Integral Deligne Cohomology for Real Varieties

INTEGRAL DELIGNE COHOMOLO GY F OR REAL V ARIETIES PEDRO F. D OS S ANTOS A N D P AULO LIMA-FILH O Abstra ct. W e develop an integr a l version of Deligne cohomology for smooth prop er real v arieties Y . F or this purp ose the role pla yed b y singular cohomology in the complex case has to be replaced b y or di- nary bigr ade d S -e quivariant c ohomol o gy , where S := Gal ( C / R ). This is th e S -equiv ariant counterpart of singular cohomolog y; cf. [LMM81]. W e establish the basic prop erties of th e theory and give a geometric interpretatio n for the groups in dimension 2 in wei ghts 1 and 2. Contents 1. In tro duction 2 2. Ordinary equiv ariant cohomology and sh ea ves 7 2.1. Ordin ary equiv ariant cohomology 7 2.2. Int egration and change of co eﬃcien t functors 10 3. Deligne cohomology for rea l v ariet ies 16 3.1. Pro du ct structure I: p ositiv e w eights 19 3.2. Pro du ct structure I I: arb itrary we igh ts 20 4. The exp on ential sequence 21 4.1. An application 24 5. The group H 2 D / R ( X ; Z (2)) 25 5.1. V ariations on the theme of H 2 , 2 Br ( Y , Z ) 25 5.2. The Deligne group H 2 D / R ( X ; Z (2)) 27 6. A remark on num b er ﬁelds 29 App end ix A. The Borel/Esnault-Vieh weg version 30 App end ix B. Proof of Th eorem 5.10 31 B.1. Co cycles for Bredon cohomology 32 B.2. The pro of 37 References 43 Date : May 2008. The ﬁrst author was supp orted in part by FC T (Po rtugal) through program POCTI. 1 2 DOS SANTOS AND LIMA-FILHO 1. Introduction In tegral Deligne cohomology H n D / C ( Y ; Z ( p )) f or a complex v ariet y Y has b een wid ely stud ied in the literature. When Y is smo oth and p rop er, it is deﬁned as the hypercohomology group H n ( Y , Z ( p ) D / C ) of the complex (1) Z ( p ) D / C : 0 → Z ( p ) → O d − → Ω 1 d − →→ · · · → Ω p − 1 , where Z ( p ) is the constant sheaf (2 π i ) p Z and Ω j is the sheaf of h olomorph ic j -forms. More generally , one can extend this setting to arbitrary complex v arieties using suitable compactiﬁcati ons and s implicial resolutions, alo ng with forms with logarithmic p oles, and the resulting theory is called Deligne- Be ˘ ılinson cohomology; see [Be ˘ ı84]. An excellen t acco unt can b e found in [EV88]. The theory is complemen ted by a homological coun terpart and together they are sho w n to satisfy Bloc h-Ogus’ f ormalism [BO74]; cf. [Gil84] and [Jan88]. In this p ap er we d ev elop an inte gral v ers ion of Deligne cohomology f or smo oth p r op er real v arieties. F or this purp ose the role pla y ed by singular co - homology H n sing ( Y , Z ( p )) in the complex case has to b e replaced by or dinary bigr ade d S -e qu ivariant c ohomolo gy H n,p Br ( Y ( C ) , Z ), where S := Gal ( C / R ). This is the S -equiv arian t coun terpart of s ingular cohomology; cf. [LMM81]. W e must emphasize that the ord inary equiv ariant cohomology of a S - space X is not obtained as the singular cohomol ogy of the Borel construction E S × S X . In fact, the “Borel v ersion” of equiv arian t cohomology is jus t H n, 0 Br ( E S × X , Z ) and , more generally , H n,p Br ( E S × X , Z ) ∼ = H n ( E S × S X ; Z ( p )), where the latter denotes cohomology with t w isted co eﬃcients Z ( p ). The bigraded version stems from the RO ( S )-graded equiv ariant cohomol ogy theories d ev elop ed b y J . P eter Ma y et al. in [LMM81 ], [LMSM86] and [Ma y96 ]. With w eigh t p = 0 ordinary equiv ariant cohomology was dev elop ed in [Bre67] and h ence, for simplicit y , we call it Br e don c ohomolo gy eve n with non-zero weig h ts, th us explainin g the notation H n,p Br ( X, Z ). F rom a motivic standp oin t, this diﬀerence can b e expressed b y sa ying that ordinary equ iv arian t cohomolo gy is to its Borel v ersion as motivi c cohomol- ogy is to its ´ etale coun terpart ( ´ et ale-motivic cohomology; cf. [MVW06]). In fact, this setting is muc h more that a mere analogy , once one observes that the top ological realization of the A 1 -homotop y catego ry of sc hemes o v er R lands in the S -equiv arian t h omotop y category . This realization carries mo- tivic cohomology to ordinary bigraded equiv arian t cohomology , and carries ´ eta le-motivic cohomology to Borel equiv arian t cohomolog y (with t wisted co eﬃcien ts). See [MV99 ] and [DI04] for details. Let A n / R denote the category of r e al holomorphic manifolds , whose ob- jects are pairs ( M , σ ) consisting of a h olomorphic manifold M together with an an ti-holomorphic in v olution σ , and whose morphisms are holomorphic maps comm u ting with the in v olutions. W e consider th ose ob jects as ha v- ing an action of S and giv e A n / R the structure of a site using equiv ari- an t op en co v ers. In order to in tro du ce Deligne cohomology for prop er real INTEGRAL DELIGNE COHOMOLOGY FOR REAL V ARIETIES 3 holomorphic manifolds, we constru ct a real v ersion Z ( p ) D / R of the Deligne complex (1) on the site A n / R and deﬁne (2) H n D / R ( M ; Z ( p )) := ˇ H n ( M eq ; Z ( p ) D / R ) . The constru ction of Z ( p ) D / R in v olv es a replacemen t of the constan t s h eaf Z ( p ) b y a co mplex Z ( p ) B r whic h computes ordinary equiv arian t cohomolog y; cf. [Y an08]. The k ey tec hnical construction is a morp hism of complexes τ p : Z ( p ) B r − → E ∗ , where E ∗ is the complex of shea v es on A n / R suc h that E j ( X ) consists of those smo oth complex-v alued diﬀerenti al j -forms in v ariant un der the sim u ltaneous action of S on X ∈ A n / R and C . Usin g this complex we deﬁne Z ( p ) D / R := Cone  Z ( p ) B r ⊕ F p E ∗ τ p − ı − − − − → E ∗  [ − 1] where F p E ∗ is the p -th piece of the Ho d ge ﬁltration. Amongst the basic prop erties of the theory are the eviden t long exact sequences and the fact that it reco v ers the usual theory for complex v arieties. More p recisely , giv en a complex pro j ectiv e v ariet y Y C , let and Y C / R denote the asso ciated r eal v ariet y obtained by restriction of scalars. Prop osition 3.2. L e t Y C b e a pr op er c omplex holomo rphic manifold and let X b e a pr op er smo oth r e al algebr aic variety. i. One has natur al isomorphisms H i D / R ( Y C / R ; A ( p )) ∼ = H i D / C ( Y C ; A ( p )) , wher e the latter deno tes the usual Delig ne c ohomolo gy of the c omplex manifold Y C . (Se e R emark 2.1 .) ii. If X C is the c omplex v ariety obtaine d f r om X by b ase extension, the c orr esp onding map of r e al varieties X C / R → X i nduc es natur al homo- morphism s H i D / R ( X ; A ( p )) → H i D / C ( X C ; A ( p )) S , wher e the latter denotes the invariants of the Deligne c ohomolo g y of the c omplex variety X C . iii. One has a long exact se quenc e (3) · · · → H j − 1 sing ( X ( C ) , C ) S → H j D / R ( X ; A ( p )) ν − → → H j,p Br ( X ( C ) , A ) ⊕ n F p H j sing ( X ( C ); C ) o S  ◦ ϕ − ı − − − − − − → H j sing ( X ( C ); C ) S → · · · Here X ( C ) d enotes the set of complex p oint s of X with the analytic top ology and  F p H j ( X ( C ); C )  S denotes the inv arian ts of the p -th level of the Ho dge ﬁltration on sin gular cohomolog y , un der the simulta neous action of S on X ( C ) and on the co eﬃcien ts C of the cohomology . W e deﬁne Deligne cohomolog y with n egativ e weig ht s p < 0 to coin- cide with ord inary equiv arian t cohomology . Using this con v en tion, we give 4 DOS SANTOS AND LIMA-FILHO 8 0 R / Z (8) Z × 0 Z × 0 7 0 R × 0 Z × 0 Z × 6 0 R / Z (6) Z × 0 Z × 0 5 0 R × 0 Z × 0 Z × 4 0 R / Z (4) Z × 0 Z × 0 3 0 R × 0 Z × 0 0 2 0 R / Z (2) Z × 0 0 0 1 0 R × 0 0 0 0 0 Z (0) 0 0 0 0 0 -1 0 -2 2 Z ( − 2) -3 Z × -4 Z × 2 Z ( − 4) -5 Z × 0 Z × -6 Z × 0 Z × 2 Z ( − 6) -7 Z × 0 Z × 0 Z × -8 Z × 0 Z × 0 Z × 2 Z ( − 8) -5 -4 -3 -2 -1 0 1 2 3 4 5 T able 1. Cohomology of a p oin t ⊕ n,p H n D / R ( X ; Z ( p )) a bigraded ring structure compatible with v arious f unc- tors in the theory and h a ving many computational pr op erties. F or example, the bigraded group stru cture of the cohomology of a p oin t X = S pec ( R ) is displa y ed in T able 1 ab o v e, and its rin g structure is displa y ed in T able 2, subsection 3.2. Using this pro du ct structure w e obtain obtain formulae for the Deligne cohomology ring of some r elev ant examples, such as: Corollary 3.9. Under the hyp othesis of Pr op osition 3.8, one has an iso- morphism of bigr ade d rings H ∗ D / R ( X ; A ( ∗ ))[ T ] / h T p +1 i ∼ = H ∗ D / R ( X × P p ; A ( ∗ )) when T is given the bigr ading (2 , 1) . A m ore general pro jectiv e bund le formula together with a th eory of c harac- teristic classes app ear in a forthcoming p ap er. The cases of w eigh ts p = 1 and p = 2 h a ve in teresting geometric inter- pretations. W e ﬁrst s ho w that Z (1) D / R is quasi-isomorphic to O × [ − 1], cf. Corollary 4.6, and deriv e as a consequence an exp onential se quenc e relating the cohomology of Z (1) B r , O and O × : Corollary 4.7. L et X b e a smo oth pr op e r r e al algebr aic v ariety. Then ther e is a long e xact se quenc e → H ∗ , 1 Br ( X ( C ); Z ) ϑ − → H ∗ ( X ; O X ) exp − − → H ∗ ( X ; O ∗ X ) → H ∗ +1 , 1 Br ( X ( C ); Z ) → INTEGRAL DELIGNE COHOMOLOGY FOR REAL V ARIETIES 5 wher e ϑ denotes the c omp osite H ∗ , 1 Br ( X ( C ); Z ) τ 1 − → H ∗ ( X ( C ); C ) G ։ H ∗ ( X ; O X ) = H ∗ , 1 ∂ ( X ( C )) G , and the latter denotes the i nvariants of the Dolb e ault c ohomolo gy of the c omplex manifold X ( C ) . When X is a curve, the exp onen tial sequence th en giv es: Prop osition 4.10. L et X b e an irr e ducible, smo oth, pr oje ctive curve over R , of ge nus g . Then P ic ( X ) ∼ = P ic 0 ( X C ) S × H 2 , 1 Br ( X ( C ) , Z ) . As a corol lary to this prop osition w e pr ovide a new pro of of Wei c hold’s The or em – a classical r esult in r eal algebraic geometry whic h determines the Picard group of a real algebraic curv e. In order to giv e a geometric in terpretation of H 2 D / R ( X ; Z (2)) for a real pro jectiv e v ariety X we ﬁrst provide in Prop osition 5.3 alternativ e in ter- pretations of the Bredon cohomology group H 2 , 2 Br ( Y , Z ) f or any S -manifold Y . Using Atiy ah’s terminology in [A ti66 ], a R e al ve ctor bund le ( E , τ ) on a S -manifold ( Y , σ ) consists of a complex v ector bund le E on Y together with an isomorphism τ : σ ∗ E → E satisfying τ ◦ σ ∗ τ = I d . No w, consider the set L 2 ( Y ) of equiv alence classes of pairs h L, q i s atisfying: p1) L is a (smo oth) complex line bu ndle on Y ; p2) q : L ⊗ σ ∗ L → 1 Y is an isomorphism of Real line bundles, wh ere L ⊗ σ ∗ L carries the tautologic al Real line b undle structur e. It follo ws that L 2 ( Y ) b ecomes a group under the tensor pro du ct of line bund les and w e sho w that this group is natur ally isomorphic to H 2 , 2 Br ( Y , Z ) . No w, let S = π 0 ( Y S ) d enote the set of connected comp onents of th e ﬁ xed p oint s et Y S , and id en tify H 0 ( Y S ; Z × ) ≡ ( Z × ) S . Giv en h L, q i ∈ H 2 , 2 Br ( Y , Z ), the restriction of q to L | Y S b ecomes a n on-degenerate h ermitian pairing, and hence it h as a well -deﬁned signature ℵ h L, q i ∈ ( Z × ) S . W e call ℵ : H 2 , 2 Br ( Y , Z ) → ( Z × ) S the e quivariant sig natur e map of Y and th e image ℵ tor ( Y ) ⊆ ( Z × ) S of the torsion subgroup H 2 , 2 Br ( Y , Z ) tor under ℵ is called the e qu ivariant signatur e gr oup of Y . In the case where Y = X ( C ) f or a r eal algebraic v ariet y X with S = π 0 ( X ( R )), w e denote th e equiv ariant signature grou p of X ( C ) simply b y ℵ tor ( X ). F or example, if X is a real algebraic curve, then ℵ tor ( X ) is the Brauer group of X ; see Section 4. Giv en a prop er real v ariet y X , let P W ∇ ( X ) denote the set of isomorph ism classes h L, ∇ , q i of triples wh ere L is a holomorphic line bun dle o v er X ( C ), ∇ is a holomorphic connection on L and q : L ⊗ σ ∗ L → 1 is a h olomorphic isomorphism of R eal line b undles satisfying th e follo win g p rop erties: (1) T h e restriction of q to X ( R ) is a p ositiv e-deﬁnite hermitian m etric. (2) As a section of ( L ⊗ σ ∗ L ) ∨ , q is parallel with resp ect to th e connection induced by ∇ . 