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Algebraic Cycles and Topology of Real Enriques Surfaces

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Compositio Mathematica 110: 215±237, 1998. 215 c 1998 Kluwer Academic Publishers. Printed in the Netherlands. Algebraic cycles and topology of real Enriques surfaces ?? FRED ERIC MANGOLTE1 ? and JOOST VAN HAMEL2 1Laboratoire de Mathematiques, Universite 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 R 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. Downloaded from https://www.cambridge.org/core. IP address: 170.106.35.229, on 26 Sep 2021 at 16:44:06, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1023/A:1000223408405 **INTERPRINT**/Preproof: A M/c 4: PIPS Nr.: 124150 MATHKAP comp4075.tex; 5/12/1997; 11:40; v.7; p.1 216 FRED ERIC MANGOLTE AND JOOST VAN HAMEL Y R classi®cation of topological types of algebraic varieties over (the mani- C 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 2n 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 2n 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. Downloaded from https://www.cambridge.org/core. IP address: 170.106.35.229, on 26 Sep 2021 at 16:44:06, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1023/A:1000223408405 comp4075.tex; 5/12/1997; 11:40; v.7; p.2 REAL ENRIQUES SURFACES 217 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 situa- tion 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 R 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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