Algebraic Cycles and Topology of Real Enriques Surfaces

Compositio Mathematica 110: 215±237, 1998. 215

c 1998 Kluwer Academic Publishers. Printed in the Netherlands.

Algebraic cycles and topology of real

Enriques surfaces ?? FREDÂ ERICÂ MANGOLTE1 ? and JOOST VAN HAMEL2 1Laboratoire de Mathematiques,Â UniversiteÂ de Savoie, 73376 Le Bourget du Lac Cedex, France; e-mail: [email protected] 2Institute of Mathematics, University of Utrecht, Budapestlaan 6, 3584 CD Utrecht, The Netherlands; e-mail: [email protected]

Received:14 May 1996; accepted in ®nal form 22 October 1996

H Y R; Z=

Abstract. For a real Enriques surface Y we prove that every homology class in 1 2 can be

R represented by a real algebraic curve if and only if all connected components of Y are orientable.

Furthermore, we give a characterization of real Enriques surfaces which are Galois-Maximal and/or Y Z-Galois-Maximal and we determine the Brauer group of any real Enriques surface . Mathematics Subject Classi®cations (1991): 14C25 14P25 14J28 13A20. Key words: Algebraic cycles, real algebraic surfaces, Enriques surfaces, Galois-Maximality, Brauer group.

1. Introduction

Y C Let Y be a complex algebraic surface. Let us denote by the set of closed points

alg

H Y C ; Z

of Y endowed with the Euclidean topology and let 2 be the subgroup

Y C ; Z

of the homology group H2 generated by the fundamental classes of Y algebraic curves on Y .If is an Enriques surface, we have

alg

Y C ; Z=H YC;Z: H2 2 One of the goals of the present paper is to prove a similar property for real Enriques

surfaces with orientable real part. See Theorem 1.1 below. R

By an algebraic variety Y over we mean a geometrically integral, separated

= f ;g C = R

scheme of ®nite type over the real numbers. The Galois group G 1 of

C Y

acts on Y , the set of complex points of , via an antiholomorphic involution and

the real part Y is precisely the set of ®xed points under this action. An algebraic R

variety Y over will be called a real Enriques surface, a real K3-surface, etc., if

Y = Y C

the complexi®cation C is a complex Enriques surface, resp. a complex K3-surface, etc. Consider the following two classi®cation problems:

? The ®rst author was supported in part by HCM contract ERBCHBG-CT-920011 (MAP-Project) and in part by HCM contract ERBCHRX-CT93-0408. ?? The second author was supported in part by EC grant ERBCHRX-CT94-0506.

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216 FREDÂ ERICÂ MANGOLTE AND JOOST VAN HAMEL

Y R

classi®cation of topological types of algebraic varieties over (the mani-

folds Y up to equivariant diffeomorphism),

Y R classi®cation of topological types of the real parts . For real Enriques surfaces the two classi®cations have been investigated recently by Nikulin in [Ni2]. The topological classi®cation of the real parts was completed by Degtyarev and Kharlamov, who give in [DKh1] a description of all 87 topological

types. Let us mention here that the real part of a real Enriques surface Y need not

Y R be connected and that a connected component V of is either a nonorientable

surface of genus 6 11 or it is homeomorphic to a sphere or to a torus.

C The problem of classifying Y up to equivariant diffeomorphism still lacks a satisfactory solution. In the attempts to solve this problem, equivariant (co)homology

plays an important role (see [Ni2], [NS], [DKh2]). It establishes for any algebraic

R G Y C

variety Y over a link between the action of on the (co)homology of and

R Y d the topology of Y . For example, if is of dimension , the famous inequalities

X 1

H Y R; Z= 6 H G; H Y C ; Z= ; r

dim 2 dim 2 (1) = r 0

2 n

X 2

H Y R ; Z= 6 H G; H Y C ; Z;

dim even 2 dim r (2) = r 0

X 1

H Y R ; Z= 6 H G; H Y C ; Z;

dim odd 2 dim r (3) = r 0 (cf. [Kr1] or [Si]) can be proven using equivariant homology.

We will say that Y is Galois-Maximal or a GM-variety if the ®rst inequality

Z Z

turns into equality, and Y will be called -Galois-Maximal,ora -GM-variety if

C inequalities (2) and (3) are equalities. When the homology of Y is torsion free,

the two notions coincide (see [Kr1, Prop. 3.6]).

R Y R 6= ;

A nonsingular projective surface Y over with is both GM and

H Y C ; Z 6= Z-GM if it is simply connected (see [Kr1, Sect. 5.3]). If 1 0, as in the case of an Enriques surface, the situation can be much more complicated. The

necessary and suf®cient conditions for a real Enriques surface Y to be a GM-variety

were found in [DKh2]; in the present paper we will give necessary and suf®cient Z conditions for Y to be -GM. See Theorem 1.2. As far as we know, this is the ®rst paper on real Enriques surfaces in which

equivariant (co)homology with integral coef®cients is studied instead of coef®cients = in Z 2.We expect that the extra information that can be obtained this way (compare for example equations (1)±(3)) will be useful in the topological classi®cation of real Enriques surfaces.

In Section 6 we demonstrate the usefulness of integral coef®cients by computing

Y Y

the Brauer group Br of any real Enriques surface . This completes the partial

Y results on the 2-torsion of Br obtained in [NS] and [Ni1]. See Theorem 1.3.

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1.1. MAIN RESULTS

alg

Y R H Y R; Z=

Let be an algebraic variety over . Denote by n 2 the subgroup

H Y R; Z=

of the homology group n 2 generated by the fundamental classes of

Y R n-dimensional Zariski-closed subsets of , see [BH] or [BCR]. We will say that these classes can be represented by algebraic cycles. The problem of determining these groups is still open for most classes of surfaces.

alg

H X C ; Z=H XC;Z For a real rational surface X we always have 2 2

alg

X R ; Z= =H XR;Z= and H1 2 12, see [Si]. For real K3-surfaces, the situation is not so rigid. In most connected components of the moduli space of real K3-

alg

H X R ; Z= > k

surfaces the points corresponding to a surface X with dim 1 2

k 6

form a countable union of real analytic subspaces of codimension k for any

X R ; Z= X

dim H1 0 2 ,where 0 is any K3-surface corresponding to a point from

H X R ; Z= that component. In some components this is only true for k< dim 1 0 2 ;

these components do not contain any point corresponding to a surface X with

alg

X R ; Z= = H X R ; Z= H1 2 1 2 , see [Ma2]. For real Abelian surfaces the

situation is similar, see [Hu] or [Ma1, Ch.V].

