Specific complex geometry of certain complex surfaces and three folds

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A Thesis Submitted for the Degree of PhD at the University of Warwick

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Specific complex geom etry of

certain complex surfaces and

three-folds

by

Yuri Dimitrov Bozhkov

Thesis submitted for the Degree of

Doctor o f Philosophy

at the University of Warwick

Contents

1 A c o n s t r u c t io n o f a l m o s t a n t i-s e lf-d u a l m e t r i c s o n K u m i n e r s u r fa c e s 1

1.1 In tro d u ctio n ... 1

1.2 T h e curvature o f h ... 3

1.3 T h e main technical l e m m a ... 8

1.4 A discussion o f a possible application o f T a u b e s ’s m e t h o d ... 10

1.5 O n the stability o f th e tangent bundles o f K u m m er s u r f a c e s ... 13

2 O n t h e c o m p le x s t r u c t u r e s o f # „ S3 x S3 1 7 2.1 In tro d u ctio n ... 17

2 .2 Elements o f Hermitian non-Kalilerian g e o m e t r y ... 23

2.3 ‘Gauduchon m etrics, stability and L i-Y a u ’ s th e o r e m ... 29

2 .4 Stability o f the tangent bundle o f # „ S3 x S 3 ... 31

2 .5 Application o f L i-Y a u ’s theorem to # „ S3 x S3 and som e direct conse­ quences ... 34

2 .6 Som e notes on the classification o f alm ost H erm itiau manifolds by Gray and H ervella... 38

2 .7 # „ S3 x S3 and its place in the the G ray-H ervella classification . . . . 46

2.8 Som e conditions on # „ S3 x S3 ... 49

2 .9 Deformations o f the H ermitian-Einstein m e t r i c ... 53

2 .1 0 Deformations o f the com plex structure ... 59

2.11 O n the deformations o f both the H erm itian metric and the complex structure ... 63

Acknowledgements

I would wish to express m y heartfelt gratitude to Prof. James Eells and Prof. Nigel

Hitchin for their continuous support and invaluable help:

I would also like to thank Prof. N arasim lian for his useful com m ents, Prof.

Audrey Todorov for proposing the problem in C h ap ter 1, Prof. A lberto Verjovsky

for his constant encouragement.

Special thanks are due to the International Centre for Theoretical Physics in

Trieste, Italy, for its financial support and gracious hospitality which m ade the com ­

pletion o f the thesis possible, and to the U niversity o f Warwick for having given m e

Declaration

T h e work in this thesis is original as far as I am aware, except where explicitly

stated to the contrary. A part o f the con tent o f Chapter 1 was published in Serdica

Declaration

T h e work in this thesis is original as far as I a m aware, except where explicitly

stated to the contrary. A part o f the content o f Chapter 1 was published in Serdica

A b s t r a c t

O n e o f the m ost im portant consequences o f Y a u 's proof o f the C a la b i’s conjecture

is the existence o f a non-trivial Ricci-flat metric on K 3 surfaces. For its explicit

construction would be o f great interest. Since it is not available yet the qualitative

description o f this metric w ould also have certain significance. In Chapter 1 we

propose an approxim ation o f th e K 3 Kahler-Einstein-C alabi-Yau m etric for K um m er

surfaces. It is obtained by glu in g 16 pieces o f the Eguchi-H anson m etric and 16

pieces o f the Euclidean m etric. T w o estimates on its curvature are proved. Then

we discuss the possibility o f application o f C .T a u b e s’s iteration sch em e for solving

anti-self-duality equations. T h e reason is that the curvature o f the m etric in question

is concentrated in small thin regions and it is alm ost anti-self-dual. It can be also

used later to deduce stability o f K um m er surfaces’ tangent bundle.

In C hapter 2 we consider a special case o f com pact 3-folds M which are diffeo-

morphic to the connected sum o f n copies o f S3 x S 3. If n > 1 03 , there is a complex

structure o f C\ = 0 on M , w hich is a uou-Kahler m anifold. W e prove that there are

no non-trivial fine bundles on M and hence we deduce that its tangent bundle is

stable with respect to any G au d uch on metric. B y a theorem o f Li an d Yau we con­

clude that there is an Hermitian-Einstein metric on M . O ur basic hypothesis is that

the Hermitian-Einstein m etric and the Gauduchon metric coincide. This is similar

to the previous situation on K 3 . T hen we consider the deform ations o f this metric,

keeping the volum e and the co m p lex structure fixed. W e seek th e place o f M in the

classification o f alm ost H erm itian manifolds by Gray and Hervella an d explore some

sorts o f conditions which can b e imposed on M and which can su b stitu te the Kiihler

one. W e also show that on Hermitian non-Kaliler manifolds w ith h 2'0 = 0 there are

Chapter 1

A construction o f almost

anti-self-dual metrics on Kum m er

surfaces

1.1

I n t r o d u c t io n

T h e existence o f K a hler-E instein, or m ore generally H erm itiau-Einstein, metrics plays

an essential role in the stu d y o f com pact com plex m anifold s. T h is is so for Yau ’s

proof o f th e C alabi’s conjecture. T h e latter states that if th e first Chern class Cj o f a

com pact Kahler m anifold M vanishes, then there exists a K a h ler metric on it, which is

Ricci-flat [7, 8]. C alabi has proved the uniqueness o f such a m etric and has suggested

how to prove its existence [7]. In this direction one can ask the question if M admits

Kahler-Einstein m etrics with regard to the sign o f c x( M ) . I f C\ < 0 , Y au has given a

positive answer to the ab ove question [64] and this has led him to som e new results

in differential and algebraic geometry [63]. T h e case Cj > 0 is not still completely

investigated, but there is a serious recent progress due t o Tian [52, 53, 54], Nadel

[43], Tian and Yau [56], A u b in [1, 2]. Y a u ’s theorem and further development o f the

problem for finding Kahler-Einstein m etrics could be considered as a generalization

o f R iem ann’s U niform ization Theorem to higher dim ensions.

O n e o f the m o st im portant consequences o f Y a u ’s p r o o f o f the C alabi's con­

jecture is the existence o f non-trivial Ricci-flat Kahler m etric on Iv3 surfaces. By

definition, a K 3 surface is a 2-dimensional com pact c o m p le x manifold whose first

Betti num ber bt = 0 a n d whose first Chern class Cj = 0 . B y a result o f Siu [48] it

follows that every K 3 surface is Kahler . F rom this fact and Y a u ’s proof one concludes

that K 3 surfaces adm it Kühler non-trivial R icci-flat metrics. T hey can be used in

the investigation o f the moduli space o f K 3 (see [57, 31]).

