of the metric, but is
clearly not an :h-symplectomorphism. Hence Blair’s question has a negative answer
for T4. Fore some special K3 surfaces such isometries (with respect to a hyper-Kahler
metric) have been shown to exist by Alekseevsky-Graev [1]. Non-Kahler examples

of this type can be given on T4 (see [5]) an on primary Kodaira surfaces which are

53

T2-bundles over T2.

Remark 3.7: It may really happen that an isometry of an almost Kahler metric is
an anti-symplectomorphism, as the following example shows:

Let M4 = 52 x S2 with the standard product metric. This metric is Kahler with
respect to the form w = wl — (.02, the diffeomorphism taking one factor into the other

is an isometry, but it is an anti-symplectomorphism of the form w.

3.5 Kiihler forms versus symplectic forms

Let M be a compact manifold admitting Kiihler structures. Let us denote by [C the
set of Kéihler forms on M and by S the set of symplectic forms on M. Obviously
KI Q S and we will be interested to detect differences between the two sets. The
following lemma gives an invariant which distinguishes cohomology classes that can
be represented by Kahler forms. It is due to Perrone in [40], but the proof we present

here is shorter.

Lemma 3.2: For any Kiihler manifold (M 2", g, J, w), the following inequality holds:
(01 U lwl'H)2 2 (Ci U [w]"‘2)([w]”), (3-9)

with equality if and only if c1 = A[w], for A E R.

Proof: Consider the bilinear form

baa]. [flD = [a] U [[3] U [60]”.

defined on real (1,1) cohomology classes with values in R. From relation (1.3) we see
that this is a symmetric form of signature (1, k — 1), where k = dimRHl'l. Let 7 be

the harmonic representative of Cl. It is a (1,1) form and since we are on a Kahler

54

manifold, it decomposes further as

Y=W+£,

where a = ficl U [w]”'1 is a constant and 5 is a trace free, harmonic, (1,1) form.
By relation (3), b([£], [6]) S 0, hence b(c1 - a[w],c1 - a[w]) S 0 and this implies the

inequality from the statement. Equality holds if and only if c1 = a[w]. C]

Using the above Lemma, we will now show that starting with a certain Kahler
form after small deformations in certain directions we will leave the space IC, but still

remain in 8.

Proposition 3.10: Let (M 2", 9, J, w) be a Kiihler manifold with c1 = A[w], for A 6 R,
A 76 0. Assume that [3 is a holomorphic (2,0) form on M. Then, for t small, t at 0,

the forms wt = w + tRe(fl) are symplectic but not Kahler forms.

Proof: Let us denote by

F(t) 3= (01 U lwtln-l)2 — (Ci U [wtl"_2)(lwtl")-

Since c1=A[,w] we clearly have F(0)= 0 and an easy computation shows that
F’ (0) = 0 as well, for any complex dimension n. We will show that F” (0) < 0, hence

F(t) < O for t 75 0, small. In fact, if n = 2

F”(t)= -2(Ci)(lRe(fl l)2)= -fl2A2([w12)([Re( )12)<0,

for any value of t, hence the result follows from the Lemma 3.2. If n 2 3, after an

elementary calculation which uses the assumption c1 = A[w] we get

F "(0) = -2A2([R¢3(fi)l2 U [wl"‘2)([wl")o

55

By 0, since Rem) is a J-anti-invariant 2-form, we see that F” (0) < 0. Thus, for
t at 0, small, we get F(t) < 0, so by Lemma 3.2 the cohomology classes [wt] cannot
contain any Kahler forms. On the other hand, wt are symplectic forms for small
values of t, since the non-degeneracy is an open condition. (As a matter of fact it can
be shown that wt are symplectic for any value of t.) D

The statement could be considerably strengthen in dimension 4.

Proposition 3.11: Let (M, 9, J, w) be a Kahler surface with cf > 0. Assume that B
is a holomorphic (2,0) form on M. Then the forms wt = w + tRe(fl) are symplectic
for any value of t E R, but for |t| suficiently large they cannot be Kahler forms.
Moreover, if the the Kahler surface satisfies c1 = A[w], for A E R, A 75 0, then the

forms w, are not Kahler for any value t 7t 0.

Proof: Consider again the function
F(t) : (01 U [L‘Jtll2 — (Ci)(lwtl2)-
Note that
w? = (w + (112.2(5))? = «22 + t’Rem)2 = (1 + gimme.

Thus wt are non-degenerate for any value of t, hence symplectic, and [wt]2 > [w]2 for

any t 71$ 0. Since Cl is a (1,1) class and fl is (2,0), c1 U [w] = Cl U [w]. Thus

F(t) = (c. u [tz — (cow) — “A“Zew’a

 

so for It] big F(t) < 0.

