aa r X i v : . [ m a t h . AG ] J a n POLARIZED REAL TORI
JAE-HYUN YANG
Abstract.
For a fixed positive integer g , we let P g = { Y ∈ R ( g,g ) ∣ Y = t Y > } be theopen convex cone in the Euclidean space R g ( g + )/ . Then the general linear group GL ( g, R ) acts naturally on P g by A ⋆ Y = AY t A ( A ∈ GL ( g, R ) , Y ∈ P g ). We introduce a notion ofpolarized real tori. We show that the open cone P g parametrizes principally polarized realtori of dimension g and that the Minkowski domain R g = GL ( g, Z )/P g may be regardedas a moduli space of principally polarized real tori of dimension g . We also study smoothline bundles on a polarized real torus by relating them to holomorphic line bundles on itsassociated polarized real abelian variety. Introduction
For a given fixed positive integer g , we let H g = { Ω ∈ C ( g,g ) ∣ Ω = t Ω , Im Ω > } be the Siegel upper half plane of degree g and let Sp ( g, R ) = { M ∈ R ( g, g ) ∣ t M J g M = J g } be the symplectic group of degree g , where F ( k,l ) denotes the set of all k × l matrices withentries in a commutative ring F for two positive integers k and l , t M denotes the transposematrix of a matrix M and J g = ( I g − I g ) . Then Sp ( g, R ) acts on H g transitively by(1.1) M ⋅ Ω = ( A Ω + B )( C Ω + D ) − , where M = ( A BC D ) ∈ Sp ( g, R ) and Ω ∈ H g . LetΓ g = Sp ( g, Z ) = {( A BC D ) ∈ Sp ( g, R ) ∣ A, B, C, D integral } be the Siegel modular group of degree g . This group acts on H g properly discontinuously.C. L. Siegel investigated the geometry of H g and automorphic forms on H g systematically.Siegel [23] found a fundamental domain F g for Γ g / H g and described it explicitly. Moreover he calculated the volume of F g . We also refer to [10], [14], [23] for some details on F g . Siegel’sfundamental domain is now called the Siegel modular variety and is usually denoted by A g .In fact, A g is one of the important arithmetic varieties in the sense that it is regarded asthe moduli of principally polarized abelian varieties of dimension g . Suggested by Siegel,I. Satake [18] found a canonical compactification, now called the Satake compactificationof A g . Thereafter W. Baily [3] proved that the Satake compactification of A g is a normalprojective variety. This work was generalized to bounded symmetric domains by W. Bailyand A. Borel [4] around the 1960s. Some years later a theory of smooth compactification ofbounded symmetric domains was developed by Mumford school [2]. G. Faltings and C.-L.Chai [7] investigated the moduli of abelian varieties over the integers and could give theanalogue of the Eichler-Shimura theorem that expresses Siegel modular forms in terms ofthe cohomology of local systems on A g . I want to emphasize that Siegel modular forms playan important role in the theory of the arithmetic and the geometry of the Siegel modularvariety A g .We let P g = { Y ∈ R ( g,g ) ∣ Y = t Y > } be an open convex cone in R N with N = g ( g + )/ . The general linear group GL ( g, R ) actson P g transitively by(1.2) A ○ Y ∶ = AY t A, A ∈ GL ( g, R ) , Y ∈ P g . We observe that the action (1.2) is naturally induced from the symplectic action (1.1). Thus P g is a symmetric space diffeomorphic to GL ( g, R )/ O ( g ) . Let GL ( g, Z ) = { γ ∈ GL ( g, R ) ∣ γ is integral } be an arithmetic discrete subgroup of GL ( g, R ) . Using the reduction theory Minkowski [16]found a fundamental domain R g , the so-called Minkowski domain for the action (1.2) of GL ( g, Z ) on P g . In fact, using the Minkowski domain R g Siegel found his fundamentaldomain F g . As in the case of H g , automorphic forms on P g for GL ( g, Z ) and geometry on P g have been studied by many people, e.g., Selberg [20], Maass [14] et al.The aim of this article is to study arithmetic-geometric meaning of the Minkowski domain R g . First we introduce a notion of polarized real tori by relating special real tori to polarizedreal abelian varieties. We realize that P g parametrizes principally polarized real tori ofdimension g and also that R g may be regarded as a moduli space of principally polarizedreal tori of dimension g . We also study smooth line bundles over a polarized real torus byrelating to holomorphic line bundles over the associated polarized abelian variety. Those linebundles over a polarized real torus play an important role in investigating some geometricproperties of a polarized real torus.We let G M ∶ = GL ( g, R ) ⋉ R g be the semidirect product of GL ( g, R ) and R g with multiplication law ( A, a ) ⋅ ( B, b ) ∶ = ( AB, a t B − + b ) , A, B ∈ GL ( g, R ) , a, b ∈ R g . Then we have the natural action of G M on the Minkowski-Euclid space P g × R g defined by(1.3) ( A, a ) ⋅ ( Y, ζ ) ∶ = ( AY t A, ( ζ + a ) t A ) , ( A, a ) ∈ G M , Y ∈ P g , ζ ∈ R g . OLARIZED REAL TORI 3