For the present K 3 are the unique sim ply-connected com pact manifolds on

which such m etrics exist. Another com pact m anifold which adm its an Einstein vac­

u um metric is th e torus T in C " . In this case the unique solution o f Einstein’s

vacuum equations is the restriction to T o f the Euclidean flat metric, but it is not so

interesting from a differential geometrical point o f view. The explicit form o f the K 3

Kähler-E instein-C alabi-Yau metric is not know n yet. The problem o f its construc­

tive description w as pointed out by Y a u [65] an d Kirby [30]. T h e construction o f

this metric in explicit form or in appropriate approximation is o f great interest for

both m athem aticians [26] and physicists [45]. Actually, N.Hitcliin has set in [26] the

problem o f finding o f the K 3 metric explicitly and proposed a m ethod o f attacking

based on tw istor theory. Later on , using also tw istor ideas, Topiwala published a new

proof of the C a la b i’s conjecture for K u m m e r surfaces [58, 59].

In this chapter we propose an approxim ation o f the K 3 metric in the particular

case o f a K u m m e r surface. It is constructed as follows.

Let T = Z4 be a lattice in C 2, which is generated by four vectors, linearly

independent over R . Consider the involution

a : T--- ► T

defined by

a ( x ) = - x

which acts on th e complex torus T — C2/r. I f we factorize the torus with respect to

the relation o f th e equivalence

if and oidy if tr(x ) = y, we shall obtain a singular surface

It is easy to see that „Y has 16 singular points and near a singular point it can be

em bedded locally in C 3. In fact, near a singular point it can b e identified locally with

the cone z 2 = x y in C 3.

T h en we blow up the 16 singular points and let K be the resulting non-singular

surface. K is said to be a K u m m er surface. O ne verifies that C\(I\) = 0 , and 6i = 0

[49]. So K is certainly o f type K 3 .

Under th e sixteen a - processes every singular point is replaced b y a copy of

C P \ the co m p lex one-dimensional p rojective space. Inside a neighbourhood (ball)

o f every distinct projective hue in K , which has radius A — A2, where A is a sufficiently

small num ber, we consider the m etric o f Eguclii- Hanson q e h ( see [9, 2 7, 15] ) and

outside the neighbourhood o f radius A - the Euclidean m etric g g . Let ( « , (i) be an

appropriate partition o f unity subordinate to the above balls. W e define

h = otgEH +

09E-It is a H erm itian metric, which is n ot K ah ler one, but in large regions ( in those

regions, where h = Qe h or h = Qe ) it is K ah ler . Moreover, it is almost anti-self-dual

because gEH and q e are anti-self-dual. T h e metric h was deduced ’’ heuristically” ( if

we use the w ords o f N .H itc liin ,[2 6 ],p .ll5 ) b y Page [46]. T h e purpose o f the present

chapter is to give a m athem atically precise description o f this metric, in particular

to obtain estim ates on its curvature (sections 1.2 - 1 .4), and to discuss the possibility

to be used for the proof o f such geom etrical properties as stability o f the Kum mer

surface’s tangent bundle. A part o f th e content covers the paper [5].

1.2

T h e cu r v a tu re o f h

W e are going to introduce som e notations and give the exact definition o f the metric

h which is th e m ain object in this chapter.

Let A b e the connection m a trix o f some connection on K . T hen there is a

well-defined operator

V /4: r ( A ' ’ ) — H A ' O A 1)

which is called covariant derivative. A s usual A p denotes the space o f exterior p-form s

on K . U sing the projection

jt : r ( A '0A *) — ► r ( A ^ ‘ )

one c a n define another operator

D A = i r o V A

which acts on the p-form s a s follows

D A<p = d p + A f\ p + ( —1 )P+V A A .

Very often , instead o f ’’ connection D A" ( o r V4 ) we shall speak about the ’’ connection

A".

Now let p be a p -fo rm o n K (0 < p < 4 ). In som e local coordinates x 1, x 2, x3, x4

it is expressed as

be an Hermitian m etric on K . R e(h ) is the corresponding Riemauuian metric which

we shall denote again b y h with n o confusion. B oth h and the Euclidean metric

determ ine Hodge operators */, and *. In local coordinates ( see [14] ) :

w here h = d et(/ttJ) and f is the fully antisym m etrical tensor. If F is a m atrix with

elem ents the p-form s <p'j, then

ip = p ix...ipd x " A ... A d x 'p.

E veryw here below we shall use Einstein’s sum m ation convention as above. Let h

,Pi,...ipd xj' A ... A <fxJ,-p

and

dxJp*' A ... A dxu

T h e L , norm s o f F are defined by

IIFIIl. = (

JK

Let { p j} j = l , . . . , 1 6 , lie the set o f the singular points o f the surface X :

Pj e

{(0 ,0 ,0 ,0 ); (0 ,0 ,0 ,1 /2 ); .. . ; ( l / 2 , 1 /2 ,1 /2 ,1 /2 )}.

Let B { be the ball o f radius A and centre at pj. W e choose

This condition provides

D ‘„ n = 0

if j ^ a. Define the function atj on X by

„ , t ) = J 1 B \ - » ' ( l \ 0 H r ? B i

and 0 < Qj < 1. Set

« ( * ) =

and &(x) = 1 — a(ar), i.e. a + /? = 1. Then

a ( x ) i 1 if x 6 B jX )7 for some j = ” \ 0 if x £ B x for every a.

After m aking 1G <7-processes, we shall denote again by B 3X the image o f the ball B x

with no confusion. In the ball = B 3X_ X, the metric o f Eguchi-Hanson has the

form

9£H ~ ( i + ij.i1 )1 + y r + T i )d

c > 0 is an arbitrary constant ([9]). In order to prove a technical lem m a (see Section

See [9]. O n the other hand our K um m er surface K defines the sam e bundle, that is

9eh is th e metric near every singular point p we need. T h en the m etric h is defined

by

h = a (x )g EH + 0 (* )g E ,

where g E is the Euclidean m etric. Let (<p, U ) be a local coordinate chart such that

p e U , p : U ---♦ R 4, <p(p) = 0.

Introduce real normal (for h ) coordinates x1, z 2, x 3, x 4. T hus

h**(p) = S'1, dh'3(p) = 0 ; <p(q) = x ,q e U , |h'1 - 6,J| < \<p\2p (p ) < |x|2p(p).

for all q € U . \x\2 is th«‘ Euchdean norm o f x 6 R 4 and p(p) is a constant which does

not depend on A. Hence

W e shall work always in a neighbourhood o f a blown up singular point p = p} , i.e. it

is sufficient to prove the estim ates on the curvature o f h only in the ball B\ o f radius

A, because outside this ball li = gE and here we have nothing to prove. W e make the

change o f variables

1.3) we choose c = 4. gEE is a Kahler metric on the bundle L — » C P1 , where L is

biliolom orphically equivalent to the cone

{ ( x , y , z ) e C 3 : z 2 = x y ) .

hij = Sij + Q ( A2). (

1

1

x l + z2 = x3 +

where

A — rt( 1 + 4 1 4 (j' : + + e * ) ( i + e ’ + e ’ );, i- )2 + / I + 41

) + H,

b =

J1 + ? +**?

