On the Brauer Group of Real Algebraic Surfaces

Let X be a real projective algebraic manifold, s numerates connected components of X(R) and 2Br(X) the subgroup of elements of order 2 of cohomological Brauer group Br(X). We study the natural homomorphism ξ :2 Br(X) → (Z/2)s and prove that ξ is epimorphic if H(X(C)/G;Z/2) → H(X(R);Z/2) is injective. Here G = Gal(C/R). For an algebraic surface X with H(X(C)/G;Z/2) = 0 and X(R) 6= ∅, we give a formula for dim2 Br(X). As a corollary, for a real Enriques surfaces Y , the ξ is epimorphic and dim2 Br(Y ) = 2s−1 if both liftings of the antiholomorphic involution of Y to the universal covering K3surface X have non-empty sets of real points (this is the general case). For this case, we also give a formula for the number snor of non-orientable components of Y which is very important for the topological classification of real Enriques surfaces. §0. Formulation of basic results In the paper of R. Sujatha and the author [N-S], the Brauer group of a real Enriques surface was studied. Here we continue the study of Brauer group with the remark that most of the results of these paper generally valid for an arbitrary smooth projective real algebraic surface. Let X be a projective algebraic variety over the field R of real numbers. Let Br(X) = H et(X ;Gm) denote the cohomological Brauer group of X . See a definition in the book of J. Milne [Mi], for example. We mention that the cohomological Brauer group is very closely related with the more interesting classical Brauer group Br(X) classifying Azumaya algebras over X (see the papers of A. Grothendieck [Gr2] and the book [Mi]). For example, it is known that Br(X) ⊂ Br(X). For curves and smooth surfaces it gives an isomorphism. But we will only consider here the cohomological Brauer group Br(X). LetX(R) denote the space of R-rational points ofX with the Euclidean topology and s denote the number of real connected components of this space. Let 2Br (X) denote the group of elements of order two in Br(X). If P ∈ X(R) is a real point of X , we get a natural map 2Br (X) →2 Br (P ) ∼= Z/2. It is shown in [CT-P] that this map does not depend from a choice of the point P in a connected component of X(R). Thus, the canonical map (0–1) 2Br (X) → (Z/2) Typeset by AMS-TEX 2 VIACHESLAV V. NIKULIN is defined. We mention that a studying of the map (0–1) and a description of the Brauer group of X is very important for calculation of a such interesting group connected with X as the Witt group W (X). See R. Sujatha [Su]. It is shown in the paper of J.-L. Colliot-Thélène and R. Parimala [CT-P] that the map (0–1) is epimorphic if X is a smooth projective algebraic surface and H(X(C);F2) = 0 (here F2 = Z/2). This is a generalization of the old result of E. Witt [W] about curves (compare with Remark 1.8 below). There doesn’t seem to be any known example of a surface where the map (0–1) fails to be epimorphic. This paper is devoted to studying of this map (0–1) and also calculating of dim 2Br (X). Our idea is to interpret 2Br (X) and the map (0–1) purely topologically, and apply to them topological considerations. We prove here the following basic result where G is the group of order two generated by the antiholomorphic involution g of X(C) defined by the structure of real algebraic variety on X . Theorem 0.1. Let X/R be an algebraic projective manifold (smooth) over the field R of real numbers. Then, the homomorphism (0–1) is epimorphic if H(X(C)/G;F2) = 0. More generally, the homomorphism (0–1) is epimorphic if the kernel of the homomorphism i : H(X(C)/G;F2) → H (X(R);F2) is equal to zero. Here i : X(R) ⊂ X(C)/G denote the embedding. Proof. See Theorem 1.6 below. In fact, in Theorem 1.6, we give a precise topological obstruction to epimorphicity of the map (0–1). This obstruction is zero if the kernel of the homomorphism i above is zero. We mention that for smooth curves X the group H(X(C)/G;F2) = 0, thus the map (0–1) is epimorphic (it is well-known, compare with [W]). For surfaces X the group H(X(R);F2) = 0, and Ker i ∗ = H(X(C)/G;F2). Now, let us show that, from Theorem 0.1, the result of J.