Y R 6= ;

THEOREM 1.1. Let Y be a real Enriques surface with . If all connected

R components of the real part Y are orientable, then

alg

Y R; Z= =H YR;Z= : H1 2 12 Otherwise,

alg

Y R; Z= = H Y R ; Z= : dim H1 2 dim 1 2 1 See Theorem 4.4 for more details. In order to state further results we should mention that the set of connected

components of the real part of a real Enriques surface Y has a natural decomposition

R =Y tY into two parts Y 12. Following [DKh1] we will refer to these two parts

as the two halves of the real Enriques surface. In [Ni1] it is shown that Y is GM

if both halves of Y are nonempty. It follows from [DKh2, Lem. 6.3.4] that if

R Y Y R precisely one of the halves of Y is empty, then is GM if and only if is nonorientable. This result plays an important role in the proof of many of the main results of that paper (see Sect. 7 in loc. cit.). In the present paper we will see in the course of proving Theorem 1.1 that a real

Enriques surface with orientable real part is not a Z-GM-variety. In Section 5 we

also tackle the nonorientable case and combining our results with the results for = coef®cients in Z 2 that were already known we obtain the following theorem.

THEOREM 1.2. Let Y be a real Enriques surface with nonempty real part.

Y Y Y

(i) Suppose the two halves Y1 and 2 are nonempty. Then is GM. Moreover,

Y R is Z-GM if and only if is nonorientable.

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218 FREDÂ ERICÂ MANGOLTE AND JOOST VAN HAMEL

Y Y Y R

(ii) Suppose one of the halves Y1 or 2 is empty. Then is GM if and only if

Z Y R is nonorientable. Moreover, Y is -GM if and only if has at least one component of odd Euler characteristic. There are examples of all cases described in the above theorem (see [DKh1,

Figure 1]).

In Section 6 we study the Brauer group Br of a real Enriques surface

Y Y using the fact that Br is isomorphic to the cohomological Brauer group

0 2

Y = H Y; G Y

Br m ,since is a nonsingular surface. In [NS] Nikulin and et

Sujatha gave various equalities and inequalities relating the dimension of the 2-

Y Y torsion of Br to other topological invariants of a real Enriques surface .Itwas

shown in [Ni1] that

; Y > s ; =

dimZ 2 Tor 2 Br 2 1

Y R where s is the number of connected components of , and that equality holds

if Y is GM. Using the results in Section 5 on equivariant homology with integral

coef®cients we can compute the whole group Br . s

THEOREM 1.3. Let Y be a real Enriques surface. Let be the number of connected

R Y R 6= ;

components of Y .If is nonorientable then

2s 1

Y ' Z= :

Br 2

R 6= ;

If Y is orientable then

2s 2

Z= Z= ;

2 4 if both halves are nonempty

Y ' Br

2 if one half is empty.

R =;

If Y then

Y ' Z= : Br 2

We were informed by the referee that Theorems 1.2 and 1.3 and a result similar to Theorem 1.1 have independently been obtained by V.A. Krasnov. His paper will be published in Izv. Ross. Akad. Nauk Ser. Math. 60 (1996), No. 5.

2. Equivariant homology and cohomology

= C =R

Since the group G Gal acts in a natural way on the complex points of an R

algebraic variety Y de®ned over , the best homology and cohomology theories for

studying the topology of Y are the ones that take this group action into account.

Y; Z= In [NS] etaleÂ cohomology H 2is used, and in [Ni1] the observation is made

Y C G; Z= that this is essentially the same as equivariant cohomology H ; 2 . In [DKh2] Degtyarev and Kharlamov do not use equivariant cohomology as

comp4075.tex; 5/12/1997; 11:40; v.7; p.4 REAL ENRIQUES SURFACES 219

such, but instead a `stabilized' form of the Hochschild-Serre spectral sequence q

2 p

X G; Z= = H G; H X; Z=

E ; 2 2. This construction, due to I. Kalinin, is

p;q

+ p

p 2

= Z= H G; M =H G; M

based on the fact that if G 2then for any group

+ p

p 1

p> M Z= H G; M =H G; M M and any 0, and if is a 2-module then even

for any p>0, so it is possible to squeeze the Hochschild±Serre spectral sequence into 1, or at most 2 diagonals. They also use the analogue of this Kalinin spectral sequence in homology. In the present paper we stick to the original equivariant cohomology supplemented with a straightforward dual construction which we call equivariant Borel±Moore homology.

First we will recall some properties of equivariant cohomology for a space with

= Z= an action of G 2. Then we will give the de®nition of equivariant Borel±Moore homology and list the properties that we are going to need. In Section 3 we give

a short treatment of the fundamental class of G-manifolds and formulate PoincareÂ

duality in the equivariant context.

G = Z=

Let X be a topological space with an action of 2. We denote the ®xed

X H X G; F G

point set of X by . In [Gr1] the groups ; are de®ned for a -sheaf

X G G X

F on , which is a sheaf with a -action compatible with the -action on .

= f ;g

Writing G 1 , this just means that we are given an isomorphism of sheaves

F ! F & & =

& : satisfying id. Now de®ne

p p G

X G; =R X;

H ; G the pth right derived functor of the -invariant global sections functor. We have

natural mappings

p p G

H X G; F ! H X; F ;

e : ; F

which are the edge morphisms of the Hochschild±Serre spectral sequence

q p+q

2 p

X G; F =H G; H X; F H X G; F

E ; ;

p;q

For us, the most important G-sheaves will be the constant sheaf 2andthe

Zk k 2 Z

constant sheaves constructed from the G-modules for . Here we de®ne

k G z =

Z , to be the group of integers, equipped with an action of de®ned by

z Ak Z= Zk

1 . We will use the notation to denote either 2or , and we will

Ak k sometimes use A instead of if is even.