( l - t - * 1 ,-t-ii‘ )(:r1.r:14 -« ,»<)

s/\ 4 - 4 i

( 1 4 - * " + * " ’ ) ( * V - * V )

1 V T + T t

---t = (1 + * " 4- + r ' ’ ).

I f we look at the p ro o f o f L e m m a 1 .1 carefully, we ran get im m ediately (1 .1 ) and

(1.2), that is, our coordinates are actually normal.

Let V ,40 denote the L evi-Civita connection corresponding to /*, which is deter­

m ined b y the Cliristoffel symbols

9 h jm _ dh }k 0 x k d x " ' , / '

1.3

T h e m a in tec h n ic a l le m m a

W e are going to prove L e m m a 1 .1 . It is well known that

l,,.. o‘h.k

o‘ h,k

m *

a2h,,

0 h „

d x k

d h ak dh,,j ' dhjA _ dhji

d x , n d x , d x , d x „ ’

.O h,, | d h „ Oju, d h ^ d h j * _ dh k K d x . d x , d x, n d x k d x , d x p n

W e see that it is sufficient to obtain upper estim ates on the quantities

I*U

|- O '

'¿w 1

Everywhere below we assume that x G B \, that is,

1. First we shall estimate |/tu'|.

(1 .3 )

Similarly we obtain

\C\ < 2, \B\ < 5 , \A\ < 22. ( 1 4 )

W e also have

A B - C 2 - D ‘ = a 2 + fl‘

Therefore

A B - C 2 - D ‘ > a 2 + ß 2 = 2 n 2 - 2<r + 1 > 1 / 4 . (1 .6 )

From (1 .2). (1 .3 ), (1 .4 ) ami (1 .0 ). it follows Ural

for som e constant C > 0 ,independent o f A

2. W e estim ate the second derivatives as follows.

where

it =

+ * ? + * a ) (*2*3 -

)]

C/*it

and

Then

| ip i < 2A*|^-| +2n(|«|(l + 4A) + M ).

It is easy to verify that |u| < 1 and |t'| < 1. Hence

In order to estim ate the first and the second derivative o f a we need the following

L e m m a 1 . 2 . ([29]) The C ° ° function « can be chosen such that there is a

positive constant L, independent o f A, fo r which the estimates

hold.

Then

Similarly we get that

P r o p o s i t i o n 1 . 1 . There exists a constant c > 0 such that for all p > 1 and

A < -¡L the follow ing estimates hold

l i d l i , < « * * • ’

(i.7)

HP+FaJi, < a i - ‘ . (1.8)

Her«* P+ = (1 + * h ) /2 ami c is independent of A.

P r o o f . T h e £ p-n orm o f Fa0 is given by

I|F||

. = {ji.

(1.9)

First we have

T hen from L e m m a 1 .1 and its proof we get that

|FaJl < ‘ ' A - 1, (1 .10 )

where the constant k is independent o f A. W e also need to estim ate y/h. From (1 .5 )

we obtain that

y /h = ( A B - C 2 - D 2)< 8 (1 .1 1 )

in the hall B \. T h e exterior o f the sixteen halls B\ does not give any contribution to

the Lp-norm o f th e curvature since there h = «/&, that is, h is flat. Then (1 .9 ) , (1 .10 )

and (1.1 1) im ply

IIFa.Hl. < I < ? * ) ' / ' < c A i - " .

In order t o obtain the second estim ate we note that in the 16 halls B X_ XJ the

metric h coincides with the metric o f Eguclii-Hanson. B ut the latter one is an anti-

self-dual m etric. Therefore P + Fa0 = 0 in B x_ x2 for any j = 1 , 1 C . It also vanishes

outside the halls B x, since in this region h coincides w ith the Euclidean flat metric

as w«‘ have ju st mention«1«!. Hence the intt'gratiou in the L p-u o rm o f P + Fa0 reduces

to tli«1 overlapping ar«‘a which consists o f the sixt«1« ! rings B x \ B x_ x, . T hus we get

||P+F *| U p < c<m st.\-2{ J *

r3d r }'t * < const.A "2(A4 - (A - A2)4),/p < c A *-2.

W e note that ( 1 .8 ) gives a little bit better estim ate than (1 .7). From (1 .7 ) for

p = 2 we see that tin* I 2-norm o f the curvature o f A0 is bou n ded by a constant, which

does n ot depend on A. For p = 2 the estim ate (1 .8 ) gives

l|C+ F * ,l k . < cV X .

H ence, we can m ake the ¿ 2-n o rm o f the self-dual part o f the curvature sufficiently

sm all choosing A to be sufficiently small. For this reason we would like to propose a

D e f in it i o n . A connection A , such that the L ^ -n orm o f the self-dual part o f the

curvature o f A is hounded from above by a sufficiently sm all number, is called almost

anti-self-dual connection.

Therefore, according to this definition, the connection Ao, determined by h,

is an alm ost anti-self-dual connection. M oreover, in large regions o f the Kurnmer

surface K it is in fact anti-self-dual.

Before concluding this section, we would like now to discuss the possibility o f

application o f C .T a u b e s’s iteration scheme [50, 51, 37] for solving anti-self-duality

equations. The essential point o f the m ethod is the following.

O n e is looking for an anti-self-dual connection in the form

A = A o + a, (1 .1 2 )

whose curvature satisfies the anti-self-duality equations

P + Fa = 0. (1 .1 3 )

T h e connection A y is fixed and a is an unknown tensor. From (1 .1 2 ) and (1 .1 3 ) it

follows that

V A. V Aau + V Aau t V M <, = - P AF Ao. (1 .1 4 )

win* re a = V \ n u and notations are the sam e as in [50].