-L. Colliot-Thélène and R. Parimala mentioned above follows. Let X be a smooth projective algebraic surface and H(X(C);F2) = 0. Then, by Poincaré duality, we have H1(X(C);F2) = 0. If X(R) 6= ∅, then any loop in X(C)/G with the beginning onX(R) has a lifting to a loop onX(C). It follows that the canonical homomorphism H1(X(C);F2) → H1(X(C)/G;F2) is epimorphic. Thus, H1(X(C)/G;F2) = 0. For the dimension 2 the quotient space X(C)/G is homeomorphic to a smooth compact 4-dimensional manifold. By Poincaré duality, we then get that H(X(C)/G;F2) = 0. This proves the statement. We apply the Theorem 0.1 to real Enriques surfaces. By a complex Enriques surface Y over C , we mean a non-singular minimal projective algebraic surface Y/C such that the invariants κ(Y ) = pg(Y ) = q(Y ) = 0. These are equivalent to irregularity q(Y ) = 0 and 2KY = 0 but KY 6= 0 where KY is the canonical class of Y . One may find all information about Enriques surfaces we need in the books [A] and [C-D]. By a real Enriques surface Y/R, we mean a projective algebraic surface Y/R such that Y ⊗RC is a complex Enriques surface. Universal covering complex surface of an Enriques surface Y (C) is a K3-surface X(C) (see [A] and [C-D]) which twice ON THE BRAUER GROUP OF REAL ALGEBRAIC SURFACES 3 covers the Enriques surface Y (C). We denote by τ the holomorphic involution on X(C) of this covering. There are precisely two liftings σ and τσ on X(C) of the antiholomorphic involution θ of Y (C) corresponding to the real structure on Y . Besides, one can see very easily that if Y (R) 6= ∅, then both σ and τσ are antiholomorphic involutions of X(C). Thus, σ and τσ define two real structures Xσ and Xτσ on the K3-surface X . We denote by Xσ(R) = X(C) , Xτσ(R) = X(C) τσ the real parts of the real K3-surfaces Xσ and Xτσ corresponding to these real structures respectively. Since τ has no fixed points on X(C), it follows that the sets Xσ(R) and Xτσ(R) have an empty intersection. From the Theorem 0.1, we get Corollary 0.2. Let Y be a real Enriques surface with the antiholomorphic involution θ, and the real part Y (R) 6= ∅. Suppose that the real parts Xσ(R) and Xτσ(R) of both liftings σ and τσ of θ to the universal covering K3-surface X(C) are non-empty. Then the canonical map (0–1) corresponding to the real Enriques surface Y is epimorphic. Proof. Y (C)/{idY (C), θ} = X(C)/{idX(C), τ, σ, τσ} = (X(C)/{idX(C), σ})/{id, τσ mod {idX(C), σ}}. Here involutions σ and τσ mod {idX(C), σ} have non-empty sets of fixed points because real parts of both involutions σ and τσ of X(C) are non-empty and are not coincided. Since for a K3-surface X , the group H1(X(C);F2) = 0, it follows like above that H1(X(C)/{idX(C), σ};F2) = 0 and H1(Y (C)/{idY (C); θ};F2) = 0. The topological space Y (C)/{idY (C); θ} is isomorphic to a smooth compact 4dimensional manifold. By Poincaré duality, then H(Y (C)/{idY (C), θ};F2) = 0. By Theorem 0.1, the map (0–1) is epimorphic for the real Enriques surface Y . The same considerations show that if one of involutions σ or τσ has an empty set of real points, then H1(Y (C)/{idY (C), θ};F2) = F2 and H (Y (C)/{idY (C), θ};F2) = F2. – If Xσ(R) 6= ∅ but Xτσ(R) = ∅ then the surface X(C)/{idX(C), σ} is a 2-sheeted universal covering of the surface Y (C)/{idY (C), θ}. Thus, the Corollary 0.2 gives precisely the case when the Theorem 0.1 may be applied to real Enriques surfaces. We discuss in Remark 1.7 below a chance of constructing a counterexample to epimorphisity of the map (0–1) using real Enriques surfaces Y above with H(Y (C)/G;F2) = F2 (equivalently, with Xσ(R) 6= ∅ but Xτσ(R) = ∅). It is not difficult to show that real Enriques surfaces with the condition H(Y (C)/G;F2) = F2 do exist. For the most part of real Enriques surfaces (from the point of view of the number of connected components of the moduli space) both involutions σ and τσ have a non-empty set of real points. But for some real Enriques surfaces one of these involutions may have an empty set of real points. The following results are devoted to a calculation of the dimension of étale cohomology groups with coefficients F2, and 2Br (X). 