6 DOS SANTOS AND LIMA-FILHO One s ees th at the tensor pro du ct endo ws P W ∇ ( X ) w ith a group structure whic h make s P W ∇ ( X ) th e k ernel of Ψ: Theorem 5.10. If X is a smo oth r e al pr oje ctive variety then one has a natur al short exact se quenc e 0 → P W ∇ ( X ) → H 2 D / R ( X ; Z (2)) Ψ − → ℵ tor ( X ) → 0 . This pap er is organized as follo ws . S ection 2 cont ains b ac kgroun d infor- mation and the key tec h nical ingredien ts for the paper , includin g Prop osition 2.8 which constructs the map of complexes τ p : A ( p ) B r − → E ∗ . In Section 3 we construct Deligne complexes A ( p ) D / R for any sub ring A ⊂ R and de- ﬁne the corresp onding Deligne cohomolo gy for prop er smo oth real v arieties. In this section we pro v e basic prop erties, introdu ce the pr o duct structure and p ro vide basic examples. In Section 4 we stud y the weigh t p = 1 case, pro ving the quasi-isomorphism Z (1) D / R ≃ O × [ − 1] together with some ap- plications. In S ection 5 w e s tudy the group H 2 D / R ( X ; Z (2)) and asso ciated in terpretations of H 2 , 2 Br ( X ( C ) , Z ). The pro of of the main result, Theorem 5.10, is delegate d to Ap p endix B. S ection 6 cont ains a remark ab out num- b er ﬁ elds, where we giv e a ring homomorphism from the Milnor K -theory of a n u m b er ﬁeld to the “diago nal” sub ring of in tegral Deligne co homology , and w e obser ve that the classical regulator of a num b er ﬁ eld can b e describ ed in terms of the image of the change -of-coeﬃcients homomorphism b et w een Deligne cohomology w ith inte gral and real co eﬃcien ts, resp ectiv ely; simi- lar computations can b e made for arbitrary Artin motiv es. In App endix A w e d escrib e the relationship b et ween Esn ault-Vieh weg’ s “Borel version” of Deligne cohomology for real v arieties and the theory discussed in this pap er. In a forthcoming series of pap ers we ﬁrst extend this theory to an i nte gr al Deligne-Beilinson c ohomolo gy the ory for arbitrary r eal v arieties, and stu dy its relation to a corresp ond ing notion of Mixe d Ho dge Structur es . Then w e p ro vide a n atural and explicit cycle map from motivic cohomology to Deligne cohomology , directly using the approac h in [MVW06]. Th is is an alternativ e d escription, ev en in the complex case, of the map s discus s ed in [Blo86], [KLMS06], [KL07]. The corresp ondin g real inte rmediate Jacobians and their r elations to real algebraic cycles are also u nder study . Ac knowledgemen ts. Th e ﬁr st auth or would like to thank T exas A&M Univ ersit y and the second author w ould lik e to thank the IS T (Instituto Sup erior T ´ ecnico, Lisb on) for their resp ectiv e w arm h ospitalit y dur ing the elab oration of parts of th is wo rk; and the second author w an ts to thank Sp encer Blo c h for inspiring con v ersation and p ointe d questions during a visit to th e Univ ersit y of Chicago. INTEGRAL DELIGNE COHOMOLOGY FOR REAL V ARIETIES 7 2. Ordinar y equiv ariant cohomo logy and s hea ves W e ﬁr st in tro duce the v arious catego ries used throughout this article. L et S := Gal ( C / R ) d enote the Galois group of C o ver R . a) T he category of smo oth manifolds w ith smo oth S -action and equiv ariant smo oth m orphisms is den oted S - M an . Let C ov ( X ) b e the set of co v erings of X ∈ S - M an by op en S -inv ariant subsets. These co v erings give S - M an a site s tr ucture whose r estriction to X is denoted X eq . b) A r e al holomor phic manifold ( M , σ ) is a smo oth complex holomorphic manifold M en d o w ed with an an ti-holomorphic inv olution σ : M → M , an d morphisms b et ween t wo such ob jects are holomorphic equiv arian t maps. These comprise the category A n / R of real holomorphic manifolds, with its eviden t site structure. c) W e d enote by S m / R the category of smo oth real algebraic v arieties. d) T he categories of smo oth holomorphic manifolds and complex algebraic v arieties are d enoted A n / C and S m / C , resp ectiv ely . Remark 2.1. (1) If M is a complex manifold, d enote M / R := M ∐ M , where M denotes M with the opp osite complex structure. The m ap σ : M / R → M / R sending a p oin t in one copy of M to the same p oint in the other cop y of M is an an ti-holomorphic inv olution. The assignmen t M 7→ M / R is a fu nctor A n / C → A n / R . (2) Given a real v ariet y X ∈ S m / R , let X ( C ) denote its set of complex- v alued p oint s with the analytic top ology . Th e natural action of S on X ( C ) which a morphism of sites from S m / R in to A n / R . (3) W e will denote a c omplex algebraic v ariet y alwa ys as X C , and w e use X C / R to denote X C seen as a real v ariet y . It follo ws that the set of complex v alued p oin ts X C / R ( C ) (o ve r R ) coincides with X C ( C ) ∐ X C ( C ). T he constructions ab o v e give a comm uting d iagram of func- tors: S m / C / /   S m / R   A n / C / / A n / R . 2.1. O rdinary equiv ariant cohomology . In [Bre67] Bredon d eﬁnes an equiv arian t cohomology theory H n G ( X ; M ) for G -spaces, where G is a ﬁ- nite group and M is a contra v ariant co eﬃcien t system. When M is a Mac key fun ctor, P . Ma y et al. [LMM81] sh o wed that this theory can b e uniquely extended to an R O ( G )-graded theory { H α G ( X ; M ) , α ∈ RO ( G ) } , called R O ( G ) -gr ade d or dinary e quivariant c ohomolo gy the ory , where RO ( G ) denotes the orthogonal representat ion ring of G . When G = S , one has RO ( S ) = Z · 1 ⊕ Z · ξ , where 1 is the trivial representat ion and ξ is the sign represent ation. In this pap er we u se the motivic notation : (4) H n,p Br ( X, M ) := H ( n − p ) · 1 + p · ξ S ( X ; M ) , 8 DOS SANTOS AND LIMA-FILHO and call H n,p Br ( X, M ) bigr ade d Br e don c ohomo lo gy . In the homotop y theoreti c approac h, one p ro v es the existence of equiv ari- an t Eilen b erg-MacLane spaces K ( M , ( n, p )) that classify Bredon cohomol- ogy (for n ≥ p ≥ 0). In other w ords, H n,p Br ( X, M ) = [ X + , K ( M , ( n , p ))] G , where the latter denotes the s et of based equiv arian t homotop y classes of maps. When n < p one uses th e susp ension axiom to deﬁne the corresp ond- ing cohomology grou p s; see [LMM81]. A qu ic k w a y to construct K ( Z , ( n, p )) is the follo wing. Let S n,p denote the one-p oin t compactiﬁcation { ( n − p ) · 1 ⊕ p · ξ } ∪ {∞} of the ind icated represent ation, and deﬁne Z 0 ( S n,p ) := Z ( S n,p ) / Z ( {∞} ), where Z ( S n,p ) d e- notes the free abelian group on S n,p , su itably top ologize d. Then Z 0 ( S n,p ) is an Eilen b erg-MacLane space K ( Z , ( n, p )); cf. [dS03b]. Example 2.2. In order to d escrib e the bigraded cohomolog y ring of a p oint B := ⊕ p,n H n,p Br ( pt, Z ) , ﬁ rst consider indeterminates ε, ε − 1 , τ , τ − 1 satisfying deg ε = (1 , 1) , deg ε − 1 = ( − 1 , − 1) , deg τ = (0 , 2) and deg τ − 1 = (0 , − 2) . Henceforth, ε and ε − 1 will alwa ys satisfy 2 ε = 0 = 2 ε − 1 . As an ab elian group, B can b e written as a direct sum (5) B := Z [ ε, τ ] · 1 ⊕ Z [ τ − 1 ] · α ⊕ F 2 [ ε − 1 , τ − 1 ] · θ where eac h sum mand is a free b igraded mo du le ov er the corresp onding ring and F 2 is the ﬁeld with tw o elemen ts (hence 2 θ = 0). The resp ectiv e bide- grees of the generators 1, α and θ are (0 , 0), (0 , − 2) and (0 , − 3) . The pro d u ct structure on B is completely d etermined b y the follo wing relations (6) α · τ = 2 , α · θ = α · ε = θ · τ = θ · ε = 0 . Note that B is n ot ﬁn itely generated as a ring, and that B has n o homoge- neous elemen ts in degrees ( p, q ) wh en p · q < 0. W e no w present an alternativ e sheaf-theoretic construction of Bredon co- homology whic h is more suitable for our pur p oses. The details of su c h construction will app ear in [Y an08 ]. Giv en U ∈ S - M an , let b U denote the full sub category of S - M an ↓ U consisting of equiv ariant ﬁnite co vering maps π S : S → U . In p articular, d { pt } is the category S -Fin ⊆ S - M an of ﬁ nite S -sets.. A top ological G -Mod u le M represen ts an ab elian M acke y pr eshe af on S - M an , in other words, the contra v arian t functor M : S - M an op → Ab send- ing U 7→ M ( U ) := Hom S - T op ( U, M ) is also cov arian t for m ap s in b U , for all U ∈ S - M an , and satisﬁes the follo wing p rop erty . Giv en a pu ll-back square Z γ − − − − → X ϕ   y   y f Y − − − − → g U INTEGRAL DELIGNE COHOMOLOGY FOR REAL V ARIETIES 9 with f ∈ b U , the diagram M ( Z ) γ ∗ ← − − − − M ( X ) ϕ ∗   y   y f ∗ M ( Y ) ← − − − − g ∗ M ( U ) comm u tes. The case where M is a subr ing a R w ith trivial S -modu le struc- ture will play a sp ecial role in th is work. Deﬁnition 2.3. Let F b e an ab elian presh eaf on S - M an , and let M b e a top ological S -mo du le. Giv en U ∈ S - M an , let F ⊗ b U M denote the co end: { M π S : S → U F ( S ) ⊗ M ( S ) } / K F ,M ( U ) in Ab, where π S ∈ b U and K F ,M ( U ) is the sub grou p generated by elemen ts of the form: ( φ ∗ F α ′ ) ⊗ m − α ′ ⊗ φ M ∗ m, when S φ − → S ′ π S ց ւ π S ′ U is a morphism in b U , α ′ ∈ F ( S ′ ) and m ∈ M ( S ). It is easy to see th at the assignment U 7− → F ⊗ b U M is a con tra v arian t functor from S - M an to Ab . Denote by F R M the resulting ab elian presh eaf on S - M an , i.e. F R M ( U ) := F ⊗ b U M . Let ∆ n denote the standard top ological n -simplex with the trivial S - action. Using the co -simplicial structure on { ∆ ∗ | n ≥ 0 } one creat es a simplicial ab elian presh eaf C • ( F ) asso ciated to an y presh eaf F , whose n -th term is C n ( F ) : U 7− → F ( ∆ n × U ) . Denote the asso ciated complex of shea v es by ( C ∗ ( F ) , d ∗ ) and use the con v en- tion in [SGA72, XVI I 1.1.5] to deﬁn e a co chain complex ( C ∗ ( F ) , d ∗ ) w here C n ( F ) := C − n ( F ) and d n : C n ( F ) → C n +1 ( F ) is deﬁned by d n = ( − 1) n d − n . A S -manifold X deﬁnes an ab elian p resheaf on S - M an Z X : U 7− → Z Hom S - M an ( U, X ) , sending U to the fr ee ab elian group on the set of s mo oth equiv arian t maps from U to X . Y oneda Lemma identiﬁes Hom AbP r eS h ( Z X, F ) = F ( X ) for an y F . In particular, if F is any ab elian presheaf and X ∈ S - M an , the pr esheaf H om ( Z X, F ) sends U to F ( X × U ) . Prop osition 2.4. i. The assignment F 7→ C ∗ ( F ; M ) is c ovariant on F . ii. L et I = [0 , 1] denote the unit interval with the trivial S -action. F or any ab elian pr eshe af F let i ∗ 0 , i ∗ 1 : H om ( Z I , C ∗ ( F )) → C ∗ ( F ) , b e the map of c omplexes induc e d by ev aluation at the end-p oints . Then ther e is a 10 DOS SANTOS AND LIMA-FILHO homoto py h F b etwe en i ∗ 0 and i ∗ 1 which is natur al on F . In p articular, the c omplexes C ∗ ( F ) have h omotop y-inv ariant c ohomolo g y pr eshe aves. In wh at f ollo ws denote ( C × ) p − 1 i := C × × · · · × 1 × · · · × C × ⊂ C × p , where 1 app ears in the i -th co ordinate. Deﬁnition 2.5. Giv en a S -manifold X , let J X,p : p M i =1 C ∗ ( Z (( C × ) p − 1 i × X )) − → C ∗ ( Z ( C × p × X )) b e the m ap induced by the inclusions and denote (7) C ∗ ( Z 0 ( S p,p ∧ X + )) := Cone( J X,p ) . W rite C ∗ ( Z 0 ( S p,p )) when X = ∅ is the emp t y manifold. Let A ⊂ R b e a subrin g endow ed with the discrete top ology . The p -th Br e don c omplex with c o eﬃcients i n A is th e complex of pr esh ea ves (8) A ( p ) B r := C ∗ ( Z 0 ( S p,p )) Z A [ − p ] , where the co end is taken lev elwise. The follo wing result is prov en in [Y an08] Theorem 2.6. L et X b e a S -manifold and let A ⊂ R b e a subring, endowe d with the discr ete top olo gy. Th en for al l p ≥ 0 and n ∈ Z ther e is a natur al isomorph ism ˇ H n ( X eq ; A ( p ) B r ) ∼ = H n,p Br ( X, A ) b etwe en the ˇ Ce ch hyp er c oho- molo gy of X e q with values in A ( p ) B r and the e quivariant c ohom olo gy gr oup H n,p Br ( X, A ) . F or all p, n ∈ Z there is a for getful fu nctor (9) ϕ : H n,p Br ( X, A ) − → H n sing ( X ; A ( p )) S , where A ( p ) is the S -submo dule A ( p ) := (2 π i ) p A ⊂ C and the in v arian ts H n sing ( X ; A ( p )) S of the sin gular cohomology of X with co eﬃcien ts in A ( p ) are tak en under the simultaneous action of S on X and A ( p ) . In the follo wing section w e p r esen t a realization of the comp osition (10) H n,p Br ( X, A ) → H n sing ( X ; A ( p )) S → H n sing ( X ; C ) S , where the latter is the c hange of co eﬃcien ts m ap. 2.2. I n tegration and c hange of coeﬃcient functors. Let A p denote the sheaf of smooth, complex v alued d iﬀeren tial p -forms on S -manifolds. Given X ∈ S - M an , denote E p ( X ) = { θ ∈ A p ( X ) | σ ∗ ( θ ) = θ } . In other w ords, E p is the su bsheaf of A p consisting of those p -form s in v arian t under the simultaneous action of S on X and C . INTEGRAL DELIGNE COHOMOLOGY FOR REAL V ARIETIES 11 Let π : E → B b e a lo cally trivial bu ndle, where B b e a smo oth m anifold, E is an orien ted manifold-with-corners and the ﬁb er is an orien table n - dimensional manifold-with-b oundary F (and with corners). L et (11) π ! : A p + n ( E ) → A p ( B ) . denote the i nte gr ation along the ﬁb er homomorph ism. If π is a map in S - M an then π ! preserve s in v arian ts, thus s ending E p + n ( E ) to E p ( B ) . The follo wing prop erties are well-kno wn. Prop erties 2.7. Let ω b e a ( n + p )-form an d let π ′ : ∂ E → X d enote the restriction of π to the b oundary of E . i. (Pro jection f orm ula) π ! ( π ∗ θ ∧ ω ) = θ ∧ π ! ( ω ) ii. (Boundary formula) dπ ! ( ω ) = π ! ( dω ) + ( − 1) p π ′ ! ( ω | ∂ E ) iii. (Pull-bac k formula) Giv en a p ull-bac k square E ′ f ′ / / π ′   E π   X ′ f / / X , then f ∗ ◦ π ! = π ′ ! ◦ f ′∗ . iv. (F unctorial it y) If X f − → Y g − → Z are smo oth ﬁbrations w ith compact ﬁb ers , then ( g ◦ f ) ! = g ! ◦ f ! . v. (Prod uct formula) Let E ′′ ρ / / ρ ′   π ′′ B B B B E π   E ′ π ′ / / X , b e a pull-bac k square where b oth π and π ′ are ﬁbrations with ﬁb er di- mensions n and n ′ , resp ectiv ely . Giv en ω ∈ A p + n ( E ) and ω ′ ∈ A q + n ′ ( E ′ ) one has π ′′ ! ( ρ ∗ 1 ω ∧ ρ ∗ 2 ω ′ ) = ( − 1) nq π ! ( ω ) ∧ π ′ ! ( ω ′ ) . In tegration along the ﬁb er can b e used to construct maps of complexes (12) τ p : A ( p ) B r → E ∗ , as follo ws. F irst consider U ∈ S - M an and 0 ≤ j ≤ p . An elemen t in A ( p ) j B r ( U ) is r epresen ted by su m s of pairs of the f orm α ⊗ m , where α = ( a, f ) with a, f and m equiv ariant maps satisfying 1. a : ∆ p − j − 1 × S → ( C × ) p − 1 i ⊂ C × p is smo oth and π : S → U is a map in b U ; 2. f : ∆ p − j × S → ( C × ) p is a smo oth map; 3. m : S → A ∈ A ( S ) is a lo cally constant. 