C .T aubes has solved the equations (1 .1 4 ) by an iterative scheme [50]. T h e

parameters o f this procedure are in the terms o f ||F40||/,p , \\P+Fa0 \\lp and /t( A 0 ) -

Ao-In order to have a convergent power series, which represents the solution o f

( 1 .1 4 ) , the iteration parameters m ust satisfy som e relations. This is so for an ’’ ap­

propriate” connection Aq, a priori constructed and depending on a sufficiently small

param eter A. W e would also like to emphasize the im portant role o f topology of

th e manifold M on which one solves the anti-self-duality ecjuations (1 .1 4 ). In the

first version o f Taubes’s m eth o d [50] M has positive-definite intersection form , while

in the im proved variant by Donaldson ([13]), the so-called alternating m eth o d , the

intersection m atrix m ay have at m ost (up to orientation) two minus signs. T h e K 3

is n ot o f this type since b% = 3 . Nevertheless, we h ave hoped that the Levi-C ivita

connection o f the metric h, introduced in the section 1.2, could be used to produce

th e ’’ appropriate” initial connection At, in the second version o f Taubes’s schem e [51],

w hich covers also the case b% = 3. T h e reason for this is the fact that its curvature

is concentrated in small thin regions o f the K u m m er surface. Moreover, it is almost

anti-self-dual.

T h e P r o p o s it io n 1 .1 provides two estim ates on the curvature o f A (, and its

self-dual part which are similar to those in [50]. However, one needs an estim ate

slightly better than (1 .8 ) to apply Taubes’s m eth o d , which for the present we have

not achieved.

1 .5

O n th e s t a b ilit y o f th e ta n g e n t b u n d le s

o f K u m m e r su rfa ces

W e shall prove stability o f th e tangent bundle o f the K u m m er surface K , supposing

th at there is an anti-self-dual connection A = A o + a , that is,

*h F A = - F A. (1 .1 5 )

Here A 0 is the Levi-Civita connection of h and A could be obtained b y T au b es’s

m eth o d (see the previous section). Choose a basis d x x ,d x2,d x :i,d x 4 o f T * which is

orthonorm al at a given point with respect to the m etric /». Denote

fx~ = dx\ A d x i + d x j A d x 4,

U t . f t . f Z ) form s i

= dx\ A d x z + d x 4 A d x2,

f£ - = dxi A d x4+ d x 2 A d x 3.

1 basis o f A* - the space o f self-dual 2 forms and ( f \ , f 2 , fz ) is

a basis o f A I -th e space o f anti-self-dual 2 form s ([14]). Since F A is anti-self-dual

((1 .1 6 )), we have

Fa = m i; + . v / ; + n ;

for some functions M , N , P . Introduce a com plex basis dzt , dz2 o f T ml,0( K ) = T m( K )

(and therefore d z \, dz2 is a basis o f T *0,I(A ')) by

dz\ = dx\ + id x 2,

dz2 = dx3 + id x 4.

Then we get that

f\ = ^ (d ziA d zi - d z2A dz2),

f 2 = ^(dz\Adz2 - d z2A d z x)

f z = ^ (d ziA d zz + d z2A d zi)

and

*(dz\ Adz\) = dz2A d z2,

*(d z2Adz2) = dz\ Adz\,

*(dz\ Adz2) = —dz\ A d z2,

*(d z2Adz\) = —dz2Adz\.

Hence

Fa = '-^ -d ziA d zt - * ^ - d z 2A dz2 + ...

and

Let uj be tin* fundam ental form o f h. In our basis it «‘an be expressed as

o> = A d i, + d z2A d z2).

Define the operator L by

L ( t j) = u jAi j

and let A be its L 2 adjoint. T hen

A = L . = w * L *

(see [62]). W e want to com pute A Fa- From (1 .1 7 ) we obtain

L * F A = -( d z i A d z i + dz2A dz2)A( )(dz2Adz2 — d zxA dz\)

M

= — — (dz\ Adz\ Adz2Adz2 — dz2A d z2A d zx A dzx) = 0.

Therefore A Fa = 0 and in this way we have proved the following

P r o p o s i t i o n 1 . 2 . The cotangent bundle T * o f the K u m m er surface K admits

an H erm itian-Einstein connection.

R e m a r k . T h e definition o f H«*rmitian-Einstein connection, degr«*e o f a sheaf,

stability property, etc. can be found in Chapter 2.

Now the proof o f stability o f the cotangent (or tangent) bundle is straightfor­

ward. Indeed, by a conformal change

h t = f h,

where / > 0 , we can get a new Hermitian m etric h i, whose fundam ental form u>i is

dB- closed:

0Bujx = o.

S«*e [21].T h is enables us to «lefin«* the degree o f th e (liolomorphic) cotangent bundle

T m o f K . U nder «'onformal changes the Hodge «-operator is invariant and therefore

the connection A is anti-self-dual with respect to the metric h x. Since the canonical

class o f our Kum uicr surface is trivial, the degree m ust vanish. O n the other hand.

according to P r o p o s i t i o n 2 . 2 , there is a H ermitian-Einstein connection on K and

from d e g (T ’ ) = 0 one can obtain in the sam e way as in page 34 that its Einstein

factor is 0. T h e latter is com patible with A F A = 0. Now if we repeat the argum ents

o f Liibke [41], we deduce that T * is sem i-stable. But it is indecomposable since we

are o n K 3 . Therefore it is stable.

In conclusion we would like to note that if P i c ( K 3 ) = 0, that is, if there are no

non-trivial hue bundles on the considered K 3 surface, then one can get the stability

Chapter 2

On the complex structures of

# n

S 3 x S 3

2.1

In tr o d u c tio n

M any aspects o f the classification theory o f com plex three-dimensional manifolds

have been clarified m ainly due to the celebrated M ori program m e. In this direction

and especially for the search o f a natural generalization o f the K 3 surfaces to higher

dim ensions the com pact three-folds with trivial canonical bundle occupy an im portant

position within the general scheme. A m o n g the examples o f such m anifolds there is

one class o f particular interest. T h ey are com pact 3-folds M with tin* following Hodge

numbers:

and whose canonical class Km — 0 ([47 , 18]). M iles Reid has called m anifolds o f this

type ’’ raksliasa" [47].

The first question which could be raised is whether such M exist. Using a

certain algebraic-geometrical procedure and C .T .C .W a ll’s classification o f 6-manifolds

[61] one can construct a com plex structure J o f the above type on the connected sum

o f u copies o f S 3 x S 3. Since this is an im portant and quite intrigueing point, we

shall give a brief description o f the construction o f J. T h e details and references can

be found in [17, 18, 47].

O ne starts with a sm ooth quintic three-fold N in C P4 which contains infinitely

m any (-1 ,-1 ) sm ooth rational pairwise disjoint curves C ,, one o f which is a fine. Recall

that a ( -1 ,-1 ) curve C in N is a curve, isom orphic to C P 1, such that the norm al bundle

o f C in N splits into 0 ( — 1) (J) 0 ( — 1). T h e existence o f such quintic threefolds N is

due to Clem ens ([10]). In [18], p. 2 9, Friedman describes a m odification o f Clem ens’

construction which provides a sim ply connected N such that [C,] span H 2(N ,S l2)

and there is a relation A,[C,] = 0 in H 2( N , SI2), where A, ^ 0 for every i. W e shall

om it this construction pointing out only that a K 3 surface is involved in it.