We let G M ( Z ) = GL ( g, Z ) ⋉ Z g be the discrete subgroup of G M . Then G M ( Z ) acts on P g × R g properly discontinuously. Weshow that by associating a principally polarized real torus of dimension g to each equivalenceclass in R g , the quotient space G M ( Z )/( P g × R g ) may be regarded as a family of principally polarized real tori of dimension g . To eachequivalence class [ Y ] ∈ GL ( g, Z )/ P g with Y ∈ P g we associate a principally polarized realtorus T Y = R g / Λ Y , where Λ Y = Y Z g is a lattice in R g .Let Y and Y be two elements in P g with [ Y ] ≠ [ Y ] , that is, Y ≠ A Y t A for all A ∈ GL ( g, Z ) . We put Λ i = Y i Z g for i = , . Then a torus T = R g / Λ is diffeomorphic to T = R g / Λ as smooth manifolds but T is not isomorphic to T as polarized real tori.The Siegel modular variety A g has three remarkable properties : (a) it is the modulispace of principally polarized abelian varieties of dimension g , (b) it has the structure of aquasi-projective complex algebraic variety which is defined over Q , and (c) it has a canonicalcompactification, the so-called Satake-Baily-Borel compactification which is defined over Q .Unfortunately the Minkowski domain R g does not admit the structure of a real algebraicvariety. Moreover R g does not admit a compactification which is defined over Q . Silhol [26]constructs the moduli space of real principally poarized abelian varieties and he shows thatit is a topological ramified covering of R g . Furthermore Silhol constructs a compactificationof this moduli space analogous to the Satake-Baily-Borel compactification. However, neitherthe moduli space nor this compactification has an algebraic structure. On the other hand,by considering real abelian varieties with a suitable level structure Goresky and Tai [9] showsthat the moduli space of real principally polarized abelian varieties with level 4 m structure ( m ≥ ) coincides with the set of real points of a quasi-projective algebraic variety definedover Q and consists of finitely many copies of the quotient G g ( m )/ P g with a discretesubgroup G g ( m ) of GL ( g, Z ) , where G g ( m ) = { γ ∈ GL ( g, Z ) ∣ γ ≡ I g ( mod 4 m ) } .This paper is organized as follows. In Section 2, we collect some basic properties aboutthe symplectic group Sp ( g, R ) to be used frequently in the subsequent sections. In Section3, we give basic definitions concerning real abelian varieties and review some propertiesof real abelian varieties. In Section 4, we discuss a moduli space for real abelian varietiesand recall some basic properties of a moduli for real abelian varieties. In Section 5 wediscuss compactifications of the moduli space for real abelian varieties and review someresults on this moduli space obtained by Silhol [26], Goresky and Tai [9]. In Section 6 weintroduce a notion of polarized real tori and investigate some properties of polarized realtori. We give several examples of polarized real tori. In Section 7 we study smooth linebundles over a real torus, in particular a polarized real torus by relating those smooth linebundles to holomorphic line bundles over the associated complex torus. To each smooth linebundle on a real torus we naturally attach a holomorphic line bundle over the associatedcomplex torus. Conversely to a holomorphic line bundle over a polarized abelian variety weassociate a smooth line bundle over the associated polarized real torus. Using these resultson line bundles, we embed a real torus in a complex projective space and hence in a realprojective space smoothly. We also review briefly holomorphic line bundles over a complextorus. In Section 8 we study the moduli space for polarized real tori. We first reviewbasic geometric properties on the Minkowski domain R g . We show that P g parameterizes JAE-HYUN YANG principally polarized real tori of dimension g and that R g can be regarded as the modulispace of principally polarized real tori of dimension g . We show that the quotient space G M ( Z )/( P g × R g ) may be considered as a family of principally polarized tori of dimension g .In Section 9 we discuss real semi-abelian varieties corresponding to the boundary points ofa compactification of a moduli space for real abelian varieties. We recall that a semi-abelianvariety is defined to be an extension of an abelian variety by a group of multiplicative type.In Section 10 we discuss briefly real semi-tori corresponding to the boundary points of amoduli space for polarized real tori. In the final section we present some problems relatedto real polarized tori which should be investigared in the near future. In the appendix wecollect and review some results on non-abelian cohomology to be needed necessarily in thisarticle. We give some sketchy proofs for the convenience of the reader.Finally I would like to mention that this work was motivated and initiated by the worksof Silhol [26] and Goresky-Tai [9]. Notations:
We denote by Q , R and C the field of rational numbers, the field of realnumbers and the field of complex numbers respectively. We denote by Z and Z + the ring ofintegers and the set of all positive integers respectively. The symbol “:=” means that theexpression on the right is the definition of that on the left. For two positive integers k and l , F ( k,l ) denotes the set of all k × l matrices with entries in a commutative ring F . For a squarematrix A ∈ F ( k,k ) of degree k , σ ( A ) denotes the trace of A . For any M ∈ F ( k,l ) , t M denotesthe transpose matrix of M . I n denotes the identity matrix of degree n . For a matrix Z ,we denote by Re Z (resp. Im Z ) the real (resp. imaginary) part of Z . For A ∈ F ( k,l ) and B ∈ F ( k,k ) , we set B [ A ] = t ABA.