There is a cup-product

p q p+q

X G; Ak H X G; Al ! H X G; Ak + l

H ; ; ;

f X ! Y and a pull-back f for any continuous equivariant mapping : , which both

have the usual properties.

p p

H G; M = H G; M ;

If X is a point, pt; which is cohomology of the

M H G; Z= group G with coef®cients in . Recall that as a graded ring, 2 is

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220 FREDÂ ERICÂ MANGOLTE AND JOOST VAN HAMEL

= [ ] isomorphic to the polynomial ring Z 2 ,where is the nontrivial element in

G; Z= H 2 . By abuse of notation, we will also use the notation for the nontrivial

1 2 2

G; Z ' Z= H G; Z ' element in H 1 2and for the nontrivial element in

2 H G; Z 2 = Z 2. This notation is justi®ed by the fact that 1 maps to

1 2 2

G; Z= 2 H G; Z H 2 under the reduction modulo 2 mapping and maps to

2 2

2 H G; Z=

2 .

! H G; Z= !

The constant mapping X pt induces a mapping 2

p G p G

X G; Z= H X ; Z= ,! H X G; Z=

H ; 2 and we have a natural injection 2 ; 2 , X

so cup-product gives us for any G-space a mapping

G G

X ; Z= H G; Z= ! H X G; Z= ; H 2 2 ; 2

which is well-known to be an isomorphism. Taking the inverse of this isomorphism

H X ; Z=

and sending to the unit element in 2 we obtain a surjective homo-

G G

X G; Z= ! H X ; Z= A = Z

morphism of rings H ; 2 2 and we de®ne for

= k 2 Z

or Z 2andany the homomorphism of rings

H X G; Ak ! H X ; Z= : ; 2

to be the composite mapping

X G; Ak H X G; Ak

H ; ;

G G

mod 2

- -

X G; Z= H X ; Z= ;

H ; 2 2

i X ,! X

where i is induced by the inclusion : . Note that coincides with the 0

mapping in [Kr3]. It is clear from the de®nition that

f ! =f !:

We use the notation

n;p n p G

H X G; Ak ! H X Z= ; : ; ; 2

for the mapping induced by . In Section 5, we will need one technical lemma which can easily be proven using the Hochschild±Serre spectral sequence.

G 2

G X 6= ; e

LEMMA 2.1. Let X be a -space with .Thenif is not surjective on

2 G 1 1

H X; Ak ! 2 H X G; Ak e ! 6=

,thereisaclass ; 1 such that 0,

A 1

! = but 0.

comp4075.tex; 5/12/1997; 11:40; v.7; p.6 REAL ENRIQUES SURFACES 221

The homology theory we are going to use is the natural dual to equivariant cohomology. For an extensive treatment of its properties, see [vH]. Here we will give a short account without proofs.

IntherestofthissectionweassumeX to be a locally compact space of ®nite

= Z= Ak

cohomological dimension with an action of G 2, and will be as above.

Ak We de®ne the equivariant Borel±Moore homology of X with coef®cients in

H X G; Ak = R R X; Z;Ak

G c

p ; Hom

p 2 Z; G

for where HomG stands for homomorphisms in the category of -modules

and c stands for global sections with compact support; this is the natural equivariant generalization of the usual Borel±Moore homology in the context of sheaf

theory (see, for example, [Iv, Ch.IX]). n

If X is homeomorphic to an -dimensional locally ®nite simplicial complex

G H X G; Ak

with a (simplicial) action of , then we can determine p ; from a double complex analogous to the double complex (1-12) in [N1], which is used for

the calculation of equivariant cohomology. Consider the oriented chain complex

1 1 1 1

C ! !C C p

!C with closed supports (i.e., the elements of are -

n p

n 10

k chains that can be in®nite). The chain complex with coef®cients in A is de®ned

1 1

C Ak = C Ak ;

p p

G H X G; Ak

and we give it the diagonal -action. Then p ; is naturally isomorphic

to the th homology group of the total complex associated to the double complex

6 6 6

1 1

1 1 1 1

- - -

C Ak C Ak C Ak

n n

1n 1 1

6 6 6

1 1

1 1 1 1

- - -

C Ak C Ak C Ak ;

n n n

where the lower left-hand corner has bidegree 0 . Note that by construction

H G; Ak = H G; Ak X

p pt; , so PoincareÂ duality holds trivially when is a point (and the proof of PoincareÂ duality in higher dimensions, as stated in

Proposition 3.1, is no more dif®cult than in the nonequivariant case). In particular,

H X G; Ak p<

p ; need not be zero for 0.

H X G; Ak X

The groups p ; are covariantly functorial in with respect to

k ! Z=

equivariant proper mappings and the homomorphisms Z 2 induce homo-

H X G; Zk ! H X G; Z= p morphisms p ; ; 2 that ®t into a long exact sequence

comp4075.tex; 5/12/1997; 11:40; v.7; p.7 222 FREDÂ ERICÂ MANGOLTE AND JOOST VAN HAMEL

! H X G; Zk ! H X G; Zk p

p ;;

! H X G; Z= ! H X G; Zk ! :

p p ; 2 1 ; (4)

As in the case of cohomology, there are natural homomorphisms

Ak G

e H X G; Ak ! H X; Ak ; p

: p ; p

which are the edge morphisms of a Hochschild±Serre spectral sequence p

E X G; Ak = H G; H X; Ak H X G; Ak :

p+q

; q ;

p;q

e e

If no confusion is likely, we use instead of p ; otherwise we will often write

k Z k+ Z=

Z 2 212

e = e ;e = e e = e

p p p

,and p , and we have similar conventions for

p p p

the edge morphisms e in cohomology.

There is a cap-product between homology and cohomology

H X G; Ak H X G; Al ! H X G; Ak l ;

p ; ; ;

! 7! \ !;

and of course we have

0 0

\ ! [ ! = \!\!;

(5)

\ ! =e \e!;

e (6)

X ! Y

and for any proper equivariant mapping f :

f \ ! = f \ f ! : (7)

H G; A Recall that is the nontrivial element in 1 . Cap-product with

X G; A

considered as an element of H ; 1 de®nes a map

s H X G; Ak ! H X G; Ak + ;

p : p ; ; 1

p 1

7! \ :

Ak Ak

e s p

It can be shown, that the p and ®t into a long exact sequence

A 1

+ p

p 1

- -

H X G; Ak H X; A p

p ;

A 1

- - -

H X H : G; Ak X G; Ak

p p ; 1 1 ; (8)

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REAL ENRIQUES SURFACES 223

Ak Ak

s e p

For p we adopt the same notational conventions as for .