Then we take k > 2 such curves C , o f degrees d,, one o f which we choose to be

a line. Since the C , are ( -1, -1) disjoint curves, they can be contracted to k ordinary

double points P ,. In this way a three-fold N is obtained. Here b y contraction we

m ean an isomorphism N \ C i —» N \ P i between complex analytic varieties. B y [17] N

has small deformations M in which the singularities disappear and if H l( N , 0 ) = 0,

then all smoothings M have trivial canonical bundle. From L e m m a 8 .1, [18], p.

25, 7Ti(iV) = 7Ti( M ) and hence M is sim ply connected. M oreover, since b y the

construction o f N the curves C , satisfy the above mentioned relation in H 2( N , U 2),

the Corollary 8.8, p. 28 in Friedman’s report [18] implies that H 2( M , Z ) = Z /d Z ,

where d is tin* greatest com m on divisor o f the d,. B ut d = 1 because one o f the

contracted curves is a line. T hus H 2( M , Z ) = 0.

The Betti numbers o f M and N are related by

and

M M ) = b s(N ) + 2k - 2»,

where k — a is th e rank o f the kernel o f ® Z [ C i ] —* H 2( N , Z ) (see [18] or [47]). As

we saw b2( M ) = 0. From the last formula and from the special construction o f

the "generic” quintic manifold TV, the third Betti num ber o f M is in fact b:t( M ) =

2 (k + 101). It can also he seen that H 3( M , Z ) is torsion-free.

Summ arizing, this com plicated algebraic-geometrical procedure provides a com ­

pact simply connected 6-m anifold M with H 2( M , Z ) — 0 , H 3( M , Z ) - torsion-free and

which possesses a com plex structure .7 with trivial canonical class.

O n the other hand, according to the classification o f C .T .C .W a ll [61] any com ­

pact oriented 6-m anifold which is sim ply connected and whose second Stiefel-W hitney

class w 2 = 0, is classified up to diffeomorphism by the third Betti num ber b3, H 2( Z ) ,

first Pontrjagin class p t and a trilinear m ap H 2( Z ) x H 2( Z ) x H 2( Z ) —* Z given

by cup product. Restricting to the case H 2( Z ) — 0 , this implies that any simply

connected manifold with H 2( Z ) = 0 and H 3( Z ) - a torsion-free Z m odule o f rank

2 n is diffeomorphic to a connected sum o f n copies o f S 3 x S3 ([61]). Hence, since

w 2( M ) = p \ (M ) — 0 , M is diffeomorphic to # „ S3 x S 3 , where n = 101 + k > 103

and there is a com p lex structure J on A i, such that its first C hern class c t(J ) = 0.

M oreover, there is also a special case o f the W a ll’s result [61], which states that a

com pact simply connected 6-m anifold with H ’ ( Z ) - torsion-free and w 2 = 0 has an

almost com plex structure if and only if w3 = 0. In the latter case there is a unique up

to liom otopy alm ost com plex structure with ci = 0. Therefore, as an almost complex

structure, J is unique up to homotopy.

R.Friedm an [18] asks the question what is the m inim al n such that there exists

a com plex structure with trivial canonical bundle on the connected sum o f n copies

o f S3 x S 3. A s we saw, it is at m ost 103. Probably another concrete example o f a

three-fold with trivial canonical class will reduce this num ber. N ote that the Calabi-

Eckmanu com plex structure on S 3 x S 3 does not have vanishing first Chern class.

There is also a lot o f interesting questions in this direction. See [18]. However we

shall stop the discussion at this point, since it is outside the framework o f the thesis.

O ur purpose in this chapter is to derive m ore information about M and the

differential-geometrical structure o f its m oduli space. Especially we would like to

enlighten the deformation theory for M which for the present does not seem to be

satisfactory.

At this stage, we have at our disposal only the Hodge d iam ond o f M , a com plex

structure .7 with c\(J) = 0 and the com pactness. Here are som e direct consequences

o f these facts.

I f we look at the H odge diam ond, we can see at first sight that A / is not a

Kahler manifold. Indeed, the inequalities

0 < br < Y , p+q=r

which hold for any com pact Hermitian manifold [19] imply that b2 = 0. Note that in

th«' Kahler case the second inequality is actually an «'quality which follows from the

Hodge decomposition o f K ahler manifolds.

A nother natural qw 'stion concerns tli«' relationship between h2 ' an<l n - the

num ber o f copies o f S 3 x S 3 in the connected su m . For any com p act complex manifold

the Eul«*r characteristic can be calculated by

dimM dimM/2

* < « ) = d ( - i

r= 0 p,q—0

where br is tin* r-th B«‘tti num ber and h,,,q is the respective Hodge number. The

second «‘quality is w«'ll-known for Kiihh'r m anifolds for tli«' sam e reason w«' pointed

out above - the particular Hodg«* decomposition o f such m anifolds. In the general

H<*rmitian case this formula can be obtained b y considering the Frolicher sp«'ctral

s«'<pien<'<*s [19] which r«*lat«' th«' cohomology groups o f Dolbeault as invariants o f the

complex structure and tin* «'«»homology groups o f D«> Rliam as topological invariants.

It «'an also be obtain«'«! by tin- Atiyah-Sing«*r in«l«*x theorem. For M it giv«*s

See the Hodge diam ond. O n the other hand, under taking a connected su m o f two

even-dimensional m anifolds AT, and N 2, the Euler characteristic behaves as follows:

X { N * # N 3) = X( N i ) + x( N 2) - 2.

See [3]. T hus

x ( # , . S3 x S 3) = x ( # , . - i S3 x 5 3) - 2 = ... = - 2 ( n - 1),

since

x ( S 3 x S 3) = x ( 5 3) x ( S 3) = 0.

Therefore

Further we note that

h2t m u - l .

where 0 is the sheaf o f germs o f the liolomorpliic vector fields over M , itp - the sheaf

o f the liolomorphic p-form s, and we have used the triviality o f the canonical bundle

which im plies 0 = Q 2. Moreover,

H ‘ (A / ,e ) =

= 0.

T h e spaces

H '( Q )

o f com plex structures advanced by Ivodaira and Spencer [34, 35]. A subsequent

theorem o f Kodaira, Nireuberg and Spencer [33] states that if a manifold has a

com plex structure J such that H 2( 0 ) = 0 , then for any infinitesimal deformation

I o f J there exists an actual deformation o f J which infinitesimally coincides with

I. Therefore the space i f2( 0 ) m ay be called obstruction space for the existence o f

infinitesimal deformations o f the complex structure. A s we saw above, it vanishes

for M , so we can apply directly the Kodaira-Nirenberg-Spencer theorem [33], from

which we conclude that the local m oduli space o f M is sm ooth, that is, the first order

deformations are unobstructed.