For a complex matrix A , A denotes the complex conjugate of A . For A ∈ C ( k,l ) and B ∈ C ( k,k ) , we use the abbreviation B { A } = t ABA.
We denote C ∗ = { ξ ∈ C ∣ ∣ ξ ∣ = } . Let Γ g = { γ ∈ Z ( g, g ) ∣ t γ J g γ = J g } denote the Siegel modular group of degree g , where J g = ( I g − I g ) is the symplectic matrix of degree 2 g . For a positive integer N , we letΓ g ( N ) = { γ ∈ Γ g ∣ γ ≡ I g ( mod N ) } denote the principal congruence subgroup of Γ g of level N and for a positive integer m , welet(1.4) Γ g ( , m ) = {( A BC D ) ∈ Γ g ∣ A, D ≡ I g ( mod 2 ) , B, C ≡ ( mod 2 m ) } . Let G g ∶ = GL ( g, Z ) and for a positive integer N let(1.5) G g ( N ) = { γ ∈ GL ( g, Z ) ∣ γ ≡ I g ( mod N ) } . OLARIZED REAL TORI 5 The Symplectic Group
For a given fixed positive integer g , we let let Sp ( g, R ) = { M ∈ R ( g, g ) ∣ t M J g M = J g } be the symplectic group of degree g .If M = ( A BC D ) ∈ Sp ( g, R ) with A, B, C, D ∈ R ( g,g ) , then it is easily seen that(2.1) A t D − B t C = I g , A t B = B t A, C t D = D t C or(2.2) t AD − t CB = I g , t AC = t CA, t BD = t DB.
The inverse of such a symplectic matrix M is given by M − = M = ( t D − t B − t C t A ) . We identify GL ( g, R ) ↪ Sp ( g, R ) with its image under the embedding A z → ( A t A − ) , A ∈ GL ( g, R ) . A Cartan involution θ of Sp ( g, R ) is given by θ ( x ) = J g x J − g , x ∈ Sp ( g, R ) , in other words,(2.3) θ ( A BC D ) = ( D − C − B A ) , ( A BC D ) ∈ Sp ( g, R ) . The fixed point set K of θ is given by K = {( A B − B A ) ∈ Sp ( g, R ) } . We may identify K with the unitary group U ( g ) of degree g via K ∋ ( A B − B A ) z→ A + iB ∈ U ( g ) . Let H g = { Ω ∈ C ( g,g ) ∣ Ω = t Ω , Im Ω > } be the Siegel upper half plane of degree g . Then Sp ( g, R ) acts on H g transitively by(2.4) M ⋅ Ω = ( A Ω + B )( C Ω + D ) − , where M = ( A BC D ) ∈ Sp ( g, R ) and Ω ∈ H g . The stabilizer at iI g is given by the compactsubgroup K ≅ U ( g ) of Sp ( g, R ) . Thus H g is biholomorphic to the Hermitian symmetricspace Sp ( g, R )/ K via Sp ( g, R )/ K Ð→ H g , xK z→ x ⋅ ( iI g ) , x ∈ Sp ( g, R ) . We note that the Siegel modular group Γ g of degree g acts on H g properly discontinuously. JAE-HYUN YANG
Now we let(2.5) I ∗ ∶ = ( − I g I g ) . We define the involution τ ∶ Sp ( g, R ) Ð→ Sp ( g, R ) by(2.6) τ ( x ) ∶ = I ∗ x I ∗ , x ∈ Sp ( g, R ) . Precisely τ is given by(2.7) τ ( A BC D ) = ( A − B − C D ) , ( A BC D ) ∈ Sp ( g, R ) . Lemma 2.1. (1) τ ( x ) = x, x ∈ Sp ( g, R ) if and only if x ∈ GL ( g, R ) . (2) τ θ = θτ. So τ ( K ) = K. (3) If A + iB ∈ U ( g ) with A, B ∈ R ( g,g ) , then τ ( A + iB ) = A − iB. Proof.