12 DOS SANTOS AND LIMA-FILHO In the d iagram C × p U ∆ p − j × U p 1 o o ∆ p − j × S 1 × π o o ˆ π i i f 5 5 l l l l l l l l l l l l l l l p 2 $ $ I I I I I I I I I I S m / / A , ˆ π is a lo cally trivial ﬁbration with ﬁ b er dimension p − j , and we consider the lo cally constan t map m ◦ p 2 = p ∗ 2 m as an elemen t in E 0 (∆ p − j × S ). Denote (13) ω p := dt 1 t 1 ∧ · · · ∧ dt p t p ∈ E p ( { C × } p ) and deﬁne (14) τ j ( α ⊗ m ) = ˆ π ! { p ∗ 2 m · f ∗ ω p } ∈ E j ( U ) where ˆ π ! : E p (∆ p − j × S ) − → E j ( U ) is the in tegration along the ﬁb er homo- morphism. Th is can b e extend ed to a homomorphism τ j : M S ∈ b U  C p − j ( Z 0 ( S p,p ))( S ) ⊗ A ( S )  − → E j ( U ) . Prop osition 2.8. F or e ach 0 ≤ j ≤ p , the map τ j ab ove factors thr ough C p − j ( Z 0 ( S p,p )) ⊗ b U A . F urthermor e, these maps induc e a morphism of c om- plexes of pr eshe aves (15) τ p : A ( p ) B r − → E ∗ . Pr o of. Giv en a morphism S φ / / π   ? ? ? ? ? ? ? S ′ π ′ ~ ~ ~ ~ ~ ~ ~ ~ ~ U in b U , consider the asso ciated diagram (16) ∆ p − j × S 1 × φ / / ˆ π + + p ( ( P P P P P P P P P P P P P P ∆ p − j × S ′ p ′ v v m m m m m m m m m m m m m m ˆ π ′ s s f          S m v v n n n n n n n n n π   = = = = = = = = φ / / S ′ π ′           A U { C × } p . Pic k α = ( a, f ) ∈ Cone( J p ) p − j ( S ′ ) and m ∈ A ( S ); cf. Deﬁn ition 2.5. By deﬁnition, τ j ( φ ∗ α ⊗ m ) = ˆ π ! ( p ∗ m · (1 × φ ) ∗ f ∗ ω p ) . INTEGRAL DELIGNE COHOMOLOGY FOR REAL V ARIETIES 13 It follo ws from Pr op erties i)–iv) of integrat ion along the ﬁ b ers that for all θ ∈ E ∗ (∆ p − j × S ′ ) one has ˆ π ! { p ∗ m · (1 × φ ) ∗ θ } = ˆ π ′ ! ( p ′∗ ( φ ∗ m ) · θ ), sin ce the top square in (16) is a pull-bac k diagram. In particular, for θ = f ∗ ω p one obtains τ j ( α ⊗ φ ∗ m ) = ˆ π ′ ! ( p ′∗ ( φ ∗ m ) · f ∗ ω p ) = τ j ( φ ∗ α ⊗ m ) . This prov es the ﬁrs t assertion in the prop osition. T o pro v e the s econd assertion, let ∂ i : ∆ p − j − 1 ֒ → ∆ p − j b e th e inclus ion of the i -th face, and pic k α ⊗ m ∈ Cone( J p ) p − j ( S ) ⊗ A ( S ), as b efore, repre- sen ting an elemen t in A ( p ) j B r ( U ). Then, using th e sign conv en tion relating the diﬀeren tial D of A ( p ) ∗ B r with the diﬀeren tial d B of C ∗ ( Z ( C × p )) one gets (17) τ j +1 ( D ( α ⊗ m )) = τ j +1  ( − d ∗ p a, a + d ∗ p f ) ⊗ m  = ˆ π ! ( p ∗ m · a ∗ ω p ) + ( − 1) j n X i =1 ( − 1) i ˆ π ! ( ˜ p ∗ m · ∂ ∗ i f ∗ ω p ) , where ˜ p : ∆ p − j − 1 × S → S is the pro jection. Sin ce a : ∆ p − j − 1 × S → C × p factors thr ough some ( C × ) p − 1 i ⊂ C × p , one has a ∗ ω p = 0 . On the other hand, (18) n X i =1 ( − 1) i ˆ π ! ( ˜ p ∗ m · ∂ ∗ i f ∗ ω p ) = n X i =1 ( − 1) i ˆ π ! (( ∂ i × 1) ∗ { p ∗ m · f ∗ ω p } ) = ˆ π ! ( { p ∗ m · f ∗ ω p } | ∂ (∆ p − j × S ) ) = ( − 1) j d ˆ π ! ( p ∗ m · f ∗ ω p ); cf. the b oundary formula (2.7)ii. It follo w s from (17) and (18) th at τ j +1 ◦ D = d ◦ τ j , for 1 ≤ j ≤ p. The t w o remaining cases are: (19) 0 / / E p +1 ( U ) A ( p ) p B r ( U ) O O τ p / / E p ( U ) d O O and (20) A ( p ) 0 B r ( U ) τ 0 / / E 0 ( U ) A ( p ) − 1 B r ( U ) d − 1 O O / / 0 O O . Giv en f : S → C × p , m : S → A with S ∈ b U , then the b oundary form ula for integrat ion along the ﬁ b ers giv es d π ! ( m · f ∗ ω p ) = 0, thus sh o wing that (19) comm utes. 14 DOS SANTOS AND LIMA-FILHO T o show that (20) comm utes, pic k f : S × ∆ p +1 − → C × p , m : S → A . Then: τ 0 ( D [ f ⊗ m ]) = τ 0 ( p +1 X k =0 ( − 1) k f ◦ (1 × ∂ k ) ⊗ m ) = p +1 X k =0 ( − 1) k ˆ π ! ( m · (1 × ∂ k ) ∗ f ∗ ω p ) = p +1 X k =0 ( − 1) k ˆ π ! ((1 × ∂ k ) ∗ { m · f ∗ ω p } ) = d ˆ π ! ( m · f ∗ ω p ) ± ˆ π ! ( d { m · f ∗ ω p )) = 0 . The functorialit y of the maps τ j with resp ect to U should b e evident.  Remark 2.9. Recall that if U is a S -manifold, one obtains a natural S - isomorphism U triv × S ∼ = − → U × S b y sendin g ( x, g ) to ( g x, g ). On the other hand , E p ( U triv × S ) is the subgroup of A p ( U triv × S ) ∼ = A p ( U triv ) × A p ( U triv ) inv ariant un der the in volution that sends ( ω 1 , ω σ ) to ( ω σ , ω 1 ) . This observ ation s h o ws that we ha v e a fun ctor F : E p ( U × S ) 7− → A p ( X ) ω 7− → ω | U × 1 satisfying the follo wing prop erties: (1) F comm utes with d iﬀeren tials; (2) If ı : U × S → U × S is the S -homeomorphism sending ( x, α ) to ( x, σ α ), then F ( ı ∗ ω ) = σ ∗ F ( ω ); (3) F ◦ p ∗ = I d , where p : U × S → U is the pro jection. T o d escrib e the pro duct structure on A ( ∗ ) B r , let Γ n,m := { σ : ∆ n + m → ∆ n × ∆ m | n ≥ 0 , m ≥ 0 } b e the triangulation of ∆ n × ∆ m inducing the Alexander-Whitney diagonal appro ximation; cf. [ES52, p. 68]. Giv en p, q ≥ 0, the (external) pairing of preshea v es Z ( C × p ) ⊗ Z ( C × q ) → Z ( C × p + q ) yields a p airing of complexes (21) C ∗ ( Z ( C × p )) ⊗ C ∗ ( Z ( C × q )) → C ∗ ( Z ( C × p + q )) in the usual manner. Denoting C ∗ ( Z ( d C × r )) := ⊕ r i =1 C ∗ ( Z (( C × ) r − 1 i )) , one sees that this p airing sends b oth C ∗ ( Z ( d C × p )) ⊗ C ∗ ( Z ( C × q )) and C ∗ ( Z ( C × p )) ⊗ C ∗ ( Z ( d C × q )) to C ∗ ( Z ( \ C × p + q )), and h ence it ind u ces a pairin g of complexes (22) C ∗ ( Z 0 ( S p,p )) ⊗ C ∗ ( Z 0 ( S q ,q )) → C ∗ ( Z 0 ( S p + q ,p + q )) . Finally , the multiplica tion A ⊗ A → A together w ith the appropriate sign con ven tions yields a p airing of complexes µ : A ( p ) B r ⊗ A ( q ) B r → A ( p + q ) B r . See the p ro of of Th eorem 2.10 b elo w. INTEGRAL DELIGNE COHOMOLOGY FOR REAL V ARIETIES 15 Theorem 2.10. The maps of c omplexes τ ar e c omp atible with multiplic a- tion. In other wor ds, for ev e ry p, q ≥ 0 and i, j ∈ Z one has a c ommutative diagr am of pr eshe aves: A ( p ) i B r ⊗ A ( q ) j B r µ / / τ p ⊗ τ q   A ( p + q ) i + j B r τ p + q   E i ⊗ E j ∧ / / E i + j Pr o of. Giv en U ∈ S - M an, elemen ts in A ( p ) i B r ( U ) and A ( q ) j B r ( U ) are rep - resen ted resp ectiv ely by α ⊗ m = ( a, f ) ⊗ m and β ⊗ n = ( b, g ) ⊗ n , wher e (1) π S : S → U and π T : T → U are in b U and π : S × U T → U denotes their ﬁ b ered pro duct; (2) a : ∆ p − i − 1 × S → C × r p − 1 , f : ∆ p − i × S → C × p smo oth equiv ariant , and m : S → A lo cally constan t; (3) b : ∆ q − j − 1 × T → C × s q − 1 , g : ∆ q − j × T → C × q smo oth and equiv ari- an t, and n : T → A lo cally constan t. The relev an t maps in the constructions that follo w are summarized in the diagram ∆ p + q − i − j × S × U T σ × 1   ED ˆ σ   ∆ p − i × S p S / / ˆ π S + + W W W W W W W W W W W W W W W W W W W W W W W W W W S m / / A ∆ p − i × ∆ q − j × S × U T i   ρ S 5 5 k k k k k k k k k k k k k k ρ T ) ) S S S S S S S S S S S S S S S 5 5 k k k k k k k k k k k k k k p / / _ _ _ _ _ _ _ _ _ _ _ S × U T π / / U (∆ p − i × S ) × (∆ q − j × T ) f × g   ∆ q − j × T ˆ π T 3 3 g g g g g g g g g g g g g g g g g g g g g g g g g g p T / / T n / / A C × p × C × q ≡ C × p + q where σ : ∆ p + q − i − j → ∆ p − i × ∆ q − j is in Γ p − i,q − j . By deﬁnition, µ ( α ⊗ m, β ⊗ n ) = ( a ∪ U g + f ∪ U b, f ∪ U g ) ⊗ m ⋆ n, where f ∪ U g := ( − 1) j ( i + p ) P σ ∈ Γ p − i,q − j ( − 1) | σ | ( f × g ) ◦ i ◦ ( σ × 1) , and m ⋆ n : S × U T → A is d eﬁned as m ⋆ n ( s, t ) = m ( s ) n ( t ) ∈ A. The elemen ts a ∪ U g an d f ∪ U b are d eﬁned similarly . 16 DOS SANTOS AND LIMA-FILHO Hence (23) ( − 1) j ( i + p ) τ p + q ( µ ( α ⊗ m, β ⊗ n )) = = X σ ( − 1) | σ | ˆ σ ! ( { ( σ × 1) ◦ p } ∗ ( m ⋆ n ) · { ( f × g ) ◦ i ◦ ( σ × 1) } ∗ ω p + q ) = X σ ( − 1) | σ | ˆ σ ! (( σ × 1) ∗ { p ∗ ( m ⋆ n ) · i ∗ ◦ ( f × g ) ∗ ω p + q } ) = X σ ( − 1) | σ | ˆ σ ! (( σ × 1) ∗ { p ∗ ( m ⋆ n ) · i ∗ ( f ∗ ω p × g ∗ ω q ) } ) . The collect ion Γ p − i,q − j giv es a tr iangulation of ∆ p − i × ∆ q − j and hence (24) X σ ( − 1) | σ | ˆ σ ! (( σ × 1) ∗ { p ∗ ( m ⋆ n ) · i ∗ ( f ∗ ω p × g ∗ ω q ) } ) = ( π ◦ p ) ! ( p ∗ ( m ⋆ n ) · i ∗ ( f ∗ ω p × g ∗ ω q )) = ( π ◦ p ) ! ( ρ ∗ S a S ∧ ρ ∗ T a T ) , where a S := p ∗ S m · f ∗ ω p and a T := p ∗ T m · g ∗ ω q . The pr o duct formula (see Pr op erties 2.7(v)) giv es ( π ◦ p ) ! ( ρ ∗ S a S ∧ ρ ∗ T a T ) = ( − 1) ( p − i ) j ˆ π S ! ( a S ) ∧ ˆ π T ! ( a T ) := ( − 1) ( p − i ) j τ p ( α ⊗ m ) ∧ τ q ( β ⊗ n ). The theorem no w f ollo ws from this last identit y together with (23) and (24).  Remark 2.11. The forgetful functor φ : H ∗ , ∗ Br ( X, A ) → H ∗ sing ( X ; C ) S de- scrib ed in (10) is the ring homomorphism ind uced b y the morphisms of complexes τ ∗ : A ( ∗ ) Br → E ∗ . 3. Deligne cohomology for rea l v ariet ies In this section we restrict our atten tion to A n / R , the category of real holomorphic manifolds, seen as a site with the top ology in duced by S - M an . In particular, heretofore the complexes A ( p ) B r , E ∗ and A ∗ will b e restricted to A n / R . The Ho dge decomp osition A n = ⊕ i + j = n A i,j is in v arian t with resp ect to the S -action on A n and this giv es a Ho dge ﬁltration { F p E ∗ } on the complex E ∗ of in v arian t smo oth form s . Deﬁnition 3.1. Let A ⊂ R b e a s ubring endo w ed with th e discrete top ology . i. Giv en p ≥ 0 , deﬁn e the p -th e quivariant D e ligne c omplex A ( p ) D / R on A n / R as: (25) A ( p ) D / R := Cone  A ( p ) B r ⊕ F p E ∗ ι p − → E ∗  [ − 1] , where ι p ( α, ω ) = τ p ( α ) − ω ; cf. (15). ii. Giv en a p rop er manifold X ∈ A n / R and p ≥ 0 , deﬁn e the Deligne c ohomolo gy of X as the ˇ Cec h hyp ercohomology groups (26) H i D / R ( X ; A ( p )) := ˇ H i  X eq ; A ( p ) D / R  , where X eq denotes the equiv arian t s ite of X . INTEGRAL DELIGNE COHOMOLOGY FOR REAL V ARIETIES 17 iii. If p < 0 , d eﬁne H i D / R ( X ; A ( p )) := H i,p Br ( X ( C ) , A ) . In other words, Deligne cohomology with negativ e weigh ts is deﬁned so as to coincide with Bredon cohomology . iv. The follo wing d iagram in tro duces notation for th e v arious natural maps arising fr om the cone (25). (27) H n,p Br ( X ( C ) , A ) ϕ / / H n sing ( X ( C ); A ( p )) S    H n D / R ( X ; A ( p ))  4 4 i i i i i i i i i i i i i i i i i ν / / H n,p Br ( X ( C ) , A ) ⊕ F p H n sing ( X ( C ); C ) S / / pr 1 O O pr 2   H n sing ( X ( C ); Z ( p )) S F p H n sing ( X ( C ); C ) S '  4 4 i i i i i i i i i i i i i i i i i Prop osition 3.2. L et Y C b e a pr op er c omplex holomorp hic manifold and let X b e a pr op er smo oth r e al algebr aic variety. i. One has natur al isomorphisms H i D / R ( Y C / R ; A ( p )) ∼ = H i D / C ( Y C ; A ( p )) , wher e the latter deno tes the usual Delig ne c ohomolo gy of the c omplex manifold Y C . (Se e R emark 2.1 .) ii. If X C is the c omplex v ariety obtaine d f r om X by b ase extension, the c orr esp onding map of r e al varieties X C / R → X i nduc es natur al homo- morphism s H i D / R ( X ; A ( p )) → H i D / C ( X C ; A ( p )) S , wher e the latter denotes the invariants of the Deligne c ohomolo g y of the c omplex variety X C . iii. One has a long exact se quenc e (28) · · · → H j − 1 sing ( X ( C ) , C ) S → H j D / R ( X ; A ( p )) ν − → → H j,p Br ( X ( C ) , A ) ⊕ n F p H j sing ( X ( C ); C ) o S  ◦ ϕ − ı − − − − − − → H j sing ( X ( C ); C ) S → · · · Remark 3.3. T he map H i D / R ( X ; A ( p )) → H i D / C ( X C ; A ( p )) S , is not an isomorphism in general. Ho wev er, it is alwa ys an isomorphism when 1 / 2 ∈ A . Example 3.4. Denote D i,p := H i D / R (Sp ec R ; Z ( p )) D i,p R := H i D / R (Sp ec R ; R ( p )) and D i,p C := H i D / C (Sp ec R ; Z ( p )) , 18 DOS SANTOS AND LIMA-FILHO and let D i,p → B i,p := H i,p Br ( pt, Z ) denote the natural map b et ween the Deligne and Bredon cohomolog y groups of a p oin t, r esp ectiv ely . See Ex- ample 2.2. T he follo w ing statemen ts follo w fr om the exact sequence in th e Prop osition ab ov e and from the deﬁn itions. Fix p ≥ 0 . i. The v anishing of singular cohomolog y in n egativ e d egrees giv es isomor- phisms D − i,p ∼ = − → B − i,p = 0 , for i > 0 . ii. D i, 0 ∼ = B i, 0 ∼ = ( Z (0) , if i = 0 0 , otherwise . iii. F or p > 0 one has an exact s equ ence 0 → D 0 ,p → B 0 ,p → H 0 ( pt ; C ) S = R → D 1 ,p → B 1 ,p → 0 . iv. F or i 6 = 0 , 1 one has D i,p ∼ = B i,p . No w, B 0 ,p ∼ = ( Z (2 k ) , if p = 2 k 0 , if p is o dd and B 1 ,p ∼ = ( Z / 2 Z , if p = 2 k + 1 0 , if p is even . Hence, D 0 ,p ∼ = 0 , for all p > 0, and for k ≥ 1 one has s hort exact sequences: (e k ) 0 → Z (2 k ) → R → D 1 , 2 k → 0 and (o k ) 0 → R → D 1 , 2 k − 1 → Z / 2 Z → 0 . Using similar exact sequences, one concludes th at (29) D 1 , 2 k − 1 R = R ֒ → D 1 , 2 k − 1 C = C is the inclusion as ﬁxed p oin t set, and the c hange of co eﬃcients homomor- phism is giv en by D 1 , 2 k − 1 = R × − → D 1 , 2 k − 1 R (30) x 7− → log | x | . It follo ws that D 1 , 2 k = R / Z (2 k ) and that D 1 , 2 k − 1 ∼ = R × . See T able 1 for a displa y of the bigraded group stru cture of D ∗ , ∗ . INTEGRAL DELIGNE COHOMOLOGY FOR REAL V ARIETIES 19 3.1. Pro duct structure I: p ositive w eigh ts. The constru ctions in this section are mo delled after [EV88]. Deﬁnition 3.5. Giv en 0 ≤ α ≤ 1 and p, q ≥ 0, deﬁne a pairing of complexes of preshea v es (31) ∪ α : A ( p ) i D / R ⊗ A ( q ) j D / R → A ( p + q ) i + j D / R b y ( a, θ , ω ) ∪ α ( a ′ , θ ′ , ω ′ ) := ( a · a ′ , θ ∧ θ ′ , ω ∧ Ξ α [ a ′ , θ ′ ] + ( − 1) i Ξ 1 − α [ a, θ ]) , where ( a, θ , ω ) ∈ A ( p ) i B r ⊕ F p E i ⊕ E i − 1 , ( a ′ , θ ′ , ω ′ ) ∈ A ( q ) j B r ⊕ F q E j ⊕ E j − 1 and for ( u, η ) ∈ A ( p ) i B r ⊕ F p E i one deﬁn es Ξ t ( u, η ) := (1 − t ) τ p ( u ) + t η . The follo wing results are straightforw ard and their pro ofs are left to the reader. Prop osition 3.6. Fix p, q ≥ 0 and let X b e a pr op er r e al holomor phic manifold. i. The p airings ∪ α ar e homo topic to e ach other, for al l 0 ≤ α ≤ 1 , and they induc e p airings ∪ : H i D / R ( X ; A ( p )) ⊗ H j D / R ( X ; A ( q )) → H i + j D / R ( X ; A ( p + q )) , satisfying a ∪ b = ( − 1) ij b ∪ a. ii. The natur al maps H ∗ D / R ( X ; A ( p )) → H ∗ ,p Br ( X, A ) , for p ≥ 0 , ar e c omp at- ible with the r esp e c tiv e multiplic ative structur es of De ligne and Br e don c ohomolo gy. iii. The natur al map ι : H ∗ D / R ( X ; A (0)) → H ∗ , 0 Br ( X ( C ) , A ) is a ring isomor- phism. W ell-kno wn facts from Bredon cohomolog y allo w the computatio n of a few more examples. Example 3.7. L et X ∈ A n / R b e a real c el lular p r op er al gebraic v ariet y , and let C H ∗ ( X ) d enote its Chow ring, seen as a bigraded ring where C H p ( X ) has degree (2 p, p ). As a b igraded ring, the Bredon cohomolo gy of X with Z -coeﬃcients is giv en by H ∗ , ∗ Br ( X ( C ) , Z ) ∼ = C H ∗ ( X ) ⊗ B ∗ , ∗ , cf. [dS LF07]. Since C H ∗ ( X ) is free and Bredon cohomology is a ge ometric c ohomolo gy the ory in the sense of [Kar00], it f ollo ws that H ∗ , ∗ Br ( X, A ) ∼ = C H ∗ ( X ) ⊗ B ∗ , ∗ A , where B ∗ , ∗ A := H ∗ , ∗ Br ( pt, A ) and A ⊂ R is a subr ing of R . F urthermore, the singular cohomology of X ( C ) with C -co eﬃcien ts is inv ariant and of Ho dge t y p e ( p, p ). Hence, the long exact sequence in the Pr op osition 3.2 an d the ab o v e observ ations giv e n atural isomorphisms: H 2 p D / R ( X ; A ( p )) ∼ = H 2 p,p Br ( X ( C ) , A ) ∼ = H 2 p ( X ( C ); A ( p )) S (32) ∼ = C H p ( X ) ⊗ A. 20 DOS SANTOS AND LIMA-FILHO In p articular, when X = P p , one has H ∗ , ∗ Br ( P p , A ) ∼ = B ∗ , ∗ A [ h ] / h h p +1 i , where h ∈ H 2 , 1 Br ( P p , Z ) is the ﬁrst Chern class of the hyperp lane bundle in Bredon cohomology . Deﬁne ξ ∈ H 2 D / R ( P p ; A (1)) as the elemen t corresp ondin g to h via the isomorphisms (32). It follo ws that H 2 j D / R ( P p ; A ( j )) ∼ = Z is generated b y ξ j , for all 0 ≤ j ≤ p and is 0 for j > p. Let ξ ∈ H 2 D / R ( X × P p ; A (1)) denote the pull-bac k u nder the pro jection X × P p → P p of th e class ξ ∈ H 2 D / R ( P p ; A (1)) d eﬁned ab o ve, and let π : X × P p → X denote the pr o jection onto X . Giv en α ∈ H i D / R ( X × P p ; A ( − r )) , with r , k ≥ 0, d eﬁ ne α ∪ ξ k ∈ H i +2 k D / R ( X × P p ; A ( k − r )) as follo ws: (33) α ∪ ξ k := ( α · h k , if k ≤ r ( α · h r ) ∪ ξ k − r , if k ≥ r. Here we are iden tifying Deligne and Bredon cohomology with weigh ts ≤ 0 , and b eing careful to diﬀerenti ate b et w een h and ξ and b et we en the pro duct α · h k in Bredon cohomology and the ∪ pr o duct for Deligne cohomology with p ositiv e weig ht s. The follo win g result is a pr eliminary v ersion of a more general pro jectiv e bund le form u la. Prop osition 3.8. L et X b e a pr op er r e al holo morphic manifold. Given inte gers p ≥ 0 and i, q ∈ Z , the map ψ : ⊕ p j =0 H i − 2 j D / R ( X ; A ( q − j )) − → H i D / R ( X × P p ; A ( q )) ( a 0 , . . . , a p ) 7− → π ∗ a 0 + π ∗ a 1 ∪ ξ + · · · + π ∗ a p ∪ ξ p is an isomorphism. Pr o of. Denote by ξ C the image of ξ u nder the cycle map in to singular co- homology with complex co eﬃcien ts. Since ξ C is of Ho dge typ e ( p, p ) and in v arian t u nder S , the map a 7→ π ∗ a ∪ ξ j C from H i − 2 j ( X ( C ); C ) to H i ( X ( C ) × P p ( C ); C ) send s F q − j H i − 2 j ( X ( C ); C ) S to F q H i ( X ( C ) × P p ( C ); C ) S . Recall that Ch er n classes are compatible under the f orgetful functor from Bredon cohomology to singular cohomology and that the pro jectiv e bun dle form ula holds in b oth theories; cf. [d S 03a]. The r esult n o w f ollo ws from the deﬁnition of ∪ ξ in Deligne cohomology and the ﬁve- lemma suitably applied to the long exact sequences in Prop osition 3.2.  3.2. Pro duct structure I I: arbitrary weigh ts. Giv en p, q ≥ 0 deﬁne the pro du ct (34) ∪ : H i D / R ( X ; A ( − p )) ⊗ H j D / R ( X ; A ( − q )) → H i + j D / R ( X ; A ( − p − q )) simply as the p ro duct in Bredon cohomology . Given 0 ≤ | r | ≤ p , one u ses (33) to d eﬁne ⋆ : H i D / R ( X ; A ( − p )) ⊗ H j D / R ( X ; A ( p − r )) → H i + j +2 p D / R ( X × P p ; A ( p − r )) INTEGRAL DELIGNE COHOMOLOGY FOR REAL V ARIETIES 21 8 0 R / Z (8) τ 3 ε 2 0 τ 2 ε 4 7 0 R × 0 τ 2 ε 3 0 6 0 R / Z (6) τ 2 ε 2 0 τ ε 4 5 0 R × 0 τ ε 3 0 4 0 R / Z (4) τ ε 2 0 ε 4 3 0 R × 0 ε 3 0 2 0 R / Z (2) ε 2 0 0 1 0 R × 0 0 0 0 1 0 0 0 0 -1 0 -2 2 τ − 1 -3 θ -4 ε − 1 θ 2 τ − 2 -5 ε − 2 θ 0 τ − 1 θ -6 ε − 3 θ 0 ε − 1 τ − 1 θ 2 τ − 3 -7 0 ε − 2 τ − 1 θ 0 τ − 2 θ -8 ε − 3 τ − 1 θ 0 ε − 1 τ − 2 θ 2 τ − 4 -3 -2 -1 0 1 2 3 4 T abl e 2. Generators for cohomology of a p oint b y sending a ⊗ b 7→ a ⋆ b := ( a ∪ ξ p ) ∪ π ∗ b . No w, let π † : H i + j +2 p D / R ( X × P p ; A ( p − r )) → H i + j D / R ( X ; A ( − r )) denote the comp osition of ψ − 1 (see Pr op osition 3.8) with the pro jection on to the last factor. Sending a ⊗ b 7→ π † ( a ⋆ b ) deﬁn es the pro d uct (35) ∪ : H i D / R ( X ; A ( − p )) ⊗ H j D / R ( X ; A ( p − r )) → H i + j D / R ( X ; A ( − r )) , Com bining Pr op osition 3.6, (34) and (35) one d eﬁnes a prod uct on Deligne cohomology which mak es H ∗ D / R ( X ; A ( ∗ )) → H ∗ , ∗ Br ( X ( C ) , A ) a natural map of bigraded rin gs. Corollary 3.9. U nder the hyp othesis of P r op osition 3.8, one has an iso- morphism of bigr ade d rings H ∗ D / R ( X ; A ( ∗ ))[ T ] / h T p +1 i ∼ = H ∗ D / R ( X × P p ; A ( ∗ )) when T is given the bigr ading (2 , 1) . The ring structur e of D ∗ , ∗ can b e easily read from T able 2. See Example 2.2 for notation. 4. The e xponent ial seque n ce In this section we show that Z (1) D / R is quasi-isomorph ic to O × [ − 1]. As an immediate corollary , we obtain an exp onential se quenc e relating th e co- homology of Z (1) B r , O and O × . Using this sequence we obtain a new pro of of Weichold’s The or em – a classical r esult in real algebraic geometry . 22 DOS SANTOS AND LIMA-FILHO Deﬁnition 4.1. Let X b e a smo oth algebraic v ariet y and let ( R ∗ X , d R ) denote the follo wing co c hain complex of ab elian shea v es on X E 0 , 0 × X ∂ log − − − → E 0 , 1 X ∂ − → E 0 , 2 X ∂ − → · · · where E 0 , 0 × X ⊂ E 0 X denotes the sub sheaf of no where zero fu n ctions and E p,q X ⊂ E p + q X denotes the su bsheaf on inv ariant ( p + q )-forms of Ho d ge t yp e ( p, q ). Remark 4.2. (1) It is easy to c hec k that ( R ∗ , d R ) is a resolution of O × . Similarly , O → E 0 , 0 ∂ − → E 0 , 1 ∂ − → · · · is a soft resolution of O . (2) T h e exp onen tial map O → O × extends to a m ap exp : E 0 , ∗ → R ∗ b et w een the resp ectiv e resolutions. Deﬁnition 4.3. Recall that an ele ment in Z (1) 1 B r ( U ) is represen ted by sums of pairs of the form f ⊗ m , w ith f and m equiv arian t maps su c h that (1) f : S → C × is smo oth and π : S → U is in b U ; (2) m : S → Z ∈ Z ( S ) is lo cally constant. Let η : Z (1) 1 B r → E 0 , 0 × b e the map send ing f ⊗ m ∈ Z (1) 1 B r ( U ) to η ( f ⊗ m ) = ˆ π ! ( f m ) . Alternativ ely , η ( f ⊗ m )( u ) = Q s ∈ π − 1 ( u ) f ( s ) m ( s ) . Remark 4.4. It follo ws from this deﬁnition that, giv en σ ∈ Z (1) 0 B r and α ∈ Z (1) 1 B r w e hav e τ 0 1 σ = log η ( σ (1)) − log η ( σ (0)) and τ 1 1 α = d log η α . Prop osition 4.5. Ther e is a quasi-isomo rphism ς : Z (1) D / R → R ∗ [ − 1] such that the c omp osite E ∗ [ − 1] → Z (1) D / R ς − → R ∗ [ − 1] induc es the map exp : C X [ − 1] → O × X [ − 1] , and the c omp osition E 0 , ∗ [ − 1] → Z (1) D / R → R ∗ [ − 1] induc es the map exp : O X [ − 1] → O × X [ − 1] . Pr o of. Let X b e a pro jectiv e real v ariet y and let ς : Z (1) D / R → R ∗ [ − 1] denote the map of complexes displa y ed in the f ollo wing diagram: / / Z (1) 0 B r   d − 1 Z (1) D / R / / Z (1) 1 B r ⊕ E 1 , 0 ⊕ E 0 d 0 Z (1) D / R / / η · exp   E 1 , 1 ⊕ E 2 , 0 ⊕ E 1 / / p 0 , 1   · · · / / 0 / / R 0 ∂ log / / R 1 ∂ / / · · · where η · exp denotes the map ( f , ω , h ) 7→ η ( f ) exp h , and p 0 , 1 denotes the pro jection from E 1 to E 0 , 1 . Similarly , for i > 1 w e set ς i = p 0 ,i , w here p 0 ,i : E i → E 0 ,i is the p ro jection. It is easy to chec k ς is a map of co c hain complexes and that the cohomol- ogy pr eshea v es of b oth complexes are concen trated in dimension 1. Hence it suﬃces to c hec k that H 1 ( ς ) is an isomorph ism on the stalks. Surje ctivity: Note that H 1 ( R ∗ [ − 1]) = H 0 ( R ∗ ) ∼ = O × and let g ∈ O × u , u ∈ X . Let h ∈ O u b e suc h that exp h = g . S et α = 0 in Z (1) B r,u (so that α can b e INTEGRAL DELIGNE COHOMOLOGY FOR REAL V ARIETIES 23 represent ed, for example, b y 1 ⊗ 0) and set w = ∂ g g ∈ E 1 , 0 u . Then we ha v e, d 0 Z (1) D / R ( α, ω , h ) = 0 and ς u ( α, ω , h ) = g . Inje ctivity: Let ( α, ω , h ) ∈ Z (1) 1 D / R ,u , u ∈ X , b e suc h that ς u ( α, ω , h ) = 0 and d 0 Z (1) D / R ( α, ω , h ) = 0. Th e ﬁrst equalit y just means that − h is a logarithm for η ( α ). F rom the second equalit y w e get ω = − dh − τ 1 1 α = − ( dh + d log η ( α )) = 0 . Let P i f i ⊗ m i b e a repr esentati v e for α (cf. Deﬁnition 4.5 ). C ho ose log f i so th at P i m i log f i = − h , shrinking neigh b orh o o ds if necessary , and deﬁne σ := ∆ 1 ∋ t 7→ exp " − t X i m i log f i #! ∈ Z (1) 0 B r,u . Then, by our c h oice of log f i , we ha v e d − 1 Z (1) D / R ( σ ) = ( α, ω , h ). The remaining assertions are evid ent.  