It is worthwhile to note that the K 3 surfaces provided one o f the earliest ex­

amples which illustrate the Kodaira-Spencer theory. Their m oduli space is sm ooth

o f (real) dimension 40. A s we pointed out in the first chapter the K 3 surfaces admit

a non-trivial Kahler-E iustein-C alabi-Yau m etric. T h e Calabi-Yau m anifolds are an­

other recent example for the theory o f deform ations o f complex structures. G .T ia n

[55] proved that the local m oduli space o f a C alabi-Y au manifold is sm ooth o f di­

mension dim I f1 ( 0 ) = dim H1 ( ÍÍ*'1-1). Since for such manifolds 0 = f l m _ l, the

obstruction space H 2( 9 ) is ). The latter one need not be zero for Kabler

manifolds. Indeed, for Kahler three-folds

H 2(Sl2)

situation on # „ 53 X S 3.

T rying to trace som e analogy between M and a K 3 surface we could say that

both possess a nice m oduli space (for M at least locally). However, as is seen from

Tian 's p ro o f or from the work o f A .T odorov [57], the C alabi-Yau metric provides an

im portant tool for the investigation o f the m oduli space. Moreover, on K 3 fixing

a com plex structure and a cohom ology class, namely the class o f the Kahler form,

determines uniquely the C alabi-Y au metric. Hence, the m oduli space o f Kahler-

Einstein metrics on K 3 is o f dim ension 57 ([4 , 36]).

Now it is an open problem to have a C alabi-Y au substitute for non-K ahler

manifolds. O n the other hand there is the notion o f stability which could suggest one

such candidate. Indeed, Ulilenbeck and Yau [60] proved the existence o f Hermitian-

Einstein m etrics in stable bundles over com pact Kahler manifolds. T h e theorem o f

Uhlenbeck and Yau [60] was generalized by Li an d Yau [38] for n on-K ahler manifolds.

The K ah ler condition is replaced by the Gauduclion condition (for the definitions see

section 2 .3 ) which holds for a large class o f Hermitian metrics. Therefore our first

aim will be to prove the stability o f the tangent bundle o f M .

In section 2.4 we prove that there are n o non-trivial fine bundles on M . This

enables us to deduce that its holomorphic tangent bundle is stable with respect to

any Gauduclion metric. T h is is one o f the m ain results in this chapter. T h en we

that there is an H ermitian-Einstein m etric on the connected su m # „ S'3 x S'3. A s far

as is known to the au thor, the only (but very im portant) application o f the Li-Yau

theorem can be found in [39], where Y au et al. give a short proof o f a fam ous theorem

o f Bogom olov.

O ur basic hypothesis is that the Hermitian-Einstein metric and th e Gauduclion

m etric coincide. T h en , fixing the com plex structure and the volum e, we consider the

deform ations o f the H erm itian-Einstein-C alabi-Yau-Li-G auduclion m etric. However,

for this purpose we need to im pose an additional condition which can be a substitute

o f the Kaliler condition. W e begin a search for different types o f m etrics which being

Hermitian-Einstein adm it no (essential) Hermitian deform ations, that is, which are

rigid for such manifolds. It is natural to suppose in the first place that such conditions

are to be found in the terms o f the torsion. W e also seek the place o f M in the Gray-

Hervf ilia classification [25] o f alm ost Hermitian manifolds. Further we m ake some

notes on the deformations o f both the metric and the com plex structure. Especially

we show that on any Hermitian uon-K ahler m anifold with H 2( 0 ) — 0 there are no

non-zero anti-sym m etric deformations o f the com plex structure. In section 2 .2 we

collect som e elements o f the Hermitian non-K ahlerian geometry, som e o f which seem

to be little known within the m athem atical community. For com pleteness o f the

exposition, in section 2 .3 we also recall the definition o f stability, H ermitian-Einstein

m etrics, Gauduchon’s condition, L i-Y a u ’s theorem.

2 .2

E le m e n ts o f H e r m itia n n o n -K a h le r ia n

g e o m e tr y

Let M be a com pact com plex manifold o f com plex dimension rn > 2 , T - its tangent

bundle and let ./ : T --- ► T be the alm ost com plex structure induced by th e complex

structure in the standard way [62]. A Riemannian metric g is called Hermitiau if at

each point x € M

g ( X , Y ) = g ( J X , J Y )

for all X , Y £ Tx . T h e fundam ental or Kahler form F o f g is given by

F ( X , Y ) = - g ( X , J Y ) .

A differential operator D : T ( T ) — ► T (T * ® T ) which satisfies

D x ( f Y ) = X f . Y + f D x Y

for every X , Y € T ( T ) and all / € is said to be a connection. It can be

extended to higher tensor powers o f the tangent and cotangent bundles in a natural

way.

There are two tensors associated to D . T h e torsion T o f D is a section in

A27 * <8> T ,

T ( X , Y ) = D X Y - D y X ~ [ X , Y ] (2 .1 )

for X , Y £ T ( T ) . T h e curvature R o f D is defined by

R ( X , Y ) Z = [Dy, D x ]Z - D [y,x]Z

for all sm ooth vector fields X , Y , Z . W ith this sign convention the sphere S 2 lias

R l212 > 0.

All tensors can be extended to the complexification o f T :

T ® C = V e T "

and to the tensor powers o f T and T * . T h e tangent bundle T a n d the holomorpliic

tangent bundle T ' are identified. Since the related m aterial is well-known (see e.g.

[62]) we shall n ot present the details her«'.

A connection is said to

a) preserve the metric </, or to be a m etric connection, or to be com patible with

«/, if D (/ = 0 , th at is,

( D g ) ( X , V, Z ) = X g ( Y , Z ) - g ( D x Y , Z ) - g ( Y , D x Z ) = 0;

b) preserve the com plex structure J or to be com patible w ith J if D J = 0 , that

is,

T h ere are two connections canonically associated to a given Hermitian metric

g. According to A.Lichnerowicz [40], the first canonical connection V o f g is uniquely

determined b y the following defining properties:

V fl = 0

V .7 = 0 (

2

2

T v = 0,

where T v denotes the torsion o f V .