It is easy to prove the above lemma. We leave the proof to the reader. ◻ We note that τ ∶ Sp ( g, R ) Ð→ Sp ( g, R ) passes to an involution (which we denote by thesame letter) τ ∶ H g Ð→ H g such that(2.8) τ ( x ⋅ Ω ) = τ ( x ) τ ( Ω ) for all x ∈ Sp ( g, R ) , Ω ∈ H g . In fact, we can see easily that the involution τ ∶ H g Ð→ H g is the antiholomorphic involutiongiven by(2.9) τ ( Ω ) = − Ω , Ω ∈ H g . Its fixed point set is the orbit i P g = GL ( g, R ) ⋅ ( iI g ) ⊂ C ( g,g ) of GL ( g, R ) , where P g = { Y ∈ R ( g,g ) ∣ Y = t Y > } is the open convex cone of positive definite symmetric real matrices of degree g in theEuclidean space R g ( g + )/ . For x ∈ Sp ( g, R ) and Ω ∈ H g , we define the set(2.10) H τxg ∶ = { Ω ∈ H g ∣ x ⋅ Ω = τ ( Ω ) = − Ω } be the locus of x -real points. If Γ ⊂ Sp ( g, R ) is an arithmetic subgroup of Sp ( g, R ) suchthat τ ( Γ ) = Γ, we define(2.11) H τ Γ g ∶ = ⋃ γ ∈ Γ H τγg . Lemma 2.2.
Let x ∈ Sp ( g, R ) and H xg be the set of points in H g which are fixed under theaction of x . Then the set H xg ∩ i P g is a proper real algebraic variety of i P g if x ≠ ± I g ∈ GL ( g, R ) . Proof.
It is easy to prove the above lemma. We omit the proof. ◻ OLARIZED REAL TORI 7 Real Abelian Varieties
In this section we review basic notions and some results on real principally polarizedabelian varieties (cf. [9, 21, 24, 25, 26]).
Definition 3.1.
A pair ( A , S ) is said to be a real abelian variety if A is a complex abelianvariety and S is an anti-holomorphic involution of A leaving the origin of A fixed. The setof all fixed points of S is called the real point of ( A , S ) and denoted by ( A , S )( R ) or simply A ( R ) . We call S a real structure on A . Definition 3.2. (1) A polarization on a complex abelian variety A is defined to be theChern class c ( D ) ∈ H ( A , Z ) of an ample divisor D on A . We can identify H ( A , Z ) with ⋀ H ( A , Z ) . We write A = V / L , where V is a finite dimensional complex vector space and L is a lattice in V . So a polarization on A can be defined as an alternating form E on L ≅ H ( A , Z ) satisfying the following conditions (E1) and (E2) :(E1) The Hermitian form H ∶ V × V Ð→ C defined by (3.1) H ( u, v ) = E ( i u, v ) + i E ( u, v ) , u, v ∈ V is positive definite. Here E can be extended R -linearly to an alternating form on V .(E2) E ( L × L ) ⊂ Z , i.e., E is integral valued on L × L. (2) Let ( A , S ) be a real abelian variety with a polarization E of dimension g . A polarization E is said to be real or S - real if (3.2) E ( S ∗ ( a ) , S ∗ ( b )) = − E ( a, b ) , a, b ∈ H ( A , Z ) . Here S ∗ ∶ H ( A , Z ) Ð→ H ( A , Z ) is the map induced by a real structure S . If a polarization E is real, the triple ( A , E, S ) is called a real polarized abelian variety . A polarization E on A is said to be principal if for a suitable basis (i.e., a symplectic basis) of H ( A , Z ) ≅ L , itis represented by the symplectic matrix J g (cf. see Notations in the introduction). A realabelian variety ( A , S ) with a principal polarization E is called a real principally polarizedabelian variety .(3) Let ( A , E ) be a principally polarized abelian variety of dimension g and let { α i ∣ ≤ i ≤ g } be a symplectic basis of H ( A , Z ) . It is known that there is a basis { ω , ⋯ , ω g } of thevector space H ( A , Ω ) of holomorphic 1-forms on A such that ( ∫ α j ω i ) = ( Ω , I g ) for some Ω ∈ H g . The g × g matrix ( Ω , I g ) or simply Ω is called a period matrix for ( A , E ) . The definition of a real polarized abelian variety is motivated by the following theorem.
Theorem 3.1.
Let ( A , S ) be a real abelian variety and let E be a polarization on A . Thenthere exists an ample S -invariant (or S -real) divisor with Chern class E if and only if E satisfies the condition (3.2).Proof. The proof can be found in [25, Theorem 3.4, pp. 81-84]. ◻ Now we consider a principally polarized abelian variety of dimension g with a level struc-ture. Let N be a positive integer. Let ( A = C g / L, E ) be a principally polarized abelian JAE-HYUN YANG variety of dimension g . From now on we write A = C g / L , where L is a lattice in C g . A level N structure on A is a choice of a basis { U i , V j } ( ≤ i, j ≤ g ) for a N -torsion points of A which is symplectic, in the sense that there exists a symplectic basis { u i , v j } of L such that U i ≡ u i N ( mod L ) and V j ≡ v j N ( mod L ) , ≤ i, j ≤ g. For a given level N structure, such a choice of a symplectic basis { u i , v j } of L determinesa mapping F ∶ R g ⊕ R g Ð→ C g such that F ( Z g ⊕ Z g ) = L by F ( e i ) = u i and F ( f j ) = v j , where { e i , f j } ( ≤ i, j ≤ g ) isthe standard basis of R g ⊕ R g . The choice { u i , v j } (or equivalently, the mapping F ) will bereferred to as a lift of the level N structure. Such a mapping F is well defined modulo theprincipal congruence subgroup Γ g ( N ) , that is, if F ′ is another lift of the level structure,then F ′ ○ F − ∈ Γ g ( N ) . A level N structure { U i , V j } is said to be compatible with a realstructure S on ( A , E ) if, for some (and hence for any) lift { u i , v j } of the level structure, S ( u i N ) ≡ − u i N ( mod L ) and S ( v j N ) ≡ v j N ( mod L ) , ≤ i, j ≤ g. Definition 3.3.