56

If c1 = A[w], we see that

 

Fm = _(cixuzewnay,

so F(t) < Ofor anytyéO. [:1

Proposition 3.12: Let (M, J) be a minimal complex surface of general type, with
b+ > 1 and no two-torsion classes in H2(X, Z). Consider w to be a Kahler form on
M and B a holomorphic (2,0) form on M. Let at be the line joining the cohomology
classes [w] and [Re(fl)], at = (1 — t)[w] + t[Re(fl)]. Then the cohomology class ao
contains a Kahler form; fort sufficiently big, but t 76 1, at contains a symplectic form,

but does not contain any Ka'hler form; an cannot contain any symplectic form.

Proof: The first two claims follow immediately from the hypothesis and the Propo-
sition 3.11. It only remains to prove that al does not contain any symplectic form.
Let us assume that wl is a symplectic form with [wl] = [a1]. First let us remark
that the canonical bundle K1 induced by wl must be isomorphic to :lzK, where K is
the canonical bundle induced by the Kahler form w. This follows from two results of
Seiberg—Witten theory. By a theorem of Taubes [44], the spinc structure induced by
K1 has non-vanishing Seiberg—Witten invariant. But for minimal complex surfaces
of general type Theorem 7.4.1 of [39] says that the only spine structures with non-
zero invariants are those induced by :l:K, where K is the canonical bundle. Hence

K1 = :l:K. But then
K1 ~[w1] = iK-a = K - [Re(fl)] = 0,

since K is a (1,1) class. By another result of Taubes from [45], this can happen if

and only if K1 is trivial, hence K is trivial. But for minimal complex surfaces it is

known that C¥(M) = K2 > 0. D

CHAPTER 4

Seiberg—Witten Invariants when

Reversing Orientation

A conjecture formulated within the Donaldson’s theory, but easily adapted to the
Seiberg-Witten context states that each compact, orientable, simply-connected 4-
manifold has with one of the orientations all the invariants equal to zero. In this
chapter we give an affirmative answer to this conjecture for a large class of com-
plex surfaces. The author has proved the conjecture for complex surfaces of negative
signature admitting a Kalhler Einstein metric. The same result was obtained inde-
pendently by N. Leung and recently, D. Kotschick proved a more general theorem.
We will state the theorem of Kotschick and indicate how the proof goes in general,

but we will treat in detail the case considered by the author.

4.1 Statement of the result

Let X be a closed, oriented 4-manifold and let X denote the manifold X with the
reversed orientation. Denote by x(X) the Euler characteristic and by o(X) the signa-
ture of X. The following conjecture is known about the Seiberg-Witten (Donaldson)

invariants and the orientation:

57

58

Conjecture: For a compact, orientable, simply-connected 4-manifold X, all Seiberg-

Witten invariants vanish either on X or on X .

Although some work has been done (see [31]), within the frame of Donaldson’s
theory the conjecture remained wide open. 'Itanslated to Seiberg—Witten invariants,
an affirmative answer to the conjecture has been recently given for a large class of
complex 4-manifolds. The author has proved [19] the statement of the conjecture for
complex surfaces with negative signature which admit Kiihler Einstein metrics. A
similar result has been also obtained independently by N. Leung, [36]. Recently, D.

Kotschick, [30] obtained a more general theorem whose statement we give below:

Theorem 4.1: (Kotschick, [30]) Let X be a complex surface of general type and
assume that X admits a non-zero Seiberg- Witten invariant (of any degree). Then X
has ample canonical bundle, cf (X) is even and the signature o(X) is non-negative.

Moreover, X has zero signature if and only if it is uniformized by the polydisk.

This result implies that the above conjecture is true for any complex surface of general
type X satisfying one of the following conditions:

(i) cf(X) is odd;

(ii) canonical bundle is not ample;

(iii) o(X) < 0.

4.2 Proof of Theorem 4.1 and Examples

Let us first remark that the result does not use simply connectedness. However,
trying to extend the conjecture for all complex surfaces of general type, ignoring the
assumption of simple connectivity does not work. Signature zero examples are easy
to obtain, as there exist Kahler surfaces with orientation reversing diffeomorphisms.

We will show that positive signature examples also exist. First we start by giving

59

the proof of the particular case of the conjecture obtained by the author, but also
indicating how Kotschick obtains the more general case (for details see [30]). Our

main purpose is to highlight the role played by the signature.

Sketch of Proof: The conclusion that of (X) must be even comes from thefact that
the dimension of the moduli space for the spinc structure with non-zero invariant on
X is even. Then Kotschick argues that the canonical bundle is ample, by showing
that there are no embedded holomorphic spheres of self-intersection -1 or -2 in X.
Indeed, if X contains an embedded sphere of negative self-intersection, non-trivial in
homology, in X it becomes sphere of positive self-intersection and this would imply
that all invariants of X vanish.

When the canonical bundle is ample, by Yau’s solution to the Calabi conjecture
[49], it follows that X admits a Kahler-Einstein metric g (this is the case treated in
[19] and [36]). Rescaling this metric, we may assume that Volg(X) = Volg(X) = 1.