G G

H X ;A ! H X G; A p The natural mapping p ;and the cap-product give

us a homomorphism

G G

H X ; Z= H G; Z= ! H X G; Z= ; 2 2 ; 2

which is an isomorphism. Taking the inverse of this isomorphism and sending the G

2 H G; Z= H X ; Z= nontrivial element 2 to the unit element in 2 we obtain

a surjective homomorphism

G G

H X G; Z= ! H X ; Z= :

; 2 2

i H X G; Z= ! H X G; Z=

n n

Furthermore, the mapping : ; 2 ; 2 induced by the

X ! X n < inclusion i : is an isomorphism for any 0, so we can de®ne a

homomorphism

H X G; Ak ! H X ; Z= : ; 2

by taking the composite mapping

mod 2 \

- -

H X G; Ak H X G; Z= H X G; Z=

< ; ; 2 0 ; 2

G G

- -

H X G; Z= H X ; Z= ;

; 2 2 X

where N is any integer greater than the (cohomological) dimension of .Weuse

H X G; Ak

n n;p

the notation n for the restriction of to ; , we write for the

G G

H X ; Z= ! H X ; Z=

composition of n with the projection 2 2 , and similar

n; de®nitions hold for n;even and odd.

It is clear from the above that

s = ; (9)

and that the mapping

H X G; Z= ! H X ; Z=

n : ; 2 2 n<

induced by is surjective if 0. Note that, together with the Hochschild±Serre

E G; Z= X

spectral sequence ; 2 , this proves equation (1). Equations (2) and (3) p;q

can be derived from the Hochschild±Serre spectral sequence with coef®cients in Z and the following proposition.

PROPOSITION 2.2. Let X be a locally compact space of ®nite cohomological

= Z=

dimension with an action of G 2.Then

H X G; Zk ! H X ; Z= n

n;even : ; even 2

n + k

is an isomorphism if n<0and is even, and

H X G; Zk ! H X ; Z= n

n;odd : ; odd 2

n + k is an isomorphism if n<0and is odd.

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224 FREDÂ ERICÂ MANGOLTE AND JOOST VAN HAMEL

H X G; Zk H X ; Z=

Observe that it is not claimed that n ; 2 is

G G

X ; Z= H X ; Z=

contained in Heven 2 (resp. odd 2 ). In fact this is often not the

2 H X G; Zk p n + k

case: for any n ; there is a mod 2 such that

= + + + + ;

n;p n;p n;p

n;p 2 2 (10)

G G

H X ; Z= ! H X ; Z=

+ p where is the Bockstein homomorphism p 1 2 2 asso-

ciated to the short exact sequence

Z= ! Z= ! Z= ! 0 ! 2 4 2 0

(compare [Kr3, Thm. 0.1]).

We will also use the symbol for the connecting homomorphism

H X G; Z= ! H X G; Zk

+ n n 1 ; 2 ; of the long exact sequence (4), and we

have

= + n + k ;

n; n+ n+ ;

even 1;even 1odd if is even (11)

= + n + k :

n+ ; n+ ; n;odd 1 odd 1even if is odd (12)

It is clear from the de®nition and the projection formula (7) that

\ ! = \!;

(13)

X ! Y G

and for any proper mapping f : of -spaces

f =f :

(14)

G; A ' A H ;A = A

There are canonical isomorphisms H0 pt; and 0 pt ,so

X !

the homomorphisms induced by the constant mapping ': pt give us for every X

compact G-space the degree maps

X G; A ! A deg : H ;

G 0

and

X; A ! A; deg: H0

which satisfy the equality

= e:

e deg deg (15)

G G

H X ; Z= H X ; Z=

Extending the degree map on 0 2 by 0 to the whole of 2 ,

we have by equation (14) that

;

deg deg mod 2 (16) G

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REAL ENRIQUES SURFACES 225

2 H X G; A for any 0 ; .

Finally, de®ne G

G 0

H X ;A = f H X ;A ! Ag;

ker deg:

G G

G 0 0

H X ; Z= = H X ; Z= \ H X ; Z=

and even 2 even 2 2 . We will record three

technical lemmas for use in Section 5. They can be proven by a careful inspection

E X G; Ak

of either the Hochschild±Serre spectral sequence p;q ; or the long

exact sequence (8) with the appropriate coef®cients.

G X 6= ; LEMMA 2.3. Let X be a compact connected -space with .Then

G 0

H X G; Z= ! H X ; Z=

2: 2 ; 2 2

is surjective if and only if the composite mapping

[

1 G 2

X G; Z= ! H X; Z= H G; H X; Z= H1 ; 2 1 2 1 2

is zero. G LEMMA 2.4. Let X be a compact connected -space. Then

G 0

H X G; Z ! H X ; Z=

2;even: 2 ; even 2

is surjective if and only if the composite mapping

e 2

[

1 G 2

- -

X G; Z H X; Z H G; H X; Z H1 ; 1 1 1 1 1

is zero.

G X 6= ; LEMMA 2.5. Let X be a locally compact connected -space with .Then

the mapping

H X G; Z ! H X ; Z=

2;odd : 2 ; 1 odd 2

is surjective if and only if the composite mapping +

e 2

1 G

- 2

H H X G; Z X; Z G; H X; Z H1 ; 1 1

is zero.

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3. The fundamental class of a G-manifold

Z= Z X A

Let again A be 2or .Let be an -oriented topological manifold of ®nite

G = f ;g

dimension d with an action of 1 . We will de®ne the fundamental class

Ak k

of X in equivariant homology with coef®cients in for even or odd.

H X; A = A A

It is well-known, that d ,andthe -orientation determines a

H X; A X X

generator of d . Observe that we do not need to require to be A

compact, since we use Borel±Moore homology. If G acts via an -orientation

G G

2 H X; A 2 H X; A

X X d

preserving involution, then d ,otherwise 1.

2 Z

By the Hochschild±Serre spectral sequence (2) we have for k an isomorphism

H X G; Ak ' H X; Ak e d

d ; , given by the edge morphisms ,sowe d

have the fundamental class

2 H X G; Ak ; X d ;

where k must have the right parity. G

PROPOSITION 3.1 (PoincareÂ duality). Let X be a -manifold with fundamental

2 H X G; Ak X

class d ; . Then the mapping

H X G; Al ! H X G; Ak l ;

; d ;

! 7! \ !; X

is an isomorphism.