A real principally polarized abelian variety of dimension g with a level N structure is a quadruple A = ( A , E, S, { U i , V j }) with A = C g / L , where ( A , E, S ) is a realprincipally polarized abelian variety and { U i , V j } is a level N structure compatible with areal structure S . An isomorphism A = ( A , E, S, { U i , V j }) ≅ ( A ′ , E ′ , S ′ , { U ′ i , V ′ j }) = A ′ is a complex linear mapping φ ∶ C g Ð→ C g such that (3.3) φ ( L ) = L ′ , (3.4) φ ∗ ( E ) = E ′ , (3.5) φ ∗ ( S ) = S ′ , that is, φ ○ S ○ φ − = S ′ , (3.6) φ ( u i N ) ≡ u ′ i N ( mod L ′ ) and φ ( v j N ) ≡ v ′ j N ( mod L ′ ) , ≤ i, j ≤ g. for some lift { u i , v j } and { u ′ i , v ′ j } of the level structures. Now we show that a given positive integer N and a given Ω ∈ H g determine naturally aprincipally polarized abelian variety ( A Ω , E Ω ) of dimension g with a level N structure. Let E be the standard alternating form on R g ⊕ R g with the symplectic matrix J g with respectto the standard basis of R g ⊕ R g . Let F Ω ∶ R g ⊕ R g Ð→ C g be the real linear mapping withmatrix ( Ω , I g ) , that is,(3.7) F Ω ( xy ) ∶ = Ω x + y, x, y ∈ R g . We define E Ω ∶ = ( F Ω ) ∗ ( E ) and L Ω ∶ = F Ω ( Z g ⊕ Z g ) . Then ( A Ω = C g / L Ω , E Ω ) is a principallypolarized abelian variety. The Hermitian form H Ω on C g corresponding to E Ω is given by(3.8) H Ω ( u, v ) = t u ( Im Ω ) − v, E Ω = Im H Ω , u, v ∈ C g . If z , ⋯ , z g are the standard coordinates on C g , then the holomorphic 1-forms dz , ⋯ , dz g havethe period matrix ( Ω , I g ) . If { e i , f j } is the standard basis of R g ⊕ R g , then { F Ω ( e i / N ) , F Ω ( f j / N )} OLARIZED REAL TORI 9 (mod L Ω ) is a level N structure on ( A Ω , E Ω ) , which we refer to as the standard N structure .Assume that Ω and Ω are two elements of H g such that ψ ∶ ( A Ω = C g / L Ω , E Ω ) Ð→ ( A Ω = C g / L Ω , E Ω ) is an isomorphism of the corresponding principally polarized abelian varieties, i.e., ψ ( L Ω ) = L Ω and ψ ∗ ( E Ω ) = E Ω . We set h = t ( F − ○ ψ ○ F Ω ) = ( A BC D ) . Then we see that h ∈ Γ g . And we have(3.9) Ω = h ⋅ Ω = ( A Ω + B )( C Ω + D ) − and(3.10) ψ ( Z ) = t ( C Ω + D ) Z, Z ∈ C g . Let Ω ∈ H g such that γ ⋅ Ω = τ ( Ω ) = − Ω for some γ = ( A BC D ) ∈ Γ g . We define the mapping S γ, Ω ∶ C g Ð→ C g by(3.11) S γ, Ω ( Z ) ∶ = t ( C Ω + D ) Z, Z ∈ C g . Then we can show that S γ, Ω is a real structure on ( A Ω , E Ω ) which is compatible withthe polarization E Ω (that is, E Ω ( S γ, Ω ( u ) , S γ, Ω ( v )) = − E Ω ( u, v ) for all u, v ∈ C g ). Indeedaccording to Comessatti’s Theorem (see Theorem 3.1), S γ, Ω ( Z ) = Z, i.e., S γ, Ω is a complexconjugation. Therefore we have E Ω ( S γ, Ω ( u ) , S γ, Ω ( v )) = E Ω ( u, v ) = − E Ω ( u, v ) for all u, v ∈ C g . From now on we write simply σ Ω = S γ, Ω . Theorem 3.2.