Corollary 4.6. The c omplexes Z (1) D / R and O × [ − 1] ar e quasi-isomorphic. Corollary 4.7 (Exp onen tial Sequence) . L et X b e a smo oth pr op er r e al al- gebr aic variety. Then ther e is a long exact se quenc e → H ∗ , 1 Br ( X ( C ); Z ) ϑ − → H ∗ ( X ; O X ) exp − − → H ∗ ( X ; O ∗ X ) → H ∗ +1 , 1 Br ( X ( C ); Z ) → wher e ϑ denotes the c omp osite H ∗ , 1 Br ( X ( C ); Z ) τ 1 − → H ∗ ( X ( C ); C ) G ։ H ∗ ( X ; O X ) = H ∗ , 1 ∂ ( X ( C )) G , and the latter denotes the i nvariants of the Dolb e ault c ohomolo gy of the c omplex manifold X ( C ) . Pr o of. F r om the hyp ercohomology long exact sequence of the cone and Lemma 4.5 we get the follo wing exact sequ en ce → H ∗ , 1 Br ( X ( C ); Z ) ⊕ ˇ H ∗ ( X ; F 1 E ∗ X ) ι 1 − → ˇ H ∗ ( X ; E ∗ X ) → H ∗ ( X ; O × X ) → whic h, in tu r n, giv es H ∗ , 1 Br ( X ( C ); Z ) τ 1 − → coker  ˇ H ∗ ( X ; F 1 E ∗ X ) → ˇ H ∗ ( X ; E ∗ X )  → H ∗ ( X ; O × X ) . No w, since Cone( F 1 E ∗ → E ∗ ) ≃ E 0 , ∗ and sin ce ˇ H ∗ ( X ; F 1 E ∗ X ) → ˇ H ∗ ( X ; E ∗ X ) is injectiv e, we ha v e cok er  ˇ H ∗ ( X ; F 1 E ∗ X ) → ˇ H ∗ ( X ; E ∗ X )  ∼ = ˇ H ∗ ( X ; E 0 , ∗ X ) ∼ = H ∗ ( X ; O X ) . The assertion ab ou t the map ϑ follo w s immediately from the construction of this sequence.  Remark 4.8. A similar exact sequence app ears in [Kr a91]. 24 DOS SANTOS AND LIMA-FILHO 4.1. An application. Giv en a real v ariet y X , d enote S = π 0 ( X ( R )). Hence H 0 ( X ( R ); Z × ) ∼ = ( Z × ) S and ˜ H 0 ( X ( R ); Z × ) ∼ = ( Z × ) S / Z × , where Z × ⊂ ( Z × ) S is the s u bgroup of constant functions. Lemma 4.9. L e t X b e an irr e ducible, smo oth, pr oje ctive curve over R , of genus g . L et c denote the numb er of c onne cte d c omp onents of X ( R ) Then H 2 , 1 Br ( X ( C ); Z ) ∼ = Z × ( Z × ) S / Z × ∼ = Z ×  ( Z / 2) c − 1 if c 6 = 0 0 if c = 0  Pr o of. By P oincar ´ e du altit y H 2 , 1 Br ( X ( Z ); Z ) ∼ = H S 0 ( X ( C ); Z ), w here the last group denotes Z -graded Bredon h omology . The follo wing exact sequence is w ell known (see [LLFM03]) 0 → H sing 0 ( X ( C ) / S ; Z ) → H Br 0 , 0 ( X ( C ); Z ) → H 0 ( X ( R ); Z × ) → 0 . If X ( R ) = ∅ the sequence giv es H S 0 ( X ( C ); Z ) ∼ = Z since under the abov e assumptions X ( C ) / S is connected. If X ( R ) 6 = ∅ then H Br 0 , 0 ( X ( C ); Z ) ∼ = Z × e H Br 0 , 0 ( X ( C ); Z ), where th e e H Br ∗ , ∗ ( − ; Z ) denotes reduced Bredon homology . There is a reduced version of the se- quence ab o v e whic h give s e H Br 0 , 0 ( X ( C ); Z ) ∼ = e H 0 ( X ( R ); Z × ) ∼ = ( Z × ) c − 1 , b e- cause e H sing 0 ( X ( C ) / S ; Z ) = 0.  Denote V := H 1 ( X C , O ) and let Λ ⊂ V be th e lattice Λ := Im { j C : H 1 sing ( X C , Z (1)) → H 1 ( X C , O ) } , so that P ic ( X C ) ∼ = V / Λ. T aking ﬁxed p oints giv es a s h ort exact sequ en ce (36) 0 → Λ S → V S → P ic 0 ( X C ) S → H 1 ( S , Λ) . This sho ws that V S / Λ S is the connected comp onen t P ic 0 ( X C ) S 0 . No w, Prop osition A.1 sho ws that H 1 , 1 Br ( X ( C ) , A ) ∼ = H 1 , 1 bor ( X ( C ) , A ), hence one can use the Lera y-Serre sp ectral sequence to conclude that the image of the natural map j : H 1 , 1 Br ( X ( C ) , Z ) → H 1 ( X C , O ) is p recisely Λ S . Prop osition 4.10. L et X b e an irr e ducible, smo oth, pr oje ctive curve over R , of ge nus g . Then P ic ( X ) ∼ = P ic 0 ( X C ) S 0 × H 2 , 1 Br ( X ( C ) , Z ) . Pr o of. By the p revious results, P ic ( X ) ﬁts in the follo win g exact sequence H 1 , 1 Br ( X ( C ) , Z ) j − → H 1 ( X C ; O ) S → P ic ( X ) c 1 − → H 2 , 1 Br ( X ( C ) , Z ) → 0 . The result f ollo ws.  Corollary 4.11 (W eic hold’s Theorem [PW91, Prop osition 1.1]) . With X as ab ove, let c denote the numb er of c onne cte d c omp onents of X ( R ) . Then P ic ( X ) ∼ = Z × ( R / Z ) g ×  ( Z / 2) c − 1 if c 6 = 0 0 i f c = 0  Pr o of. This follo ws directly from the prop osition together with Lemma 4.9.  INTEGRAL DELIGNE COHOMOLOGY FOR REAL V ARIETIES 25 5. The group H 2 D / R ( X ; Z (2)) In this section we deriv e a geometric int erpretation of the in tegral coho- mology group H 2 D / R ( X ; Z (2)) for a real pro jectiv e v ariet y X . As a moti- v ation we start with a geometric inte rpretation of th e Bredon cohomology group H 2 , 2 Br ( Y , Z ) of an arbitrary S -manifold Y . 5.1. V ariations on the t heme of H 2 , 2 Br ( Y , Z ) . As a S - manifold, the sphere S 2 , 2 is isomorphic to P 1 ( C ) with the actio n induced by a linear inv olution σ . Describing the action in homogeneous coord inates by σ : [ x 0 : x 1 ] 7→ [ x 0 : − x 1 ], one sees that S P ∞ ( P 1 ( C )) ≡ P ∞ ( C ) inherits th e linear inv olution σ ([ x 0 : x 1 : x 2 : · · · ]) = [ x 0 : − x 1 : x 2 : − x 3 : · · · ] and that this is an equiv arian t K ( Z , (2 , 2)); cf. [dS 03b]. In particular, one can deﬁne a linear isomorphism of tautological bun dles τ : σ ∗ O ( − 1) → O ( − 1) satisfying τ ◦ ( σ ∗ τ ) = 1. Giv en a S -space Y , with inv olution σ , let P 1 ( Y ) d enote the set of pairs ( L, τ ) wh ere: t1) L is a (smo oth) complex line bund le on Y; t2) τ : σ ∗ L → L is a b undle isomorphism; t3) τ ◦ ( σ ∗ τ ) = 1. Deﬁne an equiv alence relation on P 1 ( L ) b y ( L, τ ) ∼ 1 ( L ′ , τ ′ ) iﬀ there exists an isomorphism φ : L → L ′ suc h that φ ◦ τ = τ ′ ◦ σ ∗ φ . Lemma 5.1. The tensor pr o duct of line bund les induc es a gr oup structur e on the set L 1 ( Y ) := P 1 ( Y ) / ∼ 1 of e quivalenc e classes of p airs satisfying t1)–t3). F urthermor e, this gr oup is natur al ly isomorphic to H 2 , 2 Br ( Y , Z ) . Pr o of. The ﬁrst assertion is clea r and the last one follo w s from th e fact that ( CP ∞ , σ ) with the linear inv olution σ describ ed ab ov e is an equiv ariant K ( Z , (2 , 2)).  Recall that a R e al ve ctor b u nd le ( E , τ ) on a S -manifold ( Y , σ ) consists of a complex v ector b undle E on Y together with an isomorp hism τ : σ ∗ E → E satisfying τ ◦ σ ∗ τ = I d . No w, consider the set P 2 ( Y ) consisting of pairs ( L, q ) satisfying: p1) L is a (smo oth) complex line bu ndle on Y ; p2) q : L ⊗ σ ∗ L → 1 Y is an isomorphism of Real line bundles, wh ere L ⊗ σ ∗ L carries the tautologic al Real line b undle structur e; Denote ( L, q ) ∼ 2 ( L ′ , q ′ ) iﬀ there is an isomorphism φ : L → L ′ satisfying q ′ ◦ ( φ ⊗ σ ∗ φ ) = q and ob s erv e that this is an equiv alence relatio n on P 2 ( Y ). Lemma 5.2. The tensor pr o duct also induc es a gr oup structur e on the set L 2 ( Y ) := P 2 ( Y ) / ∼ 2 of i somorph ism classes of p airs ( L, q ) satisfying p1)– p2) . Finally , consider the complex of sh ea ves G 0 a − → G 1 on S - M an where (37) G 0 ( U ) = { f : U → C × | f is sm o oth } 26 DOS SANTOS AND LIMA-FILHO and (38) G 1 ( U ) = { f : U → C × | f is sm o oth and equiv arian t } , where a : G 0 → G 1 is the “transf er map” a ( f ) = f · σ ∗ f . Prop osition 5.3. Ther e ar e natur al isomorphism s H 2 , 2 Br ( Y , Z ) ∼ = H 2 , 2 b or ( Y , Z ) ∼ = L 1 ( Y ) ∼ = L 2 ( Y ) ∼ = H 1 ( Y e q ; G 0 → G 1 ) . Pr o of. The ﬁr st tw o isomorp hisms follo w from L emma 5.1 and Prop osition A.1, resp ective ly , and the last isomorph ism is a tautology . Giv en ( L, τ ) ∈ P 1 ( Y ), pic k a h ermitian metric h : L ⊗ L → 1 Y on L and deﬁne q h τ : L ⊗ σ ∗ L → 1 Y as the comp osition L ⊗ σ ∗ L 1 ⊗ τ − − → L ⊗ L → 1 Y , where τ : σ ∗ L → L is the map indu ced b y τ . It is easy to see that q h τ is an isomorph ism of Real line bundles, and hence we obtain an elemen t ( L, q h τ ) ∈ P 2 ( Y ). Supp ose that ψ : L ′ → L in d uces an equiv alence ( L ′ , τ ′ ) ∼ 1 ( L, τ ) and pic k a hermitian metric h on L . O ne sees that ( L ′ , q ψ ∗ h τ ′ ) ∼ 2 ( L, q h τ ). It follo ws that one has a w ell-deﬁned homomorphism L 1 ( Y ) → L 2 ( Y ) sending [ L, τ ] to h L, q h τ i , w here h is any c hoice of metric on L . The construction of the in v erse h omomorphism is evid ent.  Let S = π 0 ( Y S ) denote the set of connected comp onents of the ﬁ x ed p oint set Y S , and identify H 0 ( Y S ; Z × ) ≡ ( Z × ) S . Obs erv e that the K ¨ unneth form ula yields a natural isomorph ism H 2 , 2 bor ( Y S ; Z ) ∼ = H 2 ( Y S × B S ; Z (2)) ∼ = H 0 ( Y S ; Z × ) ⊕ H 2 sing ( Y S ; Z (2)). Using the ﬁrs t id entiﬁcatio n in Prop osition 5.3 one considers the comp osition H 2 , 2 Br ( Y , Z ) ≡ H 2 , 2 bor ( Y ; Z ) → H 2 , 2 bor ( Y S ; Z ) → H 0 ( Y S ; Z × ) of the r estriction map follo w ed by the evident p ro jection to obtain a natural homomorphism (39) ℵ : H 2 , 2 Br ( Y , Z ) → ( Z × ) S . This map h as a natural geometric in terpretation when one u ses the iden- tiﬁcation H 2 , 2 Br ( Y , Z ) ≡ L 2 ( Y ). Given h L, q i ∈ L 2 ( Y ), the restriction of q to L | Y S b ecomes a non-degenerate herm itian pairing, and hence it has a w ell-deﬁned signature ℵ h L, q i ∈ ( Z × ) S . I t is easy to see th at this is another description of (39). Deﬁnition 5.4. W e call ℵ : H 2 , 2 Br ( Y , Z ) → ( Z × ) S the e quiv ariant signatur e map of Y . Th e image ℵ tor ( Y ) ⊆ ( Z × ) S of the torsion sub group H 2 , 2 Br ( Y , Z ) tor under ℵ is called the e qu ivariant signatur e gr oup of Y . In the case where Y = X ( C ) for a r eal algebraic v ariet y X w ith S = π 0 ( X ( R )), we d enote the equiv arian t signature group of X ( C ) simp ly b y ℵ tor ( X ). Example 5.5. When X is a pro jectiv e algebraic curv e, it follo ws from the cohomology sequence of the pair ( E S × S X ( C ) , B S × X ( R )) that ℵ is an isomorphism. As a consequence, one obtains isomorphisms H 2 , 2 Br ( X ( C ) , Z ) ∼ = INTEGRAL DELIGNE COHOMOLOGY FOR REAL V ARIETIES 27 B r ( X ) ∼ = ℵ tor ( X ) , wh ere B r ( X ) is the Brauer group of X , since B r ( X ) ∼ = ( Z × ) S when X is an algebraic curve ; cf. [Wit34]. 5.2. T he Deligne group H 2 D / R ( X ; Z (2)) . T he Ho dge ﬁltration on singular cohomology induces a ﬁltration on Bredon cohomology , with F j H n,p Br ( X ( C ) , Z ) := ϕ − 1  − 1 F j H n sing ( X ( C ) , C ); see diagram (27) for notation. Prop osition 5.6. F or any smo oth r e al pr oje ctive variety X , one has F 2 H 2 , 2 Br ( X ( C ) , Z ) = H 2 , 2 Br ( X ( C ) , Z ) tor = im(  ) , wher e  is the cycle map fr om Deligne to Br e don c ohomo lo gy (27) . In p ar- ticular the image of the c omp osition Ψ : H 2 D / R ( X ; Z (2))  − → H 2 , 2 Br ( X ( C ) , Z ) ℵ − → H 0 ( X ( R ) , Z × ) is the e qu ivariant signatur e gr oup ℵ tor ( X ) . Pr o of. Consider diagram (27) with p = n = 2. S ince the middle row is ex- act, one concludes that im (  ) = F 2 H 2 , 2 Br ( X, Z ) , by deﬁnition. On the other hand,  ◦ ϕ : H 2 , 2 Br ( X ( C ) , Z ) → H 2 ( X ( C ); C ) factors as H 2 , 2 Br ( X ( C ) , Z ) → H 2 , 2 Br ( X ( C ) , Z ) / tor ֒ → H 2 , 2 Br ( X ( C ) , Z ) ⊗ Q ֒ → H 2 sing ( X ( C ); Z (2)) ⊗ Q ֒ → H 2 ( X ( C ); C ) . T he injectivi t y of H 2 , 2 Br ( X ( C ) , Z ) ⊗ Q ֒ → H 2 sing ( X ( C ); Z (2)) ⊗ Q follo ws from the isomorp hism H 2 , 2 Br ( X ( C ) , Z ) ∼ = H 2 , 2 bor ( X ( C ); Z ) and we ll- kno wn facts in equiv ariant cohomolog y . Since H 2 sing ( X ( C ); Z (2)) ⊗ Q ∩ F 2 H 2 ( X ( C ); C ) = 0 , one concludes th at F 2 H 2 , 2 Br ( X ( C ) , Z ) = H 2 , 2 Br ( X ( C ) , Z ) tor .  The next goal is to pr o vide a geo metric interpretation to the ke rnel of th e surjection Ψ : H 2 D / R ( X ; Z (2)) → ℵ tor ( X ). T o this p urp ose, consider triples ( L, ∇ , q ) w h ere L is a holomorphic line bu ndle o ver X ( C ), ∇ is a h olomor- phic connectio n on L and q : L ⊗ σ ∗ L → 1 is a h olomorphic isomorp hism of Real line b undles satisfying the follo w in g prop erties: (1) T h e restriction of q to X ( R ) is a p ositiv e-deﬁnite h ermitian metric on L | X ( R ) . (2) As a section of ( L ⊗ σ ∗ L ) ∨ , q is parallel with resp ect to th e connection induced by ∇ . A morp h ism b et ween t w o suc h triples f : ( L, ∇ , q ) → ( L ′ , ∇ ′ , q ′ ) consists of a line bu ndle map f : L → L ′ suc h that q ′ ◦ ( f ⊗ σ ∗ f ) = q an d ∇ ′ ◦ f = (1 ⊗ f ) ◦ ∇ . 28 DOS SANTOS AND LIMA-FILHO Deﬁnition 5.7. Give n a real v ariet y X , let P W ∇ ( X ) d enote the set of isomorphism classes h L, ∇ , q i of triples as ab ov e. T his is a group under the op eration h L, ∇ , q i ⊙ h L ′ , ∇ ′ , q ′ i := h L ⊗ L ′ , ∇ ⊗ 1 + 1 ⊗ ∇ ′ , q · q ′ i , whic h we call the diﬀer ential Pic ar d-Witt gr oup of X . The group P W ∇ ( X ) has an alternativ e deﬁnition in terms of a complex P ∗ : P 0 D − → P 1 D − → P 2 of preshea v es on A n / R . Giv en a real analytic v ariet y U , deﬁne P 0 ( U ) := O × C ( U ) = { f : U → C × | f is h olomorphic } ; and P 1 ( U ) := Ω 1 C ( U ) ⊕ O × R + ( U ) , where Ω 1 C ( U ) denotes the holomorphic 1-forms on U and O × R + ( U ) denotes the subgroup of O × C ( U ) consisting of those holomorphic Real fu nctions f (i.e. σ ∗ f = f ) whic h are p ositiv e on the r e al lo cu s U ( R ) := U S . Finally , deﬁne P 2 ( U ) as th e group of Real holomorphic 1-forms on U , in other words P 2 ( U ) := Ω 1 R ( U ) := { ψ ∈ Ω 1 C ( U ) | σ ∗ ψ = ψ } . Deﬁne D : P 0 ( U ) → P 1 ( U ) as D ( g ) := ( dg /g , g · σ ∗ g ) and D : P 1 ( U ) → P 2 ( U ) as D ( ψ , f ) := ψ + σ ∗ ψ − d f /f . Prop osition 5.8. P W ∇ ( X ) is natur al ly isomorph ic to ˇ H 1 ( X eq ; P ∗ ) . Pr o of. This is straight forwa rd.  