T h e Cheru connection D is the unique m etric connection which preserves the

complex structure and whose torsion T is a vector valued (2 ,0 ) form, that is,

D g = 0

D is also called the (standard) Hermitiau connection, or in the terminology of

A.Liclm erovicz "second canonical Hermitian connection” ([40]). For the Cliern con­

nection, in any local liolomorphic frame, tin* corresponding connection form s are

o f type ( 1 ,0 ) with values in E n d ( T ) . In local co m p lex coordinates adapted to the

complex structure, D has the well-known com ponents:

A metric is Kaliler if and only if its L evi-Civita and Cliern connections coincide.

From now on D will denote the Cliern connection o f a Hermitian metric g.

D J = 0 (2 .3 )

T ( J X , Y ) = J T ( X , Y ) .

r

Set

A ( X , Y ) = D x Y - V X K, (2 .4 )

or in a local frame o f T

where f A are the com ponents o f the L e v i-C iv ita connection V . T h e latin indices

vary from 1 to 2 m , while the greek from 1 to m . In any local frame ej o f T the

com ponents o f the L evi-Civita connection are given by

If Cj = d /d x

r , ‘ = ^ 9 k'"(e.g*m + fjS.™ -

r~9i,)-J, this is the well-known form ula for the Christoffel sym bols. However,

we need to calculate them in a local com plex fra m e adapted to the com plex structure.

I f we take to be such a fram e, then we shall have

r .,\ ( = ^ g ^ { 0 o 9 a » + ¿W o ii) = + T /A

,)-Further

r „*» = ^ 9 “ s(0™»„a - aM )

= ¿ 9 * 9 . i T . \ = \tJ

and

= f A = 0.

Now it is easy to obtain that the com ponents o f A with respect to a local complex

fram e adapted to the com plex structure are expressed by

Indeed,

A « * » = -T a % , A ft*» — An*,,

A J i = V . = ••»A,:, = A , \ = i r V , (2 .5 )

A n *» = A . * » = 0.

A „ Aa = r „ * „ - f „ \

= r . * f l - i ( r n * a + r A . ) = i r „ V

etc. See also Gauduchon’s paper [22] or liis thesis [23]. T h e third condition in (2.3)

is equivalent to D " = d ([32]). T h us th e curvature R o f D is given by

n » JL _ mJL d \ 9

IfiQgfi

R'd x x ' dz“ ’ dza '

Taking the trace in (2 .1 ) gives a form

t( Y ) = t r a c e { X — * T ( X , Y ) ) .

tis called the t o r s i o n 1 - f o r m . B y ( 2 .1 )

T h e (1 ,0 ) form ft determined by

0a = T\xa,

in a com plex fram e is called the t o r s i o n ( 1 , 0 ) f o r m . T h e forms r and 0 are related

by

t = 0 + 0 . ( 2 . 6 )

D enote also 0 ° =

D e fin it i o n 1 . M if said to be balanced if and only i f 0 = O.

D e fin it i o n 2 . M is said to be sem i-Káhlerian if and only if r = 0.

Definition 1 lias been used by M iclielson in [42] while Gauduelion has exploited

Definition 2 [20, 22]. From ( 2 .6 ) we see that M is balanced if and only if it is

sem i-Kablerian. It is easy to prove([42]) that M is balanced if and only if

p ( F m - l ) = 0 ,

where P is any o f the operators Under a conformal change g ' = eug , the

corresponding torsion form s are related by

& = 6 — (m — 1 )du,

t' = t — (m — l)rfu. ( 2 .7 )

(2 .7 ) tells us that if

dr =

then the class o f r in H ' ( M , R ) is a conformal invariant.

Now take a trace in th e curvature tensor. In contrast to the Kaliler case we

have three different Ricci tensors.

T h e second Ricci form r can be obtained in the following way. Fix tangent

vectors Z , V . Define R (Z , V ) by

(7 ? (Z , V ) ) ( X , Y ) = g ( R ( X , Y ) Z , J V ).

Hen re for -V = g f r .K = = & , V =

(Ra*,)\fi =

—igiiyRo^Xfi-T h e curvature operator R is b y definition

R ( 4 ) ( Z ,V ) = (R (Z ,V ),< i> )g.

( , ) g is the inner product o f form s o f one and the sam e type, in which we om it d e t(g ).

In the case when 4> is equal to the fundam ental form F o f g we have

= g Xtg ' ii(-')R a * )* 'g > '* = g ^ R * »

— RayX — ray r

that is, r = R ( F ) . T h e fo rm p = R T( F )-the transpose curvature operator, h as

com ponents

P\fi = R a a

\fi-It is called the first Ricci fo rm and ^ p represents the first Cliern class o f M . In fa c t,

T h e com m on trace

« = trgr = trgp

is the first scalar curvature o f </, and

v = trgs

is the second scalar curvature, where a is the third Ricci form determined b y

»Xfi =

R\aafi-Tin* first Bianchi identity

S { R ( X , Y ) Z } = S { T ( T ( X, Y ) , Z ) - (D X T ) ( Y , Z ) } ,

where S m eans sym m etrization o f X , Y , Z reads

Hence, by contraction

u - v = - Dx6 x. (2 .9 )

W e conclude this survey o f th<‘ Hermitian non-Kiihlerian geometry by n o tin g that

one can consider th e ( 2 ,1 ) torsion T with components

Ta*tH = gXyTaXff

for which

O F = i T / 2 . (2 .1 0 )

2.3

Gauduchon metrics, stability and Li-Yau’s

theorem

T o study an H erm itian manifold M it is always useful to pick a metric w ith some

special properties. Such metrics could be Kaliler , balanced, Einstein,etc.. However,

to adm it a special m etric the Hermitian manifold M m ust satisfy som e con ditions,

generally o f a topological nature. First o f all, one seeks a metric within a given

conformal class. For instance, to obtain a balanced metric , i.e. ¿)(i7’”, ~ I) = 0 , in this

way, in general is impossible. In m any cases such metrics sim ply d o n ot exist. W h a t

one can always achieve is due to the following result o f G auduclion [21, 20]:

T h e o r e m . Given any Hermitian m etric on a compact com plex manifold of

dim ension at least 2, there is a conformal m etric unique up to h om oth ety, such that

its fu ndam ental fo rm satisfies

d d ( F m~ ') = 0 . (2 .11 )

(2 .1 1 ) is also equivalent to 86 = 0 [20]. W e shall call Gauduchon m etric a metric

for which the condition (2 .11 ) holds. In his own terminology such m etrics are said to

be standard or o f null eccentricity. In fact, there are m any o f th e m - one within each

conform al class.

N .H itcliin observed in [28] that the Gauduchon metrics enab le us to extend the

notion o f stability to liolomorphic bundles on an arbitrary H erm itian m anifold M .