Let ( A , E, S ) be a real principally polarized abelian variety of dimension g .Then there exists Ω = X + i Y ∈ H g such that X ∈ Z ( g,g ) and there exists an isomorphismof real principally polarized abelian varieties ( A , E, S ) ≅ ( A Ω , E Ω , σ Ω ) , where σ Ω is a real structure on A Ω induced by a complex conjugation σ ∶ C g Ð→ C g . The above theorem is essentially due to Comessatti [6]. We refer to [24, 25] for the proofof Theorem 3.2.Theorem 3.2 leads us to define the subset H g of H g by(3.12) H g ∶ = { Ω ∈ H g ∣ ∈ Z ( g,g ) } . Assume Ω = X + i Y ∈ H g . Then according to Theorem 3.2, ( A Ω , E Ω , σ Ω ) is a real principallypolarized abelian variety of dimension g . The matrix M σ for the action of a complexconjugation σ on the lattice L Ω = Ω Z g + Z g with respect to the basis given by the columnsof ( Ω , I g ) is given by(3.13) M σ = ( − I g X I g ) . Since t M σ J g M σ = ( − I g X I g ) J g ( − I g X I g ) = − J g , the canonical polarization J g is σ -real. Theorem 3.3.
Let Ω and Ω ∗ be two elements in H g . Then Ω and Ω ∗ represent (real)isomorphic triples ( A , E, σ ) and ( A ∗ , E ∗ , σ ∗ ) if and only if there exists an element A ∈ GL ( g, Z ) such that (3.14) 2 Re Ω ∗ = A ( Re Ω ) t A ( mod ) and (3.15) Im Ω ∗ = A ( Im Ω ) t A. Proof.
Suppose ( A , E, σ ) and ( A ∗ , E ∗ , σ ∗ ) are real isomorphic. Then we can find an element γ = ( A BC D ) ∈ Γ g such that Ω ∗ = ( A Ω + B )( C Ω + D ) − . The map ϕ ∶ C g / L Ω ∗ = A Ω ∗ Ð→ A Ω = C g / L Ω induced by the map ̃ ϕ ∶ C g Ð→ C g , Z z→ t ( C Ω + D ) Z is a real isomorphism. Since ̃ ϕ ○ σ ∗ = σ ○ ̃ ϕ, i.e., ̃ ϕ commutes with complex conjugation on C g , we have C = . ThereforeΩ ∗ = ( A Ω + B ) t A = ( AX t A + B t A ) + i AY t A, where Ω = X + i Y. Hence we obtain the desired results (3.14) and (3.15).Conversely we assume that there exists A ∈ GL ( g, Z ) satisfying the conditions (3.14) and(3.15). Then Ω ∗ = γ ⋅ Ω = ( A Ω + B ) t A for some γ = ( A B t A − ) ∈ Γ g with B ∈ Z ( g,g ) with B t A = A t B. The map ψ ∶ A Ω Ð→ A Ω ∗ induced by the map ̃ ψ ∶ C g Ð→ C g , Z z→ A − Z is a complex isomorphism commuting complex conjugation σ . Therefore ψ is a real isomor-phism of ( A , E, σ ) onto ( A ∗ , E ∗ , σ ∗ ) . ◻ According to Theorem 3.3, we are led to define the subgroup Γ ⋆ g of Γ g by(3.16) Γ ⋆ g ∶ = { ( A B t A − ) ∈ Γ g ∣ B ∈ Z ( g,g ) , A t B = B t A } . It is easily seen that Γ ⋆ g acts on H g properly discontinuously by(3.17) γ ⋅ Ω = A Ω t A + B t A, where γ = ( A B t A − ) ∈ Γ ⋆ g and Ω ∈ H g . OLARIZED REAL TORI 11 Moduli Spaces for Real Abelian Varieties
In Section 3, we knew that Γ ⋆ g acts on H g properly discontinuously by the formula (3.17).So the quotient space X g R ∶ = Γ ⋆ g / H g inherits a structure of stratified real analytic space from the real analytic structure on H g .The stratified real analytic space X g R classifies, up to real isomorphism, real principallypolarized abelian varieties ( A , E, S ) of dimension g . Thus X g R is called the (real) modulispace of real principally polarized abelian varieties ( A , E, S ) of dimension g .To study the structure of X g R , we need the following result of A. A. Albert [1]. Lemma 4.1.
Let S g ( Z / ) be the set of all g × g symmetric matrices with coefficients in Z / . We note that GL ( g, Z / ) acts on S g ( Z / ) by N z→ AN t A with A ∈ GL ( g, Z / ) and N ∈ S g ( Z / ) . We put π ( N ) ∶ = g ∏ k = ( − n kk ) for N = ( n ij ) ∈ S g ( Z / ) . Then N ∈ S g ( Z / ) is equivalent mod GL ( g, Z / ) to a matrix of the form :(I) ( I λ
00 0 ) if π ( N ) = and rank ( N ) = λ or (II) ( H λ
00 0 ) with H λ ∶ = ⎛⎜⎝ ⋯ ⋮ ⋰ ⋮ ⋯ ⎞⎟⎠ ∈ Z ( λ,λ ) if π ( N ) = and rank ( N ) = λ . N ∈ S g ( Z / ) is said to be diasymmetric in Case (I) and to be orthosymmetric in Case(II). Theorem 4.1.