Assuming that the action of G is locally smooth (i.e., each ®xed point has a

d G

neighbourhood that is equivariantly homeomorphic to R with an orthogonal - G

action), the ®xed point set of X is again a topological manifold, but it need not V

be A-orientable and it need not be equi-dimensional. However, if is a connected

X V d 2

component of and has dimension 0, then it has a fundamental class V

H V; Z= 2 H X ; Z= V

d;d d

d0 2, and we have that the restriction of 0 0 2 to

X G G Y

equals V (see [vH]). If is a closed sub- -manifold of a -manifold , then the

X ! Y

embedding j : is proper, so it induces a mapping in equivariant homology.

H Y G; Ak X j

We de®ne the class in d ; represented by to be . R

Now let X be an algebraic variety de®ned over . We want to de®ne the class in

H X G; Zd d

2d ; represented by a subvariety of dimension . As in [Fu], we will distinguish two kinds of subvarieties, the geometrically irreducible subvarieties,

which are varieties over R themselves, and the geometrically reducible subvarieties,

which are irreducible over R, but which split into two components when tensored

with C . Then the complex conjugation exchanges these two components. d

Any complex algebraic variety V of dimension has a fundamental class

2 H V C ; Z C V

2d , and the complex conjugation on preserves orientation if

d j Z,!X d is even, and reverses orientation if is odd. This implies that if : is the

comp4075.tex; 5/12/1997; 11:40; v.7; p.12

REAL ENRIQUES SURFACES 227

d R

inclusion of a subvariety of dimension de®ned over ,then Z is a generator

G G

H Z C ; Zd Z H Z C ; Zd

C d

of d if is irreducible, and is generated by

+ Z Z Z

Z C

Z1 2 if is the union of two distinct complex varieties 1 and 2 of

d 2 H G; Zd Z C Z

dimension . Hence we de®ne the fundamental class 2d ;

Z + e

Z Z

of to be the inverse image of Z (resp. of ) under . The class C

1 2 2d

[Z ] 2 H X C G; Zd Z j

2d ; represented by is of course de®ned to be .Ifwe

[Z R] 2 H X R; Z=

use the notation d 2for the homology class represented by

Z , as de®ned in [BH], then indeed

[Z ] = [Z R ]:

2d;d (17)

X R

If Z; Z are subvarieties of de®ned over which are rationally equivalent

[Z ]=[Z ] d 6 X over R (see [Fu] for a de®nition), then , so we get for every dim

a well-de®ned cycle map

CH X ! H XC G; Zd d d 2;

from the Chow group in dimension d to equivariant homology. The image will be

alg

X C G; Zd denoted by H ; , and we see by equation (17), that 2 d

alg alg

H X C G; Zd = H X R ; Z= :

d;d ; 2 (18) d

2 2 d n For X nonsingular projective of dimension , this map coincides with the

composition of the mapping

CH X ! H XC G; Zn d

d ;

as de®ned in [Kr2] and the PoincareÂ duality isomorphism. As a consequence we can use the following description of the image of the cycle map in codimension 1,

H X C G; Z CH X where we use the notation alg ; 1 for the image of n 1in

cohomology. R

PROPOSITION 3.2. Let X be a nonsingular projective algebraic variety over .

O X C

Let h be the sheaf of germs of holomorphic functions on .Then

X C G; Z Halg ; 1 is the kernel of the composite mapping

2 - 2 2

H X C G; Z H X C ; Z ! H X C ; O :

; 1 h

Proof. This follows immediately from Proposition 1.3.1 in [Kr2], which states

X C G; Z that Halg ; 1 is the image of the connecting morphism

1 2

X C G; O ! H X C G; Z

H ; ; 1 h

comp4075.tex; 5/12/1997; 11:40; v.7; p.13 228 FREDÂ ERICÂ MANGOLTE AND JOOST VAN HAMEL

in the long exact sequence induced by the exponential sequence of G-sheaves

! Z !O !O : 2 !

0 1 h 0 h

4. Algebraic cycles

The following facts about real Enriques surfaces can be found in [Ni2] or [DKh1].

Y X ! Y Y C

Let be a real Enriques surface. Let C be the double covering of by

X C X C

a complex K3-surface X .Since is simply connected, is the universal

C H Y C ; Z=Z=

covering space of Y and 1 2. The complex conjugation on

C X C Y can be lifted to the covering by [Si, Th.A8.5], and this can obviously

be done in two different ways. Hence we can give X the structure of a variety

X X

over R in two different ways, which we will denote by 1 and 2.Thetwo

Y Y R

halves Y1 and 2 of mentioned in the introduction consist of the components

R X R X R

covered by X1 and 2 , respectively. All connected components of 1

and X2 are orientable, as is the case for the real part of any real K3-surface. Y

If a connected component of a half i is orientable, then it is covered by two

X R

components of i , which are interchanged by the covering transformation of

X Y X R i . A nonorientable component of i is covered by just one component of ; this is the orientation covering.

H Y C ; O =

Since for an Enriques surface h 0 (s e e [B P V, V. 2 3 ]), w e

alg

Y C G; Z = see by Proposition 3.2 and PoincareÂ duality that H2 ; 1

alg

Y C G; Z H Y R ; Z=

H2 ; 1 ,so 1 2 is the image of the mapping

= H Y C G; Z ! H Y R; Z= :

2 2;1 : 2 ; 1 1 2

n 2 Z

In order to determine the image of 2 we will de®ne n for any by

= H Y C G; Zn ! H Y R ; Z= :

n; n

n 1 : ; 1 1 2

= s

Observe, that n 1 .

LEMMA 4.1. For a real Enriques surface Y the codimension of Im 2 in

Y R ; Z=

H1 2 does not exceed 1.

Y R 6= ;

Proof. We may assume that . Using the fact that 1 is an isomor-

+ s

phism by Proposition 2.2, and both s0 and 1 are surjective by the long exact

= s

sequence (8), we see that 1 is surjective. Since 2 1 2 , it suf®ces to remark

H Y C G; Z ! H Y C G; Z

that if the cokernel of s2 : 2 ; 1 1 ; is nonzero, it

Y C ; Z=Z=

is isomorphic to H1 2.