4 VIACHESLAV V. NIKULIN We begin with the following general remark about a real algebraic variety X . We recall that the Kummer sequence (0–2) 0 → μ2 → Gm → Gm → 0 yields the exact sequence (0–3) 0 → Pic X/2Pic X → H et(X ;μ2) →2 Br (X) → 0. If X(R) is non-empty, Pic X = (Pic (X ⊗ C)) (it is well-known [Ma] and not difficult to see). Thus, from (0–3), we have (0–4) dim 2Br (X) = dim H et(X ;μ2)− dim (Pic (X ⊗C)) /2(Pic (X ⊗ C)). The dimension of the étale cohomology groupH et(X ;μ2) = H 2 et(X ;F2) is estimated using the Serre-Hochschild spectral sequence where G = Gal(C/R), (0–5) E 2 = H (G;H et(X ⊗ C;F2)) =⇒ H p+q et (X ;F2), where for a complex manifold X ⊗ C we have H et(X ⊗ C;F2) = H (X(C);F2) (see [Mi], for example). In the §2, we prove the following results which permit to calculate the dimension of the étale cohomology groups with coefficients F2 and the 2-torsion of the Brauer group for surfaces satisfying to the condition of Theorem 0.1. These results show that the class of real smooth projective surfaces X satisfying to the condition of Theorem 0.1 is very nice (easy to work with). Theorem 0.3. Let X/R be a real smooth projective algebraic surface such that X(R) 6= ∅ and H(X(C)/G;F2) = 0. Then the Serre-Hochschild spectral sequence (0–5) degenerates and dim H et(X ;F2) = 1; dim H et(X ;F2) = dim H (X(C);F2) + 1; dim H et(X ;F2) = dim H (X(C);F2) G + dim H(X(C);F2) + 1; dim H et(X ;F2) = 2 dim H (X(C);F2) G − dim H(X(C);F2) + 2 dim H(X(C);F2) + 1 dim H et(X ;F2) = 2 dim H (X(C);F2) G − dim H(X(C);F2) + 2 dim H(X(C);F2) + 2 for k ≥ 4. Using Theorem 0.3 and (0–4), (0–5), we get ON THE BRAUER GROUP OF REAL ALGEBRAIC SURFACES 5 Theorem 0.4. Let X/R be a real smooth projective algebraic surface such that X(R) 6= ∅ and H(X(C)/G;F2) = 0. Then dim 2Br (X) = 2s− 1 + h(X(C)) + h − (X(C))− ρ+(X ⊗C). Here h − (X(C)) = dim H 1,1 − (X(C)) where H − (X(C)) = {x ∈ H (X(C)) | g(x) = −x} is the set of potentially real algebraic cycles. And ρ+(X ⊗ C) = dim (Pic (X ⊗ C)⊗C). The characteristic class map gives an injection of (Pic (X ⊗C)⊗C) to H − (X(C)). For an Enriques surface, h(Y (C)) = 0 and all cycles are algebraic. For a real Enriques surface Y , we have seen above that the condition H(Y (C)/G;F2) = 0 is equivalent to the condition of Corollary 0.2. Thus, we get Corollary 0.5. Let Y be a real Enriques surface with the antiholomorphic involution θ and the real part Y (R) 6= ∅. Suppose that the real parts Xσ(R) and Xτσ(R) of both liftings σ and τσ of θ to the universal covering K3-surface X(C) are not empty. Then the Serre-Hochschild spectral sequence (0–5) degenerates and dim 2Br (Y ) = 2s− 1 where s is the number of real connected components of Y (R). We mention that for a real rational surface Z with a non-empty set of real points Z(R) the same results were known: The map (0–1) is epimorphic and dim 2Br (Z) = 2s − 1. It is also known that the Witt group W (Z) ∼= (Z) ⊕ (Z/2). See [Su]. Perhaps,the last result about the Witt group also valid for real Enriques surfaces with non-empty sets Xσ(R) and Xτσ(R). Using results of [N-S], we may prove some additional results about Brauer groups of real Enriques surfaces which also valid if one of the sets Xσ(R), Xτσ(R) is empty. In [N-S], the important invariants b(Y ) and ǫ(Y ) of a real Enriques surface Y with an antiholomorphic involution θ were introduced. Here b(Y ) = dim H(Y (C);F2) θ − dim (Pic Y ⊗ C)/2(Pic Y ⊗ C) + 1. The invariant ǫ(Y ) = 1 if the differential d 2 of the Hochschild-Serre spectral sequence (0–5) is zero, and ǫ(Y ) = 0 otherwise. We