Let ( A , E ) be a principally polarized abelian variety of dimension g . Thenthere exists a real structure S on A such that E is S -real if and only if ( A , E ) admits aperiod matrix of the following form ( I g , M + i Y ) , Y ∈ P g , where M is one of the forms (I) and (II) in Lemma 4.1. The above theorem is essentially due to Comessatti [6]. We refer to [24] or [25, Theorem2.3, pp. 78–80 and Theorem 4.1, pp. 86–88] for the proof of the above theorem.
Lemma 4.2.
Let Ω and Ω be two elements of H g such that Ω i = X i + i Y i , M i ∈ Z ( g,g ) , Y i ∈ P g , i = , . Then Ω and Ω have images, under the natural projection π g ∶ H g Ð→ X g R , in thesame connected component of X g R , if and only if rank ( M mod ) = rank ( M mod ) and π ( M mod ) = π ( M mod ) . Theorem 4.2. X g R is a real analytic manifold of dimension g ( g + )/ and has g + + [ g ] connected components. Moreover X g R is semi-algebraic, i.e., X g R is defined by a finitenumber of polynomial equalities and inequalities.Proof. The proof can be found in [21, Theorem 6.1, p. 161]. ◻ Remark 4.1.
Let Ω = M + i Y ∈ H g with M = t M ∈ Z ( g,g ) . If rank ( M mod ) = λ , then A Ω ( R ) has g − λ connected components (cf. [21, 24] ). The other invariant π ( M mod ) isan invariant related to the polarization. Recall that by Lemma 4.1, the connected components of X g R correspond to the differentpossible values of ( λ, i ) = ( rank ( M mod ) , π ( M mod )) on which we have the restriction :(4.1) 0 ≤ λ ≤ g, i = or , and i = λ is odd, i = if λ = . We denote by X g ( λ,i ) the connected components of X g R corresponding to the invariants ( λ, i ) . Definition 4.1.
Let M ∈ Z ( g,g ) be a g × g symmetric integral matrix. We say that M is of the standard form if M is of one of the forms in Lemma 4.1 (we observe that for fixed ( λ, i ) this form is unique). Now we can prove the following.
Lemma 4.3.
Let M ∈ Z ( g,g ) be a symmetric integral matrix which is of the standard formwith invariants ( λ, i ) . Let Γ g ( λ,i ) ∶ = { A ∈ GL ( g, Z ) ∣ AM t A ≡ M ( mod ) } . Then X g ( λ,i ) ≅ Γ g ( λ,i ) / P g . Proof.
Let [ Ω ] be a class in X g ( λ,i ) . By Lemma 4.1 and Lemma 4.2, there exist a symmetricintegral matrix M ∈ Z ( g,g ) with invariants ( λ, i ) of the standard form and an element Y ∈ P g such that M + i Y is a representative for the class [ Ω ] . If Y ∗ ∈ P g is such that M + i Y ∗ is also a representative for the class [ Ω ] , according to Theorem 3.2, M ≡ AM t A ( mod ) and Y ∗ = AY t A for some A ∈ GL ( g, Z ) . ◻ Theorem 4.3. X g ( λ,i ) is a connected semi-algebraic set with a real analytic structure.Proof. The proof can be found in [21, p. 160]. ◻ Let ( A , E, S ) be a real polarized abelian variety and − S be the real structure obtainedby composing S with the involution z z→ − z of A . We see that ( A , E, − S ) is also areal polarized abelian variety. In general ( A , E, − S ) is not real isomorphic to ( A , E, S ) .Therefore the following correspondence(4.2) Σ ∶ X g R Ð→ X g R , ( A , E, S ) z→ ( A , E, − S ) defines a non-trivial involution of X g R . OLARIZED REAL TORI 13
Let M ∈ Z ( g,g ) be a symmetric integral matrix which is of the standard form with invari-ants ( λ, i ) . It is easily checked that M = M. We put(4.3) Σ M ∶ = ( − M I g − ( I g + M ) M ) . It is easy to see the following facts (4.4) and (4.5).(4.4) Σ M ∈ Γ g and ( Σ M ) − = − Σ M . (4.5) ( t Σ M ) − ( − I g M I g ) t Σ M = ( I g − M − I g ) . Now we assume that Ω = M + i Y ∈ H g represents ( A , E, S ) . By (3.13), the matrices of S and − S are given by(4.6) M S = ( − I g M I g ) and M − S = ( I g − M − I g ) respectively with respect to the R -basis given by the columns of ( Ω , I g ) . By the formulas(4.5) and (4.6) we see that Σ M ( Ω ) represents the real polarized abelian variety. Lemma 4.4.
Let M ∈ Z ( g,g ) be a symmetric integral matrix which is of the standard formwith invariants ( λ, i ) and Y ∈ P g . Then we have (4.7) Σ M ( M + i Y ) = M + i ( I λ I g − λ ) Y − ( I λ I g − λ ) − . Proof.