Namely, if L is a liolomorphic hue bundle on M , its degree w ith respect to a given

Gauduchon m etric F is

d e g (£ ) = F ) = ^ j u I A r - ' .

where / is th e curvature o f any Hermitian connection on L c om p atib le with Bl, or

m ore generally for a torsion-free coherent sheaf S on M

d e g ( 5 ) = / c , ( 5 ) A F m- 1 JM

is well-defined for the Gauduchon condition since any two first C liern form s differ by

a d d - exact form . If cj = 0 then the degree vanishes.

Denote

H (S ) = d e g (S )/r a n k (S ).

D e f i n i t i o n . S is called stable iff

ft(S ') < fi(S )

O n the other hand there is the notion o f H erm itian-Einstein metric. Let ( E , h )

he a liolomorpliic vector bundle over (M , g ), where h is an Hermitian metric in

E--- ► M and g is an Hermitian metric on M . T h en the Cliern connection D o f h is

said t o be Hermitian-Einstein with respect to g if and only if

w here R (li) is the curvature o f D . W e shall also call h an Hermitian-Einstein metric,

when its Cliern connection is Hermitiau-Einstein. I f E = T - the tangent bundle o f

M , a n d if g = h, then (2 .1 2 ) reads

with r - the second Ricci form o f g = /». In the latter case if g is Kahler, (2 .1 3 ) is the

w ell-known Kaliler-Einstein condition.

In [41] Lubke proved if an indecomposible bundle E over a Kahler manifold

M a d m its an Hermitian-Einstein metric, then it is stable. Later Uhlenbeck and Yau

[GO] proved the opposite statem ent. N.H itcliin suggested that the same relationship

betw een stability and Hermitian-Einstein metrics should be also valid in the general

H erm itian setting. Buchdahl [6] proved the theorem for surfaces and Li and Yau [38]

generalized the work o f Uhlenbeck - Yau to th e.n o n -K a h ler case for all dimensions.

N am ely,

T h e t h e o r e m o f L i a n d Y a u .[38 ] Let M be a compact Hermitian manifold

■with a Gauduchon m etric , and E be a holomorphic vector bundle over M . Then E

is stable if and only if it admits an H erm itian-E instein m etric .

2.4

Stability of the tangent bundle of #„S3 x

S 3

« * * « ('■ >„s« = (2.12)

(2.13)

P r o p o s i t i o n 2 .1 . There are no non-trivial line bundles on M .

P r o o f . Consider the exponential exact sequence

0 — ►

.».)ll,m !)r>s |.)KX.i aUlMOJJOJ .HJI .(AtilJ .»M s n q x ' ( j , ) Q JO JU.MJSqUS SB A U }-n o ;M O ) OSJW s i g ‘ a u } - n o i M O ) .» .lo j.u .H ji p ir e j B . n js .m j j ¿ p B .t o j b s i (j, ) Q ‘ .q p n i i q j o j j j a « SI ¿ A JIIIg '.KMJ-1IOISMO) - f ) i p i M ( x ) O J ° J * » q * q n S - I q ilB J B <K| g ».»q

'M ^ - a o n J O ) s i g / ( x ) 0 = Ö JWi,ijs |ti.>i)ouI) s q j ij a iij m

j o j ( x ) O J0 S sw A B sq sq iis q j n s j o jA p io i m p i p i l o . » X jq iq B js .»ip jj.»,>q.» o j in .ti.iq jn s s i )t

(*»««•> t i B i j m u o j j a q » m p q B A o s p ) ‘ 6 9 I ' <l‘ [ s e ] n ! ( tH ) ( 9 '¿ ) n o p i s o d o j j j o o j j

" j t j y j u i u o y j n p n v g ) Hun o f yj.td v .u

y i u n jjqv^h- « j y f o ¿ a j p u n q y u .iliu v f ( o t y d j o m o j o y ) a y x ‘ Z ’ Z u o i j i s o d o j j

• s jo s iA ip o n ‘ s i i b i j i ‘j u o i s u .m u p -<>.) j o j y j o s^i)^uba«|Iiso n j i b.u .»q j jBijj j ji i p j p .>avn o p i s o d o j d s n j j m o i j

J V j o js p jo q j.>j ib j j q x

0 = , , V J! ■M"<1 I ’ " » « 0 = ( . o ' n ) , H

a;»n«>|j • [¿p] iii puoniBip .»ypojj .*qj jo u u o j ,>qj iiiojj

, , V = , ; M = (Z ' n ) ’ H z e j = (Z ' l \ ) , H z e j = L O ' W ) , H " " V

u q )J l\ j

i z ‘ n ) , H *= ( .o 'w ) ,h

.u oj.u.np pilB m stqdjouiosi iiB si q sinjx

, > jiu u l» js }j b x i) . » q i *>a b i| . » « n a q x

0 = ( O i , H ™ 1 X 0 = r o V P ™ , Ih m ( 0 ) , H ( C •0 = ( Z ) i H p j l - n n o o j A jd u ii s Si ;y j.u iis ( j

:o = (o

) ,

h

-»jopwqx o = ,„v »«a mjiojin )p>«qpa jqi ¿q ,|ir

n

(

o

),

h

('i

j A « q JM

Since Q is torsion-free and 0 ( T ) is reflexive, from L e m m a 1.1.16 in [44] it follows that

S is norm al and being torsion-free, we get that S is a reflexive rank 1 - sheaf. The

latter m eans, equivalently, that S is a hue bundle. But from the P r o p o s i t i o n 2 .1

we conclude that S is the trivial line bundle. Its non-vanishing section is therefore a

non-zero section o f 0 ( T ) — (-). O n the other h a n d , since the canonical bundle Km is

trivial, that is,

we have the pairing

from which we obtain

Km = A 3T * = O ,

T * <g> A 2T * — ► O

T = A 2T *

and

e 2 n 2.

So far, we have a non-zero holom orpliic 2-form . T h is contradicts the fact that

dim H ° ( M , Q 2) = dim H2 ° ( M ) = h 2 0 = 0

which follows from the Dolbeault theorem and fro m the Hodge d iam ond. Therefore

there are no rank 1 - subsheaves o f 0 with torsion-free quotient.

Now suppose E to be a rank 2 - subslieaf o f 0 and let F = 0 / F . W e have the

exact sequence

0 ---► E ---► 0 ---► F ---► 0

and also a part o f the dual long sequence

0 --- ► F m — ► n 1 --- > ....

For an arbitrary coherent sheaf A its dual .4* is reflexive ([32], Proposition 5 .1 8 ,p .160)

anti therefore F * is a rank 1 - reflexiv«* sheaf, i.e . F * is a line bundle. Again from

P r o p o s i t i o n 2 .1 F " has to b e tin* trivial lin<* bundle and to have a non-vanishing