2 H Y R; Z= PROPOSITION 4.2. Let Y be a real Enriques surface. A class 1 2

is contained in the image of 2 if and only if

\ w Y R = ; deg 1 0

Y R 2 H Y R; Z= Y R where w1 2is the ®rst Stiefel±Whitney class of .

comp4075.tex; 5/12/1997; 11:40; v.7; p.14

REAL ENRIQUES SURFACES 229

R 6= ;

Proof. Again we may assume that Y . Denote by the subspace of

Y R ; Z= \ w Y R =

H1 2 whose elements verify deg 1 0.

R w Y R = =H YR;Z= If Y is orientable, 1 0and 12.Furthermore,

we have a surjective morphism

X R ; Z= H X R ; Z= ! H Y R ; Z= ;

H1 1 2 1 2 2 1 2

X Y where the X1 and 2 are the two real K3-surfaces covering (see the beginning

of this section). This morphism ®ts in a commutative diagram

X C G; Z H X C G; Z H Y C G; Z H2 1 ; 1 2 2 ; 1 2 ; 1

2 X

X1 2

2 2

? ?

X R ; Z= H X R ; Z= H Y R ; Z= :

H1 1 2 1 2 2 1 2

H X C G; Zn ! H X R ; Z=

Here the : n ; 1 2 are de®ned in the

n 1 1 1

H X C ; Z = X

same way as n .As 1 0 for a real K3-surface , it follows from X

X1 2

Lemma 2.5, that 2 and 2 are surjective, which implies the surjectivity of 2. =

In other words, Im 2 .

R w Y R 6=

Now assume that Y is nonorientable. Then 1 0, and by non- =

degeneracy of the cap-product pairing codim 1. First we will prove that

Im 2 .

= cw Y C 2 H Y C G; Z cw Y C

Let K 1 ; 1 ,where 1 is the ®rst

C mixed characteristic class of the tangent bundle of Y as de®ned in [Kr2, 3.2].

K 2 H Y C ; Z Y

Then e is the ®rst Chern class of the canonical line bundle of ,

K = 2 H Y C G; Z

so 2e 0 (see [BPV, V.32]). This means that for any 2 ; 1

\ K = e \ eK = \ K =

we have degG deg 0, so deg 0by

Equations (16) and (13).

2 H Y R ; Z= H Y R ; Z=

The projection 2;2 of 2 to 2 2 is zero

2;0

K K 2 H Y R; Z= by equation (10) and the projection of 2 to

Y R ; Z= H 2 is zero by [Kr3, Th. 0.1]. This implies

2;1

\ K = \ K ;

deg deg 2;1

2 ;1

K =w YR =

but 1by [Kr2, Th. 3.2.1], and 2;1 2by de®nition,

\ w Y R =

so deg 2 1 0. In other words, Im 2 . Lemma 4.1 now gives

= 2

us that Im 2 .

Y R COROLLARY 4.3. With the above notation, 2 is surjective if and only if is orientable.

Theorem 1.1 in the introduction is an immediate consequence of

alg

Y R ; Z= Proposition 4.2. We can even give an explicit description of H 1 2 .

comp4075.tex; 5/12/1997; 11:40; v.7; p.15

230 FREDÂ ERICÂ MANGOLTE AND JOOST VAN HAMEL

2 H Y R; Z=

THEOREM 4.4. Let Y be a real Enriques surface. A class 1 2

\ w Y R = can be represented by an algebraic cycle if and only if deg 1 0.

5. Galois-maximality

The aim of this section is to describe which Enriques surfaces are Z-GM-varieties and/or GM-varieties in terms of the orientability of the real part and the distribution of the components over the halves. See the introduction for the de®nition of Galois- maximality and Section 4 for the de®nition of `halves'. The proof of Theorem 1.2 will consist of a collection of technical results and

explicit constructions of equivariant homology classes. For completeness we also = prove the parts of Theorem 1.2 concerning coef®cients in Z 2, although these

results are not new (see the Introduction). R

LEMMA 5.1. Let Y be an algebraic variety over .Then

+ G

Y Z e H Y C ; Z e

(i) is -GMif and only if is surjective on p and is surjective

p p

H Y C ; Z p

onto p 1 for all .

Y e H Y C ; Z= p p

(ii) is GM if and only p is surjective onto 2 for all . Z

Proof. This follows from the fact that Y is GM (resp. -GM) if and only if

E G; A A = Z= Y C

the Hochschild±Serre spectral sequence ; is trivial for 2 p;q

(resp. Z), and this can be checked by looking at the edge morphisms, since we

> G M H G; M !

have for every k 0andevery -module natural surjections

+ k k +

k 2 1

G; M H G; M ! H G; M H ,and 1 , which are isomorphisms for

k>0.

Y R 6= ;

LEMMA 5.2. Let Y be a real Enriques surface with .Then

p 2f ; ; ; g;e H Y C ; Zk p

(i) for any 0234is surjective onto p ,

p 2f ; ; g;e H Y C ; Z= p (ii) for any 034p is surjective onto 2 . Proof. This can be seen from the Hochschild±Serre spectral sequences (cf. [Krl,

Sect. 5]).

Y R 6= ; Y

COROLLARY 5.3. Let Y be a real Enriques surface with .Then

e H Y C ; Zk k = ;

is Z-GM if and only if 1 is surjective onto 1 for 0 1.

e e H Y C ; Z=

Moreover, Y is GM if and only if 1 and 2 are surjective onto 1 2 ,

Y C ; Z=

resp. H2 2 .

Y R 6= ; e

LEMMA 5.4. Let Y be a real Enriques surface with .If 2 is not

G G

Y C ; Z= e H Y C ; Z= surjective onto H2 2 ,then 1 is not surjective onto 1 2 .

comp4075.tex; 5/12/1997; 11:40; v.7; p.16 REAL ENRIQUES SURFACES 231

Proof. By PoincareÂ duality we see that if e2 is not surjective onto G

G 2 2

Y C ; Z= e H Y C ; Z= H2 2 ,then is not surjective onto 2 . Let us assume

2 1

! 2 H Y C G; Z= that e is not surjective. Then by Lemma 2.1 there exists an ; 2

! 6= ! =

such that e 0, but 0.

H Y C ; Z= 2

Now suppose e1 is surjective onto 1 2 , then there exists a

Y C G; Z= H1 ; 2 such that

e \ e ! 6= :

deg 1 0

\ ! 6=

This means that degG 0, but this contradicts

\ ! = \ ! = \ = :

deg deg deg 0 0 G

Hence e1 is not surjective.