Remark 5.9. Giv en h L, ∇ , q i in P W ∇ ( X ) one can alwa ys ﬁ nd an equiv ari- an t co v er U and a cocycle c in ˇ C 1 ( U , P ∗ ) r epresen ting h L, ∇ , q i that has the form c = (( g α 0 α 1 ) ; ( ( ψ α 0 ) , (1) )). In other words, [ c ] is determined by ( g α 0 α 1 ) , ( ψ α 0 ) satisfying: i: g α 0 α 1 ∈ Ω 0 C ( U α 0 α 1 ) and δ ( g α 0 α 1 ) = 1 ( gives the c o cycle c ondition for a holomorp hic line bund le L ); ii:  dg α 0 α 1 g α 0 α 1  = δ ( ψ α 0 ) ( gives the holomorp hic c onne ction on L ); iii: g α 0 α 1 · σ ∗ g α 0 α 1 = 1 ( give s the hermitian form q ); iv: ψ α 0 ∈ Ω 1 C ( U ) and ψ α 0 + σ ∗ ψ α 0 = 0, i.e. ψ α 0 is a holomorphic an ti-in v arian t 1-form ( q is p ar al lel ). The main r esult of this section is the follo wing. Theorem 5.10. If X is a smo oth r e al pr oje ctive variety then one has a natur al short exact se quenc e 0 → P W ∇ ( X ) → H 2 D / R ( X ; Z (2)) Ψ − → ℵ tor ( X ) → 0 . Pr o of. See App endix B.  INTEGRAL DELIGNE COHOMOLOGY FOR REAL V ARIETIES 29 6. A r emark on numbe r fiel d s Let F b e a num b er ﬁeld and let Γ R and Γ C denote the sets of real and complex embed dings of F , resp ectiv ely . One can write Γ C = Γ + C × S , where Γ + C con tains one c hosen element in eac h S -orbit of Γ C . Abusing language, wr ite H r D / R ( F ; A ( p )) instead of H r D / R ( X ; A ( p )), where X := S pec ( F ⊗ Q R ). Ob serv e that X ( C ) ≡ Γ R ` Γ C , and hence H r D / R ( F ; A ( p )) = H r D / R (Γ R ; A ( p )) × H r D / R (Γ C ; A ( p )) ≡ H r D / R ( ∗ ; A ( p )) Γ R × H r D / C ( ∗ ; A ( p )) Γ + C (40) ∼ = H r D / R ( ∗ ; A ( p )) s × H r D / C ( ∗ ; A ( p )) t , where A is a s ubring of R and Γ R = { ϕ 1 , . . . , ϕ s } and Γ + C = { η 1 , . . . , η t } . In particular, H 1 D / R ( F ; Z (1)) ≡ ( R × ) Γ R × ( C × ) Γ + C ∼ = ( R × ) s × ( C × ) t and (41) H 1 D / R ( F ; R (1)) ≡ R Γ R × R Γ + C ∼ = R s × R t ; cf. Example 3.4. T aking adjoints to the ev aluation map s F × × Γ R → R × and F × × Γ + C → C × giv es a m onomorphism (42) F × → ( R × ) Γ R × ( C × ) Γ + C ≡ H 1 D / R ( F ; Z (1)) . Since ⊕ p ≥ 0 H p D / R ( F ; Z ( p )) is a graded comm utativ e rin g this map induces a homomorphism (43) ρ : T ( F × ) → ⊕ p ≥ 0 H p D / R ( F ; Z ( p )) , where T ( F × ) is the tensor algebra of F × . Using the comm u tativit y of the diagram F × ⊗ F × ρ / / H 1 D / R ( F ; Z (1)) ⊗ H 1 D / C ( F ; Z (1)) ∪ / /  ⊗    H 2 D / R ( F ; Z (2))  ∼ =   H 1 , 1 Br ( F , Z ) ⊗ H 1 , 1 Br ( F , Z ) · / / H 2 , 2 Br ( F , Z ) , together with the d escription of th e ring structure of the Bredon cohomology of a p oint 2.2, one concludes that if a 6 = 0 , 1, then  ( a ⊗ (1 − a )) = 0. It follo ws that  descends to a h omomorp hism (44) ¯  : K M ∗ ( F ) → ⊕ p ≥ 0 H p D / R ( F ; Z ( p )) , from the Milnor K -theory ring of F to the “diagonal” subrin g of th e inte gral Deligne cohomology of F . Remark 6.1. i. If follo ws f rom the work of Bass and T ate that K M ∗ ( R ) / 2 K M ≥ 2 ( R ) ∼ = ⊕ i D i,i and K M ∗ ( R ) / 2 K M ∗ ( R ) ∼ = Z / 2[ ε ] ∼ = ⊕ i B i,i . 30 DOS SANTOS AND LIMA-FILHO ii. Since π : S pec ( F ⊗ Q R ) → S pec ( R ) is a ﬁnite co v er, one has an additive transfer homomorph ism π ! : H 1 D / R ( F ; R (1)) → H 1 D / R ( R ; R (1)) (45) ( x 1 , . . . , x s ; y 1 , . . . , y t ) 7→ x 1 + · · · + x s + 2 y 1 + · · · + 2 y t ; see (41). iii. In sub sequen t w ork w e will show th at the homomorphism (44) is a par- ticular case of a natural transformation betw een the moti vic cohomolo gy of a real v ariet y and its integ ral Deligne cohomology . It follo ws from (41) and (30) that the comp osition F × → H 1 D / R ( F ; Z (1)) → H 1 D / R ( F ; R (1)) is giv en by F × − → R s × R t (46) x 7− → (log | ϕ 1 ( x ) | , . . . , log | ϕ s ( x ) | ; log | η 1 ( x ) | , . . . , log | η t ( x ) | ) . Basic class ﬁeld theory sho ws that the image of the units o × F of the rings of in tegers of F under this map is a lattice L in the hyperp lane H : x 1 + · · · + x s + 2 y 2 + · · · + 2 y t = 0, i.e., the k ernel of the transfer homomorp hism (45). Therefore, the Eu clidean v olume of this lattice in H is giv en b y V ol ( L ) = √ s +4 t 2 t R, wh ere R is the classical regulator of F . Appendix A. The Bor e l/Esnaul t-Viehwe g v ersion Giv en any equ iv arian t cohomology theory h ∗ on S -spac es, one can d eﬁne its corresp ondin g Bor el version h ∗ bor as h ∗ bor ( U ) := h ∗ ( U × E S ) , wh ere E S is a con tractible S − C W -complex on which S acts freely . In p articular, one has an asso ciated Borel cohomology theory H n,p bor ( X, A ). T h is theory is (0 , 2) p er io dic and can b e more easily calculated than Br ed on cohomology , via Lera y-Serre sp ectral sequences E r,s ( p ) := H r ( S , H s ( X ; A ( p ))) ⇒ H r + s,p bor ( X ( C ) , A ) . It is easy to see that the natur al map Z 0 ( S p,p ) → F ( E S , Z 0 ( S p,p )) is an equiv arian t homotopy equiv alence, thus giving the follo win g r esult. Prop osition A.1. L et Y b e a S - sp ac e. F or al l p ≥ 0 and n ≤ p one has a natur al isomorphism H n,p Br ( Y , A ) ∼ = H n,p b or ( Y , A ) . In order to translate this constr u ction int o our con text, denote X := (Sp ec C ) / R and let E • S := N ( X → Sp ec R ) b e the nerve of the co v er X → Sp ec R . T his is a smo oth simplicial real p ro jectiv e v ariet y with the prop erty that E • S ( C ) is a simplicial ob j ect in A n / R whose geometric re- alizati on | E • S ( C ) | is a mo del for E S . In particular, Z ( E • S ( C )) deﬁnes a simplicial ab elian presheaf on S - M an whose asso ciated complex (graded in negativ e degrees) is den oted Z ( E S ) ∗ INTEGRAL DELIGNE COHOMOLOGY FOR REAL V ARIETIES 31 Deﬁnition A.2. Giv en a complex of presheav es F ∗ on S - M an, deﬁne its asso ci ate d Bor el c omplex as F ∗ bor := H om ( Z ( E S ) ∗ , F ∗ ) , and let ι F ∗ : F ∗ → F ∗ bor denote the natural map ind uced by the pro jection E • S → Sp ec R . In p articular, giv en p ∈ Z one can deﬁne the Bor el version of the De lig ne c omplex as A ( p ) bor D / R :=  A ( p ) D / R  bor and the Borel v ersion of the Bredon complex as A ( p ) bor B r . Corresp ondingly , giv en X ∈ A n / R deﬁne its Bor el version of D eligne c ohomolo gy as H i D / R , b or ( X ; A ( p )) := ˇ H i ( X eq ; A ( p ) bor D / R ) , and the Borel v ersion of Bredon cohomology as H i,p bor ( X, A ) := ˇ H i ( X eq ; A ( p ) bor B r ) It follo ws from the deﬁnitions that A ( p ) bor D / R = Con e  A ( p ) bor B r ⊕ F p E ∗ bor → E ∗ bor  [ − 1] . Prop osition A.3. Given p ∈ Z one has map of exact triangles on A n / R : A ( p ) D / R / /   A ( p ) B r ⊕ F p E ∗ / /   E ∗   / / A ( p ) D / R [1]   A ( p ) b or D / R / / A ( p ) b or B r ⊕ F p E ∗ b or / / E ∗ b or / / A ( p ) b or D / R [1] Corollary A.4. L et X b e a r e al analytic manifold suc h that X S = ∅ . Then ι : H i D / R ( X ; A ( p )) → H i D / R , b or ( X ; A ( p )) is an isomorphism for al l i and p . Pr o of. Since m ultiplication by 2 is in v ertible in E ∗ and preserve s the ﬁltra- tion { F p E ∗ } one concludes that ι : F p E ∗ → F p E ∗ bor is a qu asi-isomorphism for all p ∈ Z . The result no w follo w s from the ﬁ v e-lemma and th e f act the same result h olds for Bredon cohomology .  Remark A.5. W hen X is a pro jectiv e smo oth real v ariet y , the asso ci - ate d Bor el version of the D eligne c ohomolo gy gr oups H i D / R , b or ( X ; A ( p )) de- ﬁned ab o ve coincide with the Deligne cohomology for real v arieties in tro- duced b y Es nault and Vieh w eg in [EV88]. How ev er, in general, the group s H i D / R ( X ; A ( p )) and H i D / R , b or ( X ; A ( p )) are rather d istinct. Appendix B. Proof of The o rem 5.10 W e will need the f ollo wing tw o tec hnical lemmas. Lemma B.1. L et ı : Z → E 0 denote the natur al inclusion of she aves on S - M an . Given any r e al nu mb er λ one c an ﬁnd a map of pr eshe aves ξ λ : Z → Z (2) 0 B r such that for e ach U ∈ S - M an , the c omp osition Z ( U ) ξ λ − → Z (2) 0 B r ( U ) τ − → E 0 ( U ) c oincides with λ · ı . 32 DOS SANTOS AND LIMA-FILHO Pr o of. Let a > 0 b e a p ositiv e real num b er and let I a denote the interv al [ a, 1] if a < 1 and [1 , a ] if a ≥ 1. Let φ a,i : ∆ 2 → R × × R × ⊂ C × × C × , i = 1 , 2 b e smo oth maps that give an orien ted triangulation of the rectangle I a × I a ⊂ R × × R × . It follo ws that Z ∆ 2 ( φ ∗ a, 1 ω 2 + φ ∗ a, 2 ω 2 ) = (log a ) 2 . Let p U : U → ∗ denote the pro jection to the p oint, where U is a S - manifold. Giv en ν ∈ Z ( U ), deﬁn e ξ λ ( ν ) ∈ Z (2) 0 B r ( U ) by ξ λ ( ν ) := ( ( p ∗ U φ a, 1 + p ∗ U φ a, 2 ) ⊗ ν , if λ > 0 and a = exp √ λ − ( p ∗ U φ a, 1 + p ∗ U φ a, 2 ) ⊗ ν , if λ < 0 and a = exp p | λ | . It is clear that ξ λ is a h omomorp hism satisfying the d esired conditions.  Lemma B .2 (The p erio d argumen t) . Given α ∈ Z ( p ) B r ( U ) 0 satisfying d B α = 0 , then τ ( α ) ∈ Z ( p ) ( U ) . In other wor ds, τ ( α ) ∈ E 0 ( U ) is a lo c al ly c onstant e qu ivariant function with values i n Z ( p ) . Pr o of. The co cycle α is r epresen ted by an elemen t of the form ( a, P i f i ⊗ ν i ), w h ere f i : S i × ∆ p → ( C × ) p is smo oth and equiv arian t, p i : S i → U is an equiv arian t co vering map, and ν i : S i → Z is equiv ariant and locally constan t. Since dτ ( α ) = τ ( d B α ) = 0, we kn o w that τ ( α ) is an equiv arian t lo cally constan t function. Giv en x 0 ∈ U and y ∈ p − 1 i ( x 0 ) ⊂ U i the restriction of f i to y × ∆ p is a smo oth prop er map, and hence one obtains a smo oth in tegral p -simplex f i # [ y × ∆ p ] in ( C × ) p with b oundary ∂ f i # [ y × ∆ p ] = f i # [ y × ∂ ∆ p ] . It follo ws that T α,x 0 := X i X y ∈ π − 1 i ( x 0 ) ν i ( y ) f i # [ y × ∆ p ] is a smo oth inte gral p -cycle on ( C × ) p and a simple insp ection sho ws that R T α,x 0 ω p = τ ( α )( x 0 ). On the other hand , since T α,x 0 represent s an integ ral homology class in ( C × ) p , then R T α,x 0 ω p is a p erio d of ω p o ver an in tegral homology class and hence it lies in Z ( p ).  B.1. C o cycles for Bredon cohom ology. As a p reparation to the main argumen ts of n ext section, we describ e th e isomorphism (47) Φ : ˇ H 2 ( X ( C ) eq ; Z (2) B r ) → ˇ H 1 ( X ( C ) eq ; G 0 → G 1 ) in terms of ˇ Cec h co cycles; cf. Prop osition 5.3. If U is a S -manifold, w e denote by U triv the same space with th e trivial S -actio n. Giv en in tegers n ≥ j ≥ 0 let D n,j ⊂ ( n − j ) · 1 ⊕ j · ξ denote th e unit ball in R n with the action in duced by the repr esen tation. W e sa y th at a S -manifold U has T yp e S if it is equiv arian tly isomorphic to ( D n, 0 ) triv × S . W e sa y that U has Ty p e P if U ∼ = D n,j , for some j ≥ 0. Let Y b e a S -manifold of d imension n . A go o d c over for Y is an op en co ve r V = { V α | α ∈ Λ } suc h that all non-empt y intersectio ns are contract ible. W e INTEGRAL DELIGNE COHOMOLOGY FOR REAL V ARIETIES 33 ma y ev en assume that these in tersections are homeomorphic to disks. W e sa y that V is e quivariantly go o d if the group p erm utes th e op en sets in the co ver. A cov er w ith these prop erties alw ays yields an equiv arian t go o d co ver U , i.e. a co v er by S -in v arian t op en sets ha ving the prop ert y that all element s U α 0 ··· α k := U α 0 ∩ · · · ∩ U α k in the n er ve of U ha v e either Typ e S or T yp e P . F urtherm ore, if one of the elemen ts α 0 , . . . , α k has Typ e S , then the in tersectio n α 0 · · · α k is r equired to ha ve Type S . Also, if all α 0 , . . . , α k ha v e T yp e P , then intersec tion α 0 · · · α k m ust also ha ve Type P . Abusing language, w e sa y that the ind ex α 0 · · · α k has Type S or T yp e P , accordingly . Remark B.3. Using totally con v ex balls for a Riemannian metric so that σ acts via isometries, one sees that an y S -manifold Y has an equiv arian t go o d co v er. Also, th ese co v er s form a coﬁnal family amongst the family of all equiv ariant co ve rs of Y . Lemma B.4 (Th e lo cal obstruction argument ) . L et Y b e a S -manifold whose p ath-c omp onents ar e al l c ontr actible. Then, for al l p ≥ 0 , the c omplex Z ( p ) B r ( Y × S ) is acyclic, i.e. H j ( Z ( p ) B r ( Y × S ) ) = 0 for al l j 6 = 0 . In p articular, Z ( p ) B r ( U ) is acyclic if U is a S -manifold of T yp e S and Z ( p ) B r ( U × S ) is acyclic if U has T yp e P . Pr o of. Using the isomorph ism U × S ∼ = U triv × S in Remark B.3(ii), pic k one p oint in eac h path comp onent of Y and obtain an equiv arian t strong deformation retractio n Y × S ≃ π 0 ( Y ) × S , wh ere π 0 ( Y ) is giv en the d iscrete top ology . It follo ws from P rop osition 2.4 that one has a quasi-isomorphism Z ( p ) B r ( Y × S ) ≃ Z ( p ) B r ( π 0 ( Y ) × S ). Now, the cohomology of th e latter complex give s the b igraded Bredon cohomolog y groups H ∗ ,p Br ( π 0 ( Y ) × S , Z ) ∼ = H ∗ sing ( π 0 ( Y ); Z ( p )). The result follo w s.  