have the following results from [N-S] about these invariants: (0–6) dim 2Br (Y ) = b(Y ) + ǫ(Y ) if Y (R) 6= ∅, (0–7) b(Y ) ≥ 2s− 2 for any Y . Thus, by (0–6) and (0–7), for any real Enriques surface Y , (0–8) dim 2Br (Y ) ≥ 2s− 2 + ǫ(Y ). Additionally to Corollary 0.5 and (0–6)—(0–8), we prove 6 VIACHESLAV V. NIKULIN Theorem 0.6. Let Y be a real Enriques surface. Then: (i) The inequality (0–7) is an equality, i.e. b(Y ) = 2s−2, iff the Hochschild–Serre spectral sequence (0–5) degenerates. In particular, by Corollary 0.5, b(Y ) = 2s− 2 if Xσ(R) 6= ∅ and Xτσ(R) 6= ∅. (ii) dim 2Br (Y ) = 2s − 1 if the Hochschild–Serre spectral sequence (0–5) degenerates and Y (R) 6= ∅. In particular, it is true if Xσ(R) 6= ∅ and Xτσ(R) 6= ∅. (iii) dim 2Br (Y ) ≥ 2s− 1. In §3, we give an application of results of [N-S] and Corollary 0.5 and Theorem 0.6 to a topological studying of real Enriques surfaces Y . Let Y be a real Enriques surface with the antiholomorphic involution θ and Y (R) 6= ∅. Let snor be the number of non-orientable connected components of Y (R). We denote by Γ = {id, τ, σ, τσ} ∼= (Z/2) the group acting on the K3-surface X(C) (we use notation above). Let us suppose that the both real parts Xσ(R), Xτσ(R) are non-empty. Then we give a formula connecting the number snor with some invariants of the action of the group Γ on the lattice H(X(C);Z) with the intersection pairing. We mention that it is not clear that one can express the number snor using the action of Γ on the lattice H(X(C);Z). This formula is very important for the topological classification of real Enriques surfaces (see [N4]). See §3 for details. I am grateful to O. Gabber for assuring me that the statement of Proposition 1.1 below should be true. I am grateful to R. Sujatha for very useful discussions, in particular, for pointing out me on the results of E. Witt from [W]. A preliminary variant of this paper [N3] was written during my stay in the University of Notre Dame (USA) at 1991—1992. I am grateful to the University of Notre Dame for hospitality. §1. The proof of the Theorem 0.1. We recall that if X is a topological space with an action of a group G and A is a G-sheaf of groups on X , then the group H(X ;G,A) of equivariant cohomology (or Galois cohomology) is defined. See A. Grothendieck [Gr1, Ch. 5]. It is the right derived functor RΓ to the functor A 7→ Γ(X ;A) of G-invariant sections. This composition of functors A 7→ Γ(X ;A) and M 7→ M for a G-module M defines two spectral functors which tend to this cohomology: I 2 = H (X/G;H(G;A)) =⇒ H(X ;G,A) where H(G;A) = Rf ∗ A is the sheaf corresponding to the presheaf on X/G: (1–1) U 7−→ H(π(U);G,A). Here π : X → X/G is the quotient map. And (1–2) II 2 = H (G;H(X ;A)) =⇒ H(X ;G,A). The following statement is fundamental for us. This is analogous to the wellknown connection between étale and ordinary cohomology for a complex algebraic manifold Z and a finite abelian group B (see [Mi, Ch. III, §3]): The morphism of the ordinary site (Euclidean topology) to the étale site gives rise an isomorphism (1–3) H et(Z;B) ∼= H(Z(C);B). ON THE BRAUER GROUP OF REAL ALGEBRAIC SURFACES 7 Proposition 1.1. Let X/R be a real algebraic manifold and G = Gal(C/R). Let A be a sheaf in étale topology of X such that this sheaf is a constant G-sheaf A on X ⊗ C, where A is a finite abelian group with an action of the group G. Then there exists the canonical isomorphism H et(X ;A) ∼= H(X(C);G,A) together with the canonical isomorphism of the Hochschild-Serre spectral sequence and the Grothendieck spectral sequence II, which is defined by the canonical isomorphism E 2 = H (G;H et(X ⊗C;A)) ∼= II p,q 2 = H (G;H(X(C);A)) induced by the canonical isomorphism (1–3). Proof. We follow to the proof of the isomorphism (1–3) for complex algebraic manifolds. See [Mi,Ch. III, §3], for example. Let X(C)cx be a small site X(C) an E of morphisms of complex analytic spaces over X(C), which are