Y R 6= ;

PROPOSITION 5.5. Let Y be a real Enriques surface with .Then

Z e e

(i) Y is -GM if and only if 1 and 1 are nonzero. e

(ii) Y is GM if and only if 1 is nonzero.

e e Y Z Y (iii) If e1 is zero then 1 and 1 are zero. In particular, if is -GM, then is also GM.

Proof.IfY is an Enriques surface,

Y C ; Z=H YC;Z= =Z= ;

H1 122

e e

so e1 and 1 are surjective if and only if they are nonzero. By Lemma 5.4, 2 is

6= surjective if e1 0, so we obtain the ®rst two assertions from Corollary 5.3. The

last assertion follows from the commutative diagram

H Y C G; Zk Y C ; Zk

H1 ; 1

? ?

H Y C G; Z= H Y C ; Z= :

1 ; 2 1 2 2

Y R 6= ; e =

LEMMA 5.6. Let Y be a real Enriques surface with .Then 1 0if

and only if Y is orientable.

Y R Proof. We know from Corollary 4.3, that 2 is surjective if and only if

[

Y C ; Z = Z= H Y C ; Z is orientable. Since H1 2, the mapping 1

G; H Y C ; Z

H 1 is an isomorphism, so Lemma 2.5 gives us that 2 is surjec-

+ = tive if and only if e1 0.

comp4075.tex; 5/12/1997; 11:40; v.7; p.17

232 FREDÂ ERICÂ MANGOLTE AND JOOST VAN HAMEL

Y Y

LEMMA 5.7. If the two halves Y1 and 2 of a real Enriques surface are non-

empty, then e1 0.

X Y

Proof.Let be the K3-covering of C ,let be the deck transformation of this

X C

covering and denote by 1 and 2 the two different involutions of lifting the

Y C X R p

i i

involution of .Let i be the set of ®xed points under and let be a

X R i = ;

point in i for 1 2.

X C p p

Let l be an arc in connecting 1 and 2 without containing any other

R X R L l; l; l;l

point of X1 or 2 . Then the union of the four arcs 1 2is

L=L

homeomorphic to a circle, and we have that . This implies that the image

L Y C

of in is again homeomorphic to a circle; we choose an orientation on .

Now G acts on via an orientation reversing involution, so represents a class

] H Y C G; Z X C ! Y C

[ in 1 ; 1 .Since is the universal covering, and the

inverse image of is precisely , hence homeomorphic to a circle, the class of

Y C ; Z e [] 6=

is nonzero in H1 ,so 1 0.

;Y Y

LEMMA 5.8. If exactly one of the halves Y1 2of a real Enriques surface is

= Y R

empty, then e1 0 if and only if is orientable.

= e = Y R

Proof.Ife1 0, we have 1 0 by Proposition 5.5 and then is orientable

R X R = ; X R ! Y R by 5.6. Conversely, if Y is orientable and 2 ,then 1

H X R ; Z= !

is the trivial double covering, so it induces a surjection 1 2

0 0

H Y R ; Z= H ; Z=

2 ,where 2 denotes the kernel of the degree map as

X C ; Z= = H X C de®ned in Section 2. Since H1 2 0, the mapping : 2 1 ;

G; Z= ! H X R ; Z=

2 1 2 is surjective by Lemma 2.3. Now the functoriality

of with respect to proper equivariant mappings (Equation (14)) implies

H Y C G; Z= ! H Y R ; Z=

2 : 2 ; 2 2

is surjective, and Lemma 2.3 then gives that e1 is zero.

;Y Y

LEMMA 5.9. If exactly one of the halves Y1 2of a real Enriques surface is

6= Y R

empty, then e1 0 if and only if has components of odd Euler characteristic.

= ; Proof. Assume Y2 . By Lemma 2.4, it suf®ces to show that

H Y C G; Z ! H Y R; Z=

2;even : 2 ; even 2

is surjective if and only if Y has no components of odd Euler characteristic.

Although Y need not be orientable, we can apply the K3-covering as in the

previous lemma and prove that the image of 2;even contains a basis for the subgroup

H Y R ; Z= \ H Y R; Z=

0 2 even 2 ,so 2;even is surjective if and only if

H Y C G; Z ! H Y R; Z=

2;2 : 2 ; 2 2

Y R; Z=

is surjective. We will use that H2 2 is generated by the fundamental

R classes of the connected components of Y .

comp4075.tex; 5/12/1997; 11:40; v.7; p.18

REAL ENRIQUES SURFACES 233

Y R V

Pick a component V of .If is orientable, it gives a class in

Y C G; Z V H Y R; Z=

H2 ; , which maps to the fundamental class of in 2 2 .

[V ] V

Now assume V is nonorientable. Let be the fundamental class of in

H Y R ; Z= [V ] V H Y C G; Z=

2 2 ,let G be the class represented by in 2 ; 2 ,and

= [V ] H Y C G; Z = ;

let G be the Bockstein image in 1 ; 1 .Then 12

[V ] =[V] [V ] H Y C G; Z G

2;2 by equation (11), so is in the image of 2 ; under

e =

2;2 if and only if 1 0.

e =i [V ] i V ! Y C

From the construction of we see that 1 ,where :

[V ] 2 H V; Z [V ]

is the inclusion and 1 is the Bockstein image of . Therefore

V X C ! Y C

e1 can be represented by a circle embedded in .Since is the

L X C

universal covering, e1 is zero if and only if the inverse image of in has

X R V

two connected components. Let W be the component of 1 covering .Then

V L W V W is the orientation covering of and .If has odd Euler characteristic,

then it is the connected sum of a real projective plane and an orientable compact V surface. We see by elementary geometry that L is connected. If has even Euler characteristic, it is the connected sum of a Klein bottle and an orientable compact

surface, and we see that L has two connected components.

Proof of Theorem 1.2. By Proposition 5.5, the ®rst part of the theorem follows from Lemma 5.6 and Lemma 5.7, and the second part of the theorem follows from Lemma 5.8 and Lemma 5.9.