Let U := { U α | λ ∈ Λ } b e an “equiv arian t goo d co v er” of Y and let h =  ( h i α 0 ··· α j ) | i + j = 2 , j ≥ 0  b e a ˇ Cec h cocycle repr esen ting an element [ h ] ∈ H 2 ( Y eq ; Z (2) B r ) . The co cycle condition giv es: (48) d B h 2 α 0 = 0 and δ h i α 0 ··· α j = ( − 1) j d B h i − 1 α 0 ··· α j +1 for all i ≤ 1 , where d B and δ are th e diﬀeren tials in the Bredon and ˇ Cec h complexes, resp ectiv ely . MAIN GO AL: We wil l ﬁnd a r epr esentative (( g α 0 α 1 ) , ( ρ α 0 )) ∈ ˇ C ( U , G 0 → G 1 ) for Φ ([ H ]) , satisfying: δ ( g α 0 α 1 ) = 1;  g α 0 α 1 · σ ∗ g α 1 α 1  = δ ( ρ α 0 ); ( σ ∗ ρ α 0 ) = ( ρ α 0 ) . 34 DOS SANTOS AND LIMA-FILHO Remark B.5. Let [ h ] b e as ab o ve. G iv en a ﬁ xed p oin t x 0 ∈ Y S , let U α 0 b e an element of the co v er (necessarily of T yp e P ) cont aining x 0 . Since Z ( p ) B r has homotop y inv arian t cohomolo gy preshea v es, one has natural iso- morphisms H j ( Z (2) B r ( U α 0 ) ∼ = B j, 2 . It follo ws from (48) that h 2 α 0 represent s a class in H 2 ( Z (2) B r ( U α 0 )) ∼ = B 2 , 2 ∼ = Z × , thus giving an elemen t in Z × . Notice that this elemen t is the same for any p oint x ∈ U α 0 ∩ Y S and it is easy to s ee that it dep ends only on the class [ h ] ∈ H 2 , 2 Br ( Y S , Z ) . Hence, the r esulting m ap S → Z × dep end s only on [ h ], and this is an additional description of the signature m ap in terms of ˇ Cec h-Bredon co cycles. The ident it y δ ( h 0 α 0 α 1 α 2 ) = ( d B h − 1 α 0 ··· α 3 ) gives δ ( τ h 0 α 0 α 1 α 2 ) = 0 and hence, since E 0 is a soft sheaf, one can ﬁn d f 0 α 0 α 1 ∈ E 0 ( U α 0 α 1 ) suc h that (49) ( τ h 0 α 0 α 1 α 2 ) = δ ( f 0 α 0 α 1 ) . Let p : Y × S → Y denote the p ro jection. Giv en any α 0 · · · α j in the n erv e of the cov ering U , w e use th e same notation p : U α 0 ··· α j × S → U α 0 ··· α j to denote th e corresp onding pro jection. F or an y presheaf F on S - M an and h ∈ F ( U α 0 ··· α j ) let p ∗ h ∈ F ( U α 0 ··· α j × S ) denote the pull-bac k of h under p . T yp e S case: Recall that if α k is of Typ e S , for some k = 1 , . . . , j , then so is U α 0 ··· α j . Step 1: If α 0 is of T yp e S , then the lo c al obstruction ar gument (Lemma B.4) implies th at one can ﬁnd t 1 α 0 ∈ Z (2) 1 B r ( U α 0 ) such that (50) d B t 1 α 0 = h 2 α 0 . Step 2: Assu me that b oth α 0 and α 1 are of Typ e S . S in ce (51) d B  ( h 1 α 0 α 1 ) − δ ( t 1 α 0 )  = δ ( h 2 α 0 ) − δ ( h 2 α 0 ) = 0 , b y the co cycle condition, the lo cal obstruction argum en t guarantee s the existence of t 0 α 0 α 1 ∈ Z (2) B r ( U α 1 α 1 ) such that (52) d B t 0 α 1 α 1 = ( h 1 α 0 α 1 ) − δ ( t 1 α 0 ) . Step 3: Here we only consider α 0 · · · α j in the n erv e of the co v er U for wh ic h all α k ’s are of Typ e S . O bserve that d B  ( h 0 α 0 α 1 α 2 ) + δ ( t 0 α 0 α 1 )  = − δ ( h 1 α 0 α 1 ) + δ ( d B t 0 α 0 α 1 ) = − δ ( h 1 α 0 α 1 ) + δ (( h 1 α 0 α 1 ) − δ ( t 1 α 0 )) = 0 . Applying τ one obtains d  ( τ h 0 α 0 α 1 α 2 ) + δ ( τ t 0 α 0 α 1 )  = 0, and hence L emm a B.2 (the p erio ds ar gument ) sho ws that ( ν α 0 α 1 α 2 ) := ( τ h 0 α 0 α 1 α 2 ) + δ ( τ t 0 α 0 α 1 ) consists of lo cally constan t equiv arian t functions ν α 0 α 1 α 2 : U α 0 α 1 α 2 → Z (2). Using (49 ) one wr ites (53) ( ν α 0 α 1 α 2 ) = δ  f 0 α 0 α 1 + τ t 0 α 0 α 1  . INTEGRAL DELIGNE COHOMOLOGY FOR REAL V ARIETIES 35 Step 4: Denote ˆ g α 0 α 1 := f 0 α 0 α 1 + τ t 0 α 0 α 1 and choose a square ro ot i of − 1. Deﬁne (54) g α 0 α 1 = exp  1 2 π i ˆ g α 0 α 1  and ρ α 0 = 1 . The co cycle condition (55) g α 0 α 1 g α 1 α 2 g α 2 α 0 = 1 when all α ’s are of Typ e S follo ws from (53). Since ˆ g α 0 α 1 − σ ∗ ˆ g α 0 α 1 = 0 one concludes that (56) g α 0 α 1 · σ ∗ g α 0 α 1 = 1 = ρ β /ρ α on U α 0 α 1 and by deﬁnition (57) σ ∗ ρ α = ρ α . T yp e P case: Recall that if α k is of Typ e P , for all k = 1 , . . . , j , then so is U α 0 ··· α j . Step 1: If α 0 is of T yp e P , then the lo cal obstru ction argument (Lemma B.4) giv es some ˆ t 1 α 0 ∈ Z (2) 1 B r ( U α 0 × S ) such that d B ˆ t 1 α 0 = p ∗ h 2 α 0 . Giv en any α 0 deﬁne (58) T 1 α 0 = ( p ∗ t 1 α 0 , if α 0 is of Typ e S ˆ t 1 α 0 , if α 0 is of Typ e P , and observ e that d B T 1 α 0 = p ∗ h 2 α 0 . Step 2: If either α 0 or α 1 is of Typ e P then (59) d B  ( p ∗ h 1 α 0 α 1 ) − δ ( T 1 α 0 )  = 0 . Since B 1 , 2 = 0, it follo ws from the lo cal obstruction argumen t (see Remark B.5) that one obtains ˆ t 0 α 0 α 1 ∈ Z (2) B r ( U α 0 α 1 × S ) such that (60) d B ˆ t 0 α 0 α 1 = ( p ∗ h 1 α 0 α 1 ) − δ ( T 1 α 0 ) . Step 3: Here w e consid er th ose α 0 · · · α j in the nerve of th e co ver U for whic h one of the α k ’s is of Typ e P . In this case one has d B  ( p ∗ h 0 α 0 α 1 α 2 ) + δ ( ˆ t 0 α 0 α 1 )  = 0 . Applying τ one obtains d  ( τ p ∗ h 0 α 0 α 1 α 2 ) + δ ( τ ˆ t 0 α 0 α 1 )  = 0. It follo ws from Lemma B.2 that ( ν α 0 α 1 α 2 ) := ( τ p ∗ h 0 α 0 α 1 α 2 )+ δ ( τ ˆ t 0 α 0 α 1 ) consists of equiv ariant lo cally constant fun ctions U α 0 α 1 α 2 × S → Z (2). In view of Remark 2.9 w e can also consider ν α 0 α 1 α 2 as a non-equiv arian t lo cally constant function from U α 0 α 1 α 2 to Z (2). Step 4: Let ı : U × S → U × S b e as in Remark 2.9. O bserv e that (61) d B  ı ∗ T 1 α 0 − T 1 α 0  = ı ∗ d B T 1 α 0 − d B T 1 α 0 = h 2 α 0 ı ∗ p ∗ h 2 α 0 − h 2 α 0 = 0 . It follo w s from the lo c al obstruction ar gument that (62) ı ∗ T 1 α 0 − T 1 α 0 = d B ˜ γ 0 α 0 , 36 DOS SANTOS AND LIMA-FILHO for some ˜ γ 0 α 0 ∈ Z (2) 0 B r ( U α 0 × S ). In particular, one has d B  ˜ γ 0 α 0 + ı ∗ ˜ γ 0 α 0  = 0, and the p erio d ar gument sho ws that (63) τ  ˜ γ 0 α 0  + ı ∗ τ  ˜ γ 0 α 0  = ˜ C α 0 , for some equiv ariant lo cally constant function ˜ C α 0 on U α 0 × S with v alues in Z (2). Deﬁne ˜ g α 0 α 1 := p ∗ f 0 α 0 α 1 + τ ˆ t 0 α 0 α 1 . Th en ( ˜ g α 0 α 1 ) − ı ∗ ( ˜ g α 0 α 1 ) − δ ( τ ( ˜ γ 0 α 0 )) = τ ˆ t 0 α 0 α 1 − ı ∗ τ ˆ t 0 α 0 α 1 − δ ( τ ( ˜ γ 0 α 0 )) (64) = τ  ˆ t 0 α 0 α 1 − ı ∗ ˆ t 0 α 0 α 1 − δ ( ˜ γ 0 α 0 )  . On the other hand , equations (60) and (62) give d B  ˆ t 0 α 0 α 1 − ı ∗ ˆ t 0 α 0 α 1 − δ ( ˜ γ 0 α 0 )  = δ  − T 1 α 0 + ı ∗ T 1 α 0 − d B ( ˜ γ 0 α 0 )  (65) = 0 , and h ence Lemma B.2 s h o ws that ( ˜ g α 0 α 1 ) − ı ∗ ( ˜ g α 0 α 1 ) − δ ( τ ( ˜ γ 0 α 0 )) consists of equiv arian t lo cally constan t functions fr om U α 0 α 1 × S to Z (2). Using the notation in Remark 2.9, deﬁn e ˆ g α 0 α 1 = F ( ˜ g α 0 α 1 ) = f 0 α 0 α 1 + F ( τ ( ˆ t 0 α 0 α 1 )), and γ 0 α 0 := F τ ( ˜ γ 0 α 0 ). With the s ame choice of square r o ot i of − 1 as in the Type S case, deﬁn e (66) g α 0 α 1 = exp  1 2 π i ˆ g α 0 α 1  and ρ α 0 = exp  1 2 π i γ 0 α 0  . Step 3 shows that (67) g α 0 α 1 g α 1 α 2 g α 2 α 0 = 1 , and (64) toge ther with sub sequen t remarks and prop erties of the functor F sho w that (68) g α 0 α 1 · σ ∗ g α 0 α 1 = ρ α 1 /ρ α 0 on U α 0 α 1 . Finally , (63) and R emark 2.9 giv e F ( τ ( ˜ γ 0 α 0 )) + σ ∗ F ( τ ( ˜ γ 0 α 0 )) = F ( ˜ C α 0 ) := C α 0 . In particular, C α 0 is an equiv arian t locally constan t function on U α 0 with v alues in Z (2), and hence  1 2 π i γ 0 α 0  − σ ∗  1 2 π i γ 0 α 0  = 1 2 π i  F ( τ ( ˜ γ 0 α 0 )) + σ ∗ F ( τ ( ˜ γ 0 α 0 ))  = 1 2 π i C α 0 ∈ Z (1) ( U α 0 ) . This giv es (69) σ ∗ ρ α 0 = ρ α 0 . In other w ords (( g α 0 α 1 ) , ( ρ α 0 )) d eﬁnes a 1-cocycle in ˇ C ∗ ( U , G 0 → G 1 ). It is easy to c hec k th at this pr o cess w ould send a ˇ Cec h cob ou n dary to a cob ound - ary in ˇ C ∗ ( U , G 0 → G 1 ), realizing the isomorphism (47). INTEGRAL DELIGNE COHOMOLOGY FOR REAL V ARIETIES 37 B.2. T he pro of. W e now pro ve Theorem 5.10. Pr o of. Let U b e an equiv ariant go o d co ver of X ( C ) and ﬁx a co cycle c = ( c r,s ) r + s =2 ∈ ˇ C 2 ( U ; Z (2) D / R ) representing a class in H 2 D / R ( X ; Z (2)). Since c r,s = 0 for r > 2, the co cycle condition is giv en b y (70) D c 2 , 0 = 0 and D c r − 1 ,s +1 = ( − 1) s δ c r,s , r ≤ 1 . Recall that Z (2) D / R = Con e  Z (2) B r ⊕ F 2 E ∗ ι 2 − → E ∗  [ − 1]. W rite c 2 , 0 =  ( h 2 α 0 ) , ( ω 2 α 0 ) , ( θ 1 α 0 )  (71) c 1 , 1 =  ( h 1 α 0 α 1 ) , 0 , ( θ 0 α 0 α 1 )  (72) c 0 , 2 =  ( h 0 α 0 α 1 α 2 ) , 0 , 0  (73) c − i, 2+ i =  ( h − i α 0 ...α 2+ i ) , 0 , 0  , i ≥ 1 . (74) Since D c 2 , 0 = 0, one h as for all α 0 ∈ Λ : d B h 2 α 0 = 0 , (b y deﬁnition) ; (75) dω 2 α 0 = 0; (76) τ ( h 2 α 0 ) − ω 2 α 0 + dθ 1 α 0 = 0 . (77) Similarly , D c 1 , 1 = δ c 2 , 0 giv es th e iden tities: ( d B h 1 α 0 α 1 ) = δ ( h 2 α 0 ) (78) 0 = δ ( ω α 0 ) (79) − ( τ h 1 α 0 α 1 ) − ( dθ 0 α 0 α 1 ) = δ ( θ 1 α 0 ) , (80) D c 0 , 2 = − δ c 1 , 1 giv es: ( d B h 0 α 0 α 1 α 2 ) = − δ ( h 1 α 0 α 1 ) (81) ( τ h 0 α 0 α 1 α 2 ) = δ ( θ 0 α 0 α 1 ) (82) and for all r ≤ 0 one h as : ( d B h − r − 1 α 0 ...α 3+ r ) = ( − 1) r δ ( h − r α 0 ...α r +2 ) . (83) The assignment c := ( c r,s ) 7→ h := ( h i α 0 ··· α j ) giv es the cycle map from Deligne to Bredon cohomolo gy . Assume that c ∈ k er Ψ, hence Ψ([ c ]) = ℵ ([ h ]) = 0. Therefore the ˇ Cec h -Bredon co cycle h is “unobstru cted” and w e can apply the argument s in Type S Case ab o v e; see (50). F urth er m ore, the data in th e ˇ Cec h -Deligne complex giv es a natural choice for the f 0 α 0 α 1 in tro duced in (49 ). More precisely , one can choose f 0 α 0 α 1 := θ 0 α 0 α 1 ; cf. (82) . It follo ws that one can tak e g α 0 α 1 := exp  1 2 π i ˆ g α 0 α 1  , with (84) ˆ g α 0 α 1 = θ 0 α 0 α 1 + τ ( t 0 α 0 α 1 ) , to obtain a co cycle f or the line b undle asso ciated to [ h ]; cf. (54) . Ho we v er, one can ﬁn d an equiv alent holomorphic co cycle as follo ws. 38 DOS SANTOS AND LIMA-FILHO W rite the 1-form a α 0 := θ 1 α 0 + τ t 1 α 0 = a 1 , 0 α 0 + a 0 , 1 α 0 as a sum of their (1 , 0) and (0 , 1) parts, resp ectiv ely , where t 1 α 0 α 1 is introdu ced in (50). Since da α 0 = d  θ 1 α 0 + τ t 1 α 0  = dθ 1 α + dτ ( h 2 α 0 ) = ω 2 α 0 , cf. (50) and (77) , and ω 2 α 0 is a form of t yp e (2 , 0) one concludes that ¯ ∂ a 0 , 1 α 0 = 0 (85) ∂ a 0 , 1 α 0 = − ¯ ∂ a 1 , 0 α 0 (86) ∂ a 1 , 0 α 0 = ω 2 α 0 (87) It follo ws fr om the ¯ ∂ -P oincar ´ e lemma that one can ﬁnd f 0 α 0 ∈ E 0 ( U α 0 ) (equi- v arian t) such that (88) ¯ ∂ f 0 α 0 = a 0 , 1 α 0 . No w deﬁ n e (89) (˜ g α 0 α 1 ) := ( ˆ g α 0 α 1 ) + δ ( f 0 α 0 ) . Hence, ¯ ∂ ˜ g α 0 α 1 = { d ˆ g α 0 α 1 } 0 , 1 + δ ( ¯ ∂ f 0 α 0 ) =  dθ 0 α 0 α 1 + τ ( d B t 0 α 0 α 1 )  0 , 1 + ¯ ∂ f 0 α 0 =  dθ 0 α 0 α 1 + τ ( h 1 α 0 α 1 ) − δ ( t 1 α 0 )  0 , 1 + δ ( ¯ ∂ f 0 α 0 ) =  − δ ( θ 1 α 0 ) − ( τ h 1 α 0 α 1 ) + ( τ h 1 α 0 α 1 ) − δ ( t 1 α 0 )  0 , 1 + δ ( ¯ ∂ f 0 α 0 ) = − δ  θ 1 α 0 + τ t 1 α 0  0 , 1 + δ ( ¯ ∂ f 0 α 0 ) = − δ ( a 0 , 1 α 0 ) + δ ( ¯ ∂ f 0 α 0 ) = 0 , cf. (88). Deﬁning (90) g α 0 α 1 := exp  1 2 π i ˜ g α 0 α 1  one obtains a holomorphic stru cture ( g α 0 α 1 ) for L . See Remark 5.9(i). No w, deﬁne (91) ψ α 0 := 1 2 π i  ∂ f α 0 − a 1 , 0 α 0  = 1 2 π i  d f α 0 − θ 1 α 0 − τ t 1 α 0  . INTEGRAL DELIGNE COHOMOLOGY FOR REAL V ARIETIES 39 It f ollo ws f r om (88) and (86) that ψ α 0 is a holomorphic 1-form. F urth er m ore,  dg α 0 α 1 g α 0 α 1  = 1 2 π i { ( d ˆ g α 0 α 1 ) + δ ( d f α 0 ) } = 1 2 π i  ( dθ 0 α 0 α 1 ) + ( τ d B t 0 α 0 α 1 ) + δ ( ∂ f α 0 ) + δ ( ¯ ∂ f α 0 )  = 1 2 π i  − δ ( θ 1 α 0 ) − ( τ h 1 α 0 α 1 ) + τ  ( h 1 α 0 α 1 ) − δ ( t 1 α 0 )  + δ ( ∂ f α 0 ) + δ ( ¯ ∂ f α 0 )  = 1 2 π i      − δ (( θ 1 α 0 ) + ( τ t 1 α 0 ) | {z } a α 0 ) + δ ( ∂ f α 0 ) + δ ( a 0 , 1 α 0 )      = 1 2 π i  δ ( ∂ f α 0 − a 1 , 0 α 0 )  = δ ( ψ α 0 ) . See Remark 5.9(ii). Remark B.6. It follo w s f r om (91), (50) and (77) that dψ α 0 = − 1 2 π i  dθ 1 α 0 + dτ ( t 1 α 0 )  = − 1 2 π i  dθ 1 α 0 + τ ( d B t 1 α 0 )  = − 1 2 π i  dθ 1 α 0 + τ ( h 2 α 0 )  = − 1 2 π i ω 2 α 0 . Therefore, ( ψ α 0 ) deﬁnes a holomorphic connection ∇ on the holomorphic line bundle L associated to ( g α 0 α 1 ). Since b oth ˆ g α 0 α 1 and f α 0 are equiv ariant functions, it follo ws that σ ∗ g α 0 α 1 · g α 0 α 1 = 1 , and th is deﬁn es a holomorphic isomorphism q : L ⊗ σ ∗ L → 1 whic h b ecomes a p ositiv e deﬁnite hermitian f orm on X ( R ). (See R emark 5.9(iii).) Finally , the id en tit y σ ∗ ψ α 0 + ψ α 0 = 0 shows that q is p arallel with resp ect to the connection on L ⊗ σ ∗ L indu ced b y ∇ . S ee Remark 5.9(iv). W e ha v e thus asso ciated to c , with [ c ] ∈ k er Ψ, a triple ( L, q , ∇ ) of ele- men ts satisfying the conditions in Deﬁnition 5.7. This give s a wel l-deﬁned homomorphism (92) Φ : k er Ψ − → P W ∇ ( X ) . W e no w pro ceed to s ho w that this is in fact an isomorp h ism. 40 DOS SANTOS AND LIMA-FILHO Lemma B.7. Ther e is a natur al tr ansformation Θ : P W ∇ ( − ) → H 2 , 2 Br ( − , Z ) such that for al l U ∈ S - M an the fol lowing diagr am c ommutes k er Ψ U i / / Φ   H 2 D / R ( U ; Z (2))    P W ∇ ( U ) Θ U / / H 2 , 2 Br ( U, Z ) . Pr o of. Using the int erpretation of H 2 , 2 Br ( U, Z ) as equiv alence classes [ L, q ] of pairs as in Prop osition 5.3, then Θ U simply send s h L, ∇ , q i to [ L, q ].  