local isomorphisms. (We use notation of [Mi].) An open subset U ⊂ X(C) is a local isomorphism. It follows that we have the morphism of sites X(C)cx → X(C). Every covering of X(C) in the site X(C)cx has a refinement covering of X(C) in the site X(C). It follows that the morphism X(C)cx → X(C) gives an isomorphism of cohomology (1–4) H(X(C);A) ∼= H(X(C)cx;A). Like above, we can define G-equivariant cohomology H(X(C)cx;G,F) for the site X(C)cx as the right derived functor to the composition of functors F → Γ(X(C)cx;F) G for a G-sheaf F on the site X(C)cx. This equivariant cohomology also has a spectral sequence II(X(C)cx) with the beginning II 2 (X(C)cx) = H (G;H(X(C)cx;F)) =⇒ H (X(C)cx;G,F). One can calculate equivariant cohomology using invariant coverings (see [Gr1, Ch.5]). Any G-invariant covering of X(C) in siteX(C)cx also contains a refinement G-invariant covering of X(C) in the site X(C). It follows that we also have the canonical isomorphism of equivariant cohomology: (1–5) H(X(C)cx;G,A) ∼= H (X(C);G,A), together with the isomorphism of the corresponding spectral sequences II of these cohomology (1–6) II 2 (X(C)cx) = H (G;H(X(C)cx;A)) ∼= II p,q 2 (X(C)) = H (G;H(X(C);A)). defined by the isomorphism (1–4). Further, we use standard results about étale cohomology (see [Mi]). Since X ⊗ C → X is an étale covering with Galois group G, it follows that a sheaf F on the site Xet corresponds to a G-sheaf F on the site (X ⊗ C)et. Étale cohomology is a right derived functor to the composition of functors F → (G−mod Γ(X ⊗C;F )) and M → M. 8 VIACHESLAV V. NIKULIN Here M is a G-module. The Hochschild-Serre spectral sequence E 2 = H (G;H et(X ⊗C;F )) =⇒ H p+q et (X ;F ) corresponds to the composition of these functors. Every étale morphism Y → X ⊗ C gives a morphism Y (C) → X(C) in the site X(C)cx. This defines the morphism of sites X(C)cx → Xet. From the remarks above, this morphism defines the homomorphism of cohomology (1–7) H et(X ;A) → H (X(C)cx;G,A) = H (X(C);G,A) together with the homomorphism of spectral sequences (1–8) E r → II p,q r defined by the homomorphism E 2 = H (G;H et(X ⊗C;A)) → II p,q 2 = H (G;H(X(C)cx;A)) = H(G;H(X(C);A)). (1–9) The last homomorphism is defined by the homomorphism (1–10) H et(X ⊗ C;A) → H (X(C)cx;A) = H (X(C);A), which is the isomorphism (1–3). It follows that (1–9), (1–8) and (1–7) are isomorphisms too. This finishes the proof. We mention that D. A. Cox [C] had shown that étale homotopy type of a real algebraic manifold is defined by Euclidean topology. We recall that for a compact manifold M and a constant sheaf of modules, one can calculate sheaf cohomology using simplicial triangulation of the manifold M . One can prove this using the canonical isomorphisms between sheaf, Čhech, Alexander–Spanier, singular and simplicial triangulation cohomology for compact manifolds. See [Gr1] and [Sp], for example. Similarly, for a compact manifold M with an action of the group G and an Abelian G-group A, one can prove that equivariant sheaf, Čhech, Alexander–Spanier, singular and simplicial triangulation cohomology are isomorphic. Thus, for this case, we can calculate equivariant cohomology using the following elementary procedure (this definition is used in the book [Bro], for example): We consider some G-equivariant simplicial triangulation K of M . Thus, K is a G-equivariant simplicial complex. We may suppose that K/G is a simplicial triangulation ofM/G and the fixed partK is a simplicial triangulation ofM. We consider the corresponding chain complex Cn = Cn(K;Z) and the corresponding cochain complex C(K;A), where C(K;A) = Hom(Cn, A). We organize a cochain complex for calculation of group cohomology of C = C(K,A). For the group G = {1, g} of order two this is a complex (1–11) 0 → C 1−g −−→ C 1+g −−→ C 1−g −−→ C 1+g −−→ C 1−g −−→ · · · ON THE BRAUER GROUP OF REAL ALGEBRAIC SURFACES 9 Thus, we get a double complex (1–12) .. .. .. .. .. .. · · · d x