6. The Brauer group R Let Y be a nonsingular projective algebraic variety de®ned over .Let

0 2

Y =H Y; G

Br m

e t

n; Y n

be the cohomological Brauer group of Y , and let Tor Br be the -torsion

of Br . We have a canonical isomorphism

n; Y Tor Br

mod n

2 - 2

H fH Y C G; Z Y C G; Z=n g; ' Coker alg ; 1 ; 1 (19)

as can be deduced from the Kummer sequence

- - - -

G G ;

m m 1 n 1

and the well-known identi®cations

k k

H Y; ' H Y C G; Z=n

n ;1

e t and

H Y; G ' Y :

m Pic

comp4075.tex; 5/12/1997; 11:40; v.7; p.19 234 FREDÂ ERICÂ MANGOLTE AND JOOST VAN HAMEL

It can be checked, that the mapping

2;0 2 0

H Y C G; Z= ! H Y R; Z= : ; 2 2

induces a well-de®ned homomorphism

0 0

; Y ! H Y R ; Z= :

Tor 2 Br 2 (20)

6 Y

If dim Y 2, in particular if is a real Enriques surface, we may identify

Y Y

Br with the classical Brauer group Br (see [Gr2, II, Th. 2.1]). Two of the = main problems considered in [NS] and [Ni1] are the calculation of dimZ 2 Tor(2,

Br(Y )) and the question whether the mapping (20) is surjective for every real

Enriques surface Y . Both problems were solved for certain classes of real Enriques surfaces. The second problem has been completely solved in [Kr3], where it is

shown that the mapping (20) is surjective for any nonsingular projective surface Y

de®ned over R (see Remark 3.3 in loc. cit.). The results in Section 5 will help us

to solve the ®rst problem for every Enriques surface Y by determining the whole

group Br . R LEMMA 6.1. Let Y be a nonsingular projective algebraic variety de®ned over such that

2 2

Y C G; Z = H Y C G; Z : Halg ; 1 ; 1

Then

0 3

Y ' H Y C G; Z : Tor Br Tor ; 1

Proof. By the hypothesis and the isomorphism (19) there is for every integer

n> 0 a short exact sequence

2 2 0

Y C G; Z Z=n ! H Y C G; Z=n ! n; Y ; H ; 1 ; 1 Tor Br

hence we deduce from the long exact sequence in equivariant cohomology associ-

ated to the short exact sequence

Z ! Z ! Z=n ! ; 0 ! 1 1 1 0

that we have for every n>0 a natural isomorphism

0 3

n; Y ' n; H Y C G; Z :

Tor Br Tor ; 1

Y = Proof of Theorem 1.3. By [Gr2, 1.2 and II, Thm. 2.1] we have Br

0 3

Y = Y H Y C G; Z = Tor(Br Tor(Br . On the other hand, Tor ; 1

comp4075.tex; 5/12/1997; 11:40; v.7; p.20 REAL ENRIQUES SURFACES 235

3 3

Y C G; Z H Y C ; Z=Z= H ; 1 since 2. Hence, by Lemma 6.1 and PoincareÂ

duality

Y ' H Y C G; Z :

Br 1 ; 1

k =Z

Now consider the long exact sequence (8) for A

+ s

e1 1

- -

H Y C;Z ! H Y C G; Z H Y C G; Z !

11;1 0 ;

k =Z

It follows from Proposition 2.2 and the long exact sequence (8) for A 1

H Y C G; Z ! H Y R ; Z=

that : ; 2 induces an isomorphism

! H Y R ; Z= : im s1 even 2

We obtain an exact sequence +

- 0

Z= ! H Y C G; Z ! H Y R ; Z= ! :

21;1 even 2 0 (21)

R 6= ; e 6= H Y C G; Z '

If Y is nonorientable, then 1 0byLemma5.6,so 1 ; 1

2s 1

2 , which proves the ®rst part of the theorem.

R 6= ; e = Now assume Y is orientable. Then 1 0 by Lemma 5.6, so we get

from (21) an exact sequence

2s 1

Z= ! H Y C G; Z ! Z= ! :

0 ! 2 1 ; 1 2 0

2s 2 2

Y C G; Z ' Z= Z= Z= Hence H1 ; 1 2 or 2 4 . In order to decide between these two possibilities, consider the following com-

mutative diagram with exact rows

- -

H Y C G; Z= Y C G; Z H Y C G; Z

2 ; 2 H1 ; 1 1 ; 1

e e 2 e

1 1

? ? ?

- -

Y C ; Z= H Y C ; Z H Y C ; Z

H2 2 1 1

6 6 6

+ +

e e 2 e

1 1

- -

H H Y C G; Z= Y C G; Z Y C G; Z:

H2 ; 2 1 ; 1 ;

Y C G; Z

We have that H1 ; 1 is pure 2-torsion if and only if is surjective.

e = We claim that is surjective if and only if 1 0. Together with Lemmas 5.9 and 5.7 this would prove the second part of the theorem.

comp4075.tex; 5/12/1997; 11:40; v.7; p.21

236 FREDÂ ERICÂ MANGOLTE AND JOOST VAN HAMEL

= e = e 6=

Let us prove the claim. Since e1 0, we have 2 0. If 1 0, an easy

diagram chase shows that is not surjective. On the other hand the following

diagram can be shown to be commutative.

Y C G; Z

H2 ;

P s

P 2

P P

mod2 P

Y C G; Z= H Y C G; Z

H2 ; 2 1 ; 1

+ +

e = s e = In other words, Im s2 Im .Nowker 1 Im 2 ,soif 1 0, then is

surjective.

R = ;

Finally, we will consider the short exact sequence (21) for the case Y .

G Y C k H Y C G; Z= =

Then acts freely on ,sowehaveforall that k ; 2

H Y C =G; Z=

k 2 . By the remarks made in the introduction of Section 4, this

Y C G; Z= =Z= Z=

means that H1 ; 2 22, and we can see from the long exact

k =Z= e = e =

sequence (8) for A 2that 1 0. This implies that 1 0 (see Proposi-

Y C G; Z = Z= tion 5.5.iii), hence H1 ; 1 2.

Acknowledgements Large parts of this paper were written during visits of the ®rst author to the Vrije Universiteit, Amsterdam and of the second author to the UniversiteÂ Montpellier II. We want to thank J. Bochnak and R. Silhol for the invitations, and the Thomas Stieltjes Instituut and the UniversiteÂ Montpellier II for providing the necessary funds. We are grateful to A. Degtyarev and V. Kharlamov for giving us preliminary versions of their papers.

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