Injectivit y of Φ : Let c ′ = ( h ′ , ω ′ , θ ′ ) b e a ˇ Cec h co cyle repr esen ting a class [ c ′ ] ∈ ker Ψ U , and supp ose that Φ U ([ c ′ ]) = 0. Since 0 = Θ U ◦ Φ U ([ c ′ ]) =  ◦ i ([ c ]), one concludes that [ h ′ ] = 0 ∈ H 2 , 2 Br ( U, Z ) an d hence, one can ﬁnd t ∈ T ot( ˇ C ∗ ( U , Z (2) B r )) 1 suc h that ˇ d B t = h . It follo ws that c := c ′ − ˇ D ( t , 0 , 0) = (0 , ω , θ ) repr esen ts [ c ′ ] and is simply given by ( ω 2 α 0 ) , ( θ 0 α 0 α 1 ) and ( θ 1 α 0 ), where ω 2 α 0 ∈ F 2 E 2 ( U ), θ 0 α 0 α 1 ∈ E 0 ( U α 0 α 1 ) and θ 1 α 0 ∈ E 1 ( U α 0 ). Let ( g α 0 α 1 ) , ( ψ α 0 ) b e constru cted as in (9 0) and (91), represent ing Φ([ c ]), and observe that h = 0 allo ws one to tak e t 1 α 0 = 0, t 0 α 0 α 1 = 0 and f 0 α 0 α 1 = 0 (cf. S TEPS 1 and 2, and (49)) in their deﬁn ition. Assuming that Φ([ c ]) = 0, one can ﬁn d ( ρ α 0 ) satisfying: ρ α 0 · σ ∗ ρ α 0 = 1 , and ρ α 0 is holomorphic (93) ( g α 0 α 1 ) = δ ( ρ α 0 ) (94) ψ α 0 = dρ α 0 ρ α 0 . (95) The latter equation giv es 0 = dψ α 0 = − 1 2 π i ω 2 α 0 ; cf. Remark B.6. No w, the co cycle condition on c = (0 , 0 , θ ) b oils do wn to ( dθ 1 α 0 ) = 0 (96) ( dθ 0 α 0 α 1 ) + δ ( θ 1 α 0 ) = 0 (97) δ ( θ 0 α 0 α 1 ) = 0 . (98) Cho ose ˆ ρ α 0 suc h that exp ˆ ρ α 0 = ρ α 0 and ˆ ρ α 0 + σ ∗ ˆ ρ α 0 = 0, and let f 0 α 0 b e as in (88). Hence, b y deﬁn ition one has ( g α 0 α 1 ) =  exp 1 2 π i  ( θ 0 α 0 α 1 ) + δ ( f 0 α 0 ))   = δ ( ρ α 0 ) and (99) ψ α 0 = 1 2 π i  df 0 α 0 − θ 1 α 0  = dρ α 0 ρ α 0 = d ˆ ρ α 0 . It follo w s that one can ﬁnd n α 0 α 1 ∈ Z ( U α 0 α 1 ) such that (100) 1 2 π i  ( θ 0 α 0 α 1 ) + δ ( f 0 α 0 ))  = δ ( ˆ ρ α 0 ) + (2 π i n α 0 α 1 ) . INTEGRAL DELIGNE COHOMOLOGY FOR REAL V ARIETIES 41 Using Lemma B.1 deﬁne (101) ξ α 0 α 1 := ξ (2 πi ) 2 ( n α 0 α 1 ) ∈ Z (2) 0 B r ( U α 0 α 1 ) and deﬁne (102) b 0 α 0 = 2 π i ˆ ρ α 0 − f 0 α 0 ∈ E 0 ( U α 0 ) . It is easy to see that δ ( ξ α 0 α 1 ) = 0 and that (100) and (102) give (103) τ ( ξ α 0 α 1 ) + δ ( b 0 α 0 ) = ( θ 0 α 0 α 1 ) . One obtai ns a 1-co c hain t := (( ξ α 0 α 1 ) , 0 , ( b 0 α 0 )) ∈ T ot  ˇ C ∗ ( U , Z (2) B r )  1 that satisﬁes ˇ D ( t ) = c ; cf. (99), (102), (103) and (91). Therefore [ c ] = 0, th us showing the injectivit y of Φ. Surjectivit y of Φ : Represen t h L, ∇ , q i ∈ P W ∇ ( X ) b y a cocycle ( G α 0 α 1 ) , ( ψ α 0 ) satisfying the conditions of R emark 5.9, and let h ∈ T ot  ˇ C ∗ ( U ; Z (2) B r )  2 b e a co cycle represent ing the element Θ( h L, ∇ , q i ) = [ L, q ] ∈ H 2 , 2 Br ( X, Z ). Note that this cocycle is “unobstru cted”, in the sense of Type S ca se . Hence one can ﬁnd t 1 α 0 , t 0 α 0 α 1 and f 0 α 0 α 1 as in (50), (51) and (49), resp ectiv ely . By deﬁnition, ˆ g α 0 α 1 := f 0 α 0 α 1 + τ ( t 0 α 0 α 1 ) so that g α 0 α 1 := exp  1 2 π i ˆ g α 0 α 1  is a co cycle rep r esen ting the isomorphism cla ss of L as a smo oth line bund le and satisfying the condition g α 0 α 1 · σ ∗ g α 0 α 1 = 1, w h ic h giv es the d esired q , p ositiv e deﬁn ite o ver X ( R ). Therefore, one can ﬁn d smo oth fun ctions ρ α 0 suc h that (104)  G α 0 α 1 g α 0 α 1  = δ ( ρ α 0 ) and ρ α 0 · σ ∗ ρ α 0 = 1 . No w, ﬁnd ˆ G α 0 α 1 and ˆ ρ α 0 suc h that exp ˆ G α 0 α 1 = G α 0 α 1 , exp ˆ ρ α 0 = ρ α 0 and satisfying ˆ G α 0 α 1 + σ ∗ ˆ G α 0 α 1 = 0 , ˆ ρ α 0 + σ ∗ ˆ ρ α 0 = 0. It follo ws fr om (104 ) th at one can ﬁnd n α 0 α 1 ∈ Z ( U α 0 α 1 ) suc h that ( ˆ G α 0 α 1 ) = 1 2 π i { ˆ g α 0 α 1 + δ ( ˆ ρ α 0 ) } + (2 π i ) n α 0 α 1 . Hence: (105) 2 π i ˆ G α 0 α 1 = f 0 α 0 α 1 + τ t 0 α 0 α 1 + δ ( ˆ ρ α 0 ) + (2 π i ) 2 n α 0 α 1 . W ant to ﬁ nd ω α 0 , θ 1 α 0 , θ 0 α 0 α 1 satisfying: dω α 0 = 0 (C.1) δ ( ω α 0 ) = 0 (C.2) δ ( θ 0 α 0 α 1 ) = ( τ h 0 α 0 α 1 α 2 ) (C.3) ( dθ 0 α 0 α 1 ) + δ ( θ 1 α 0 ) + τ ( h 1 α 0 α 1 ) = 0 (C.4) ω α 0 = dθ 1 α 0 + τ h 2 α 0 . (C.5) Note that (C.5) implies (C.1). 42 DOS SANTOS AND LIMA-FILHO It follo ws from (105) that 2 π iδ ( ψ α 0 ) = 2 π id ˆ G α 0 α 1 = d f 0 α 0 α 1 + τ  d B t 0 α 0 α 1  + δ ( d ˆ ρ α 0 ) = d f 0 α 0 α 1 + τ  h 1 α 0 α 1 − δ ( t 1 α 0 )  + δ ( d ˆ ρ α 0 ) = d f 0 α 0 α 1 + τ h 1 α 0 α 1 + δ  − τ t 1 α 0 + d ˆ ρ α 0  . Therefore, (106) 0 = ( d f 0 α 0 α 1 ) + ( τ h 1 α 0 α 1 ) + δ  − 2 π iψ α 0 − τ t 1 α 0 + d ˆ ρ α 0  Deﬁne θ 1 α 0 := − 2 π iψ α 0 − τ t 1 α 0 + d ˆ ρ α 0 (107) θ 0 α 0 α 1 := f 0 α 0 α 1 (108) ω α 0 := − 2 π idψ α 0 , (109) and observ e that the latter is an in v arian t closed form of Ho dge t yp e (2 , 0), since ψ α 0 is holomorphic. W e n o w pro ceed to sh ow th at these forms satisfy (C.1)–(C.5). • dθ 1 α 0 = − 2 πidψ α 0 − τ ( d B t 1 α 0 ) = ω α 0 − τ ( h 2 α 0 ). This gives (C.5) and (C.1), as w ell. • δ ( ω α 0 ) = − 2 π iδ ( dψ α 0 ) = − 2 π idδ ( ψ α 0 ) = − 2 π id  dG α 0 α 1 G α 0 α 1  = − 2 π id ( d ˆ G α 0 α 1 ) = 0. Th is giv es (C.2). • δ ( θ 0 α 0 α 1 ) = δ ( f 0 α 0 α 1 ) = ( τ h 0 α 0 α 1 α 2 ). Th is giv es (C.3). • It follo ws from (1 06) and (107) that 0 = ( d f 0 α 0 α 1 )+ ( τ h 1 α 0 α 1 )+ δ ( θ 1 α 0 α 1 ) whic h, together with (108) , implies (C.4 ). It follo ws that c := ( h , ω , θ ) giv es a co cycle in T ot  ˇ C ∗ ( U ; Z (2) D / R )  2 suc h that [ c ] ∈ ker Ψ. Finally , one needs to v erify that Φ([ c ]) = h L, ∇ , q i . A t this p oin t, this is a mere tautology . F ollo wing the steps in th e deﬁn ition of Φ, one constructs ( γ α 0 α 1 ) , ( ξ α 0 ) repr esen ting Φ([ c ]). W e ﬁrst ﬁnd f 0 α 0 ∈ E 0 ( U α 0 ) suc h that ¯ ∂ f 0 α 0 =  θ 1 α 0 + τ ( t 1 α 0 )  0 , 1 ; cf. (88). Note that, b y deﬁn ition (107), one has θ 1 α 0 + τ ( t 1 α 0 ) = − 2 π iψ α 0 + d ˆ ρ α 0 , and since Ψ α 0 has Hod ge t yp e (1 , 0), one concludes that { θ 1 α 0 + τ ( t 1 α 0 ) } 0 , 1 = ¯ ∂ ˆ ρ α 0 . Hence, we can c ho ose f 0 α 0 = ˆ ρ α 0 . By deﬁn ition, γ α 0 α 1 = exp  1 2 π i { ˆ g α 0 α 1 + δ ( ˆ ρ α 0 ) }  = exp  1 2 π i { ˆ g α 0 α 1 + δ ( ˆ ρ α 0 ) } + (2 π in α 0 α 1 )  = exp( ˆ G α 0 α 1 ) = G α 0 α 1 . INTEGRAL DELIGNE COHOMOLOGY FOR REAL V ARIETIES 43 Also, ξ α 0 = 1 2 π i  ∂ ˆ ρ α 0 − { θ 1 α 0 + τ t 1 α 0 } 1 , 0  = 1 2 π i  d ˆ ρ α 0 − { θ 1 α 0 + τ t 1 α 0 }  = 1 2 π i (2 π iψ α 0 ) = ψ α 0 .  Referen ces [Ati6 6] M. F. Atiya h, K -the ory and r e ali ty , Quart. J. Math. O xford Ser. (2) 17 (1966), 367–386 . [Be ˘ ı84] A. A. Be ˘ ılinson, Higher r e gulators and values of L -functions , Curren t p roblems in mathematics, V ol. 24, I togi Nauk i i T ekh niki, Aka d. N auk SSS R Vsesoyuz. Inst. N auc hn. i T ekh n . Inform., Moscow , 198 4, pp. 181–238. [Blo86] Sp encer Bloch, A lgebr aic cycles and the Be ˘ ılinson c onje ctur es , The Lefschetz centennial conference, P art I (Mexico City , 1984), Contemp. Math., vol. 58, Amer. Math. S oc., Providence, RI, 1986, pp . 65–79. [BO74] Sp encer Bloch and Arthur Ogus, Gersten ’s c onje ctur e and the homolo gy of schemes , A nn. S ci. ´ Ecole Norm. Sup. (4) 7 (1974), 181–201 (1975). [Bre67] Glen E. Bredon, Equivariant c ohomolo gy the ories , Lecture Notes in Mathe- matics, No. 34, Sprin ger-V erlag, Berlin, 1967. [DI04] Daniel Dugger a nd Daniel C. Isaksen, T op olo gic al hyp er c overs and A 1 - r e alizations , Math. Z . 246 (2004 ), no. 4, 667–689. [dS03a] Pedro F. dos San tos, Algeb r aic cycles on r e al varieties and Z / 2 -e quivariant homotopy the ory , Pro c. Lond on Math. Soc. (3) 86 (2003), no. 2, 513–544. [dS03b] , A note on the e quivariant Dol d-Thom the or em , J. Pure Ap p l. Algebra 183 (2003), no. 1-3, 299–312. [dSLF07] P edro F. dos Santos and P aulo L ima-Filho, Bi gr ade d e quivariant c ohomolo gy of r e al quadrics , Preprint, 2007. [ES52] Samuel Eilen b erg and N orman S teenrod , F oundations of algebr ai c top olo gy , Princeton Universit y Press, Princeton, New Jersey , 1952. [EV88] H´ el ` ene Esnault and Eck art Viehw eg, Deli gne-Be ˘ ıl inson c ohomolo gy , Be ˘ ılinson’s conjectures on special val ues of L -functions, Perspect. Math., vol. 4, Academic Press, Boston, MA, 1988, pp. 43–91. [Gil84] Hen ri Gillet, Deligne homolo gy and Ab el-Jac obi maps , Bull. Amer. Math. So c. (N.S.) 10 (1984), no. 2, 285–288. [Jan88] Uw e Jannsen, Deligne homolo gy, Ho dge- D -c onj e ctur e, and motives , Be ˘ ılinson’s conjectures on special val ues of L -functions, Perspect. Math., vol. 4, Academic Press, Boston, MA, 1988, pp. 305–372. [Kar00] N. A . Karp enko, Cohomolo gy of r elative c el lular sp ac es and of isotr opic ﬂag varieties , A lgebra i An aliz 12 (2000), n o. 1, 3–69. [KL07] Matt Kerr and James D. Lewis, The Ab el-Jac obi m ap for higher Chow gr oups. II , I nv ent. Math. 170 (200 7), no. 2, 355–420. [KLMS06] Matt Kerr, James D. Lewis, and Stefan M ¨ uller-Stach, The Ab el-Jac obi map for higher Chow gr oups , Compos. Math. 142 (2006), no. 2, 374–396. [Kra91] V. A. Krasnov, Char acteristic cl asses of ve ctor bund l es on a r e al al gebr aic variety , I zv. Akad. Nauk S SSR Ser. Mat. 55 (1991), no. 4, 716–746. [LLFM03] H. Blaine La wson, Paulo Lima-Filho, and Marie-Louise Mic helsohn, Algebr aic cycles and the classic al gr oups. I . Re al cycles , T op ology 42 (2003), no. 2, 467– 506. [LMM81] G. Lewis, J. P . May , and J. McClure, Or dinary RO ( G ) -gr ade d c ohomolo gy , Bull. Amer. Math. So c. (N .S.) 4 (1981), no. 2, 208–212. 44 DOS SANTOS AND LIMA-FILHO [LMSM86] L. G. Lewis, Jr., J. P . May , M. S teinberger, and J. E. McClure, Equivariant stable homotopy the ory , Lecture Notes in Mathematics, vol. 1213, Sp ringer- V erlag, Berlin, 1986, With contributions by J. E. McClure. [Ma y96] J. P . Ma y , Equi variant homotopy and c ohomolo gy the ory , CBM S Regional Con- ference S eries in Mathematics, vol. 91, Pub lished for the Conference Board of the Mathematical S ciences, W ashington, DC, 1996, With contributions b y M. Cole, G. Co mezana, S. Costenoble, A. D. Elmendorf, J. P . C. Gree nlees, L. G. Lewis, Jr., R . J. Piacenza, G. T riantaﬁllou, and S. W aner. [MV99] F abien Morel and V ladimir V o evodsky , A 1 -homotopy the ory of schemes , I nst. Hautes ´ Etudes Sci. Publ. Math. (1999), n o. 90, 45–143 ( 2001). [MVW06] Carl o Mazza, Vladimir V o evodsky , and Charles W eib el, L e ctur e notes on mo- tivic c ohomolo gy , Cla y Ma thematics Monographs, v ol. 2, A merican Mathemat- ical So ciety , Providence, RI, 2006. [PW91] C. P edrini and C. W eib el, Invariants of r e al curves , R end. Sem. Mat. Univ . P olitec. T orino 49 (1991), no. 2, 139–173 (1993). [SGA72] Th´ eorie des top os et c ohomolo gie ´ etale des sch ´ emas. Tome 1: Th´ eorie des top os , S pringer-V erlag, Berlin, 1972, S´ eminaire de G´ eom´ etrie Alg´ ebrique du Bois-Marie 1963–1964 (SGA 4), Dirig´ e par M. Artin, A . Grothendieck, et J. L. V erd ier. Avec la collab oration de N. Bourbaki, P . D eligne et B. Saint-Donat, Lecture Notes in Mathematics, V ol. 269. [Wit34] E. Witt, Zerle gung r e el er al gebr aischer Funktionen in Quadr ate, Schiefk¨ orp er ¨ ub er r e el lem Funktionenk¨ orp er , J. reine angew. Math. 171 (1934), 4–11. [Y an08] H. Y an g, RO ( G ) -gr ade d e quivariant c ohomol o gy the ory and she aves , Ph.d. the- sis, T exas A& M Universit y , 2008. Dep ar t a m ento d e Ma tem ´ atica, Instituto Superi or T ´ ecnico, Por tugal E-mail addr ess : pedro.f.san tos@math.ist .utl.pt Dep ar tme nt of Ma thema ti cs, Texas A&M University, USA E-mail addr ess : plfilho@mat h.tamu.edu

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