Because g is a Kahler-Einstein metric on X, we have

82

3271'2 '

 

caX) = (4.1)

Denote by L the determinant line bundle of the spinc structure on X with non-zero

Seiberg—Witten invariant. A theorem of LeBrun [34] implies that

2

 

c1(L)2(X) < S

_ 32”,. (4.2)

On the other hand, from the dimension formula of the Seiberg-Witten moduli space,
61(L)2(X) Z 300?) + 2X0?) = -30(X) + 2><(X) = (43)

S2

327r2’

 

= —60(X) + 3o(X) + 2x(X) = —6o(X) + c¥(X) = —6o(X) +

60

where in the last equality we used (4.1).

Relations (4.2) and (4.3) imply that o(X) 2 0. For the equality case, the reader
is referred to [36]. We will just say that o(X) = 0 implies equality in (4.2) and this
equality holds if and only if there exists a Kahler-Einstein structure (g, J,w) on X
as well. But then a holonomy argument implies that X is covered by the product of

two disks. El

Next we show that the conclusion about the signature in the theorem of Kotschick
is sharp. We achieve this by giving examples of bi-symplectic 4-manifolds X (i.e. both

X and X are symplectic) and invoking the following important result of Taubes:

Theorem: (Taubes, [44]) Let (X, w) be a closed, symplectic 4-manifold with b+ 2 2.

Then the Seiberg-Witten invariant of the canonical class is equal to 3:1.

For zero signature, the simplest examples are products of two Riemann surfaces,
X = 21 x 232. Ifw,- is a volume form on 23,-, i = 1, 2, then w = w1 +w2 and (I) = w1 —w2
are symplectic forms on X inducing opposite orientations. If we take 21,22 with the
genus of each at least 2, then X = 21 x 22 is a complex surface of general type. If
w1,w2 are volume forms corresponding to hyperbolic metrics on each surface, then
the product metric on X is Kahler-Einstein metric compatible with both w and (2.

Now let us consider the case of positive signature.

Theorem 4.2: There are examples of complex surfaces of general type having non-

zero Seiberg—Witten invariants with both orientations.

Proof: Let us remark that the product examples of bi—symplectic 4-manifolds that we
discussed above belong to a larger class of manifolds admitting symplectic structures
with both orientations. ”Almost” all locally trivial fibre bundles F —) X 4 -> 2, where
F and E are closed Riemann surfaces, admit bi-symplectic structure. To see this we
just have to repeat Thurston’s construction of symplectic forms [46].

The only restriction is [F] 919 0 in H2(X , R). If this is satisfied, Thurston shows

61

that there exists a, closed 2-form on X, which restricts to a symplectic form on each
fiber, Fx, :1: E X. Taking o a symplectic form on the base 2, for e > 0 small enough,
w = 7r*o + ea is a symplectic form on X, where 1r is the projection 7r : X —) 2. The
induced volume form is

wAw=err*oAa+620(Aa.

But then, for (3 possibly smaller, (I) = 7r*o — ea is also a symplectic form on X and

wsz—err*oAa+e2aAa

gives the opposite orientation.

Many other examples of bi-symplectic 4-manifolds with signature zero can be
obtained in this way. For instance, if we take F —-) X 4 —) E to be a holomorphic fibre
bundle, then it is shown easily that the signature of the total space must be zero.

However, the signature of the total space of a fibre bundle is not always zero.
Independently, Kodaira [28] and Atiyah [6] constructed a class of examples of non-
zero signature. In fact, with one of the orientations, the total space X is a complex
surface of general type, and with this orientation the signature is positive. Here is a
short description of the examples. Take R to be a Riemann surface which has a fixed
point free holomorphic involution denoted by 7 (any surface of odd genus has fixed
point free holomorphic involutions). Let G be the cover of R corresponding to the

homomorphism

7T1(R) —-) H1 (R; Z) —-) H1(R; Z2),

and let f : C —-> R be the covering map. In C x R consider the divisor I‘ = A U A’,
where A = graph( f), A’ = graph(r o f). From the way the covering f was chosen,
I‘ induces an even class in H2(C x R, Z). Denote by X the 2-fold cover of G x R

62

branched over I‘. Note that X fibers over both C and R, but X is not a holomorphic
fibre bundle. As for the signature of X, using the general formula for the signature

of branched covers, we get

o(X) = 20(C x R) — gr . r,
where I‘ - l" is the self-intersection of the branch locus in C x R. Since 0(C x R) = 0

and
F-I‘zA-A+A'-A'=2A-A=2X(C)<0,

it follows that o(X) > 0. C]
It is worth remarking that the existence of symplectic forms inducing both orien-
tations may be used in this case to show that the canonical bundle of X is ample,

therefore X admits a Kahler-Einstein metric.

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