Using the fact that M = M , by a direct computation, we get(4.8) Σ M ( M + i Y ) = M ( I g + M ) − + i ( I g − M ) Y − ( I g + M ) − . It is easily checked that(4.9) ( I g + M ) − = I g − M = ( I λ I g − λ ) . The formula (4.7) follows immediately from (4.8) and (4.9). ◻ Proposition 4.1.
The map Σ ∶ X g R Ð→ X g R defined by (4.10) Σ ([( A , E, S )]) ∶ = [( A , E, − S )] , [( A , E, S )] ∈ X g R is a real analytic involution of X g R . For each connected component X g ( λ,i ) , we have Σ ( X g ( λ,i ) ) = X g ( λ,i ) . Hence Σ leaves the connected components of X g R globally fixed.Proof. Let M ∈ Z ( g,g ) be a symmetric integral matrix which is of the standard form withinvariants ( λ, i ) . We denote by H g ( M ) the connected component of H g containing thematrices of the form M + i Y ∈ H g with Y ∈ P g . According to (4.5) and Lemma 4.4, we seethat Σ M defines an involution of H g ( M ) . Since H g ( M ) is mapped onto X g ( λ,i ) , we obtainthe desired result. ◻ Compactifications of the Moduli Space X g R In this section we review the compactification X g R of X g R obtained by R. Silhol [26] andthe Baily-Borel compactification of Γ g ( m )/ H g which is related to the moduli space of realabelian varieties with level 4 m structure.First of all we recall the Satake compactification of the Siegel modular variety A g ∶ = Γ g / H g . Let(5.1) D g ∶ = { W ∈ C ( g,g ) ∣ W = t W, I g − W W > } be the generalized unit disk of degree g which is a bounded realization of H g . In fact, theCayley transform Φ g ∶ D g Ð→ H g defined by(5.2) Φ g ( W ) ∶ = i ( I g + W )( I g − W ) − , W ∈ D g is a biholomorphic mapping of D g onto H g which gives the bounded realization of H g by D g [23, pp. 281-283]. The inverse Ψ g of Φ g is given by(5.3) Ψ g ( Ω ) = ( Ω − i I g )( Ω + i I g ) − , Ω ∈ H g . We let T = √ ( I g I g iI g − iI g ) be the 2 g × g matrix represented by Φ g . Then T − Sp ( g, R ) T = {( P QQ P ) ∈ C ( g, g ) ∣ t P P − t QQ = I g , t P Q = t QP } . Indeed, if M = ( A BC D ) ∈ Sp ( g, R ) , then T − M T = ( P QQ P ) , where(5.4) P = {( A + D ) + i ( B − C )} and(5.5) Q = {( A − D ) − i ( B + C )} . For brevity, we set G ∗ = T − Sp ( g, R ) T. Then G ∗ is a subgroup of SU ( g, g ) , where SU ( g, g ) = { h ∈ C ( g, g ) ∣ t hI g,g h = I g,g , det h = } , I g,g = ( I g − I g ) . In the case g =
1, we observe that T − Sp ( , R ) T = T − SL ( R ) T = SU ( , ) . If g > , then G ∗ is a proper subgroup of SU ( g, g ) . In fact, since t T J g T = − i J g , we get G ∗ = { h ∈ SU ( g, g ) ∣ t hJ g h = J g } . OLARIZED REAL TORI 15
Let P + = {( I g Z I g ) ∣ Z = t Z ∈ C ( g,g ) } be the P + -part of the complexification of G ∗ ⊂ SU ( g, g ) . Since the Harish-Chandra decomposition of an element ( P QQ P ) in G J ∗ is ( P QQ P ) = ( I g QP − I g ) ( P − QP − Q P ) ( I g P − Q I g ) , the P + -component of the following element ( P QQ P ) ⋅ ( I g W I g ) , W ∈ D g of the complexification of G J ∗ is given by (( I g ( P W + Q )( QW + P ) − I g )) . We note that QP − ∈ D g . We get the Harish-Chandra embedding of D g into P + (cf. [12,p. 155] or [19, pp. 58-59]). Therefore we see that G ∗ acts on D g transitively by(5.6) ( P QQ P ) ⋅ W = ( P W + Q )( QW + P ) − , ( P QQ P ) ∈ G ∗ , W ∈ D g . The isotropy subgroup at the origin o is given by K = { ( P P ) ∣ P ∈ U ( g ) } . Thus G ∗ / K is biholomorphic to D g . The action (2.4) is compatible with the action (5.6)via the Cayley transform (5.2).In summary, Sp ( g, R ) acts on D g transitively by(5.7) ( A BC D ) ⋅ W = ( P W + Q )( QW + P ) − , ( A BC D ) ∈ Sp ( g, R ) , W ∈ D g , where P and Q are given by (5.4) and (5.5). This action extends to the closure D g of D g in C g ( g + )/ . For an integer s with 0 ≤ s ≤ g , we let(5.8) F s ∶ = { W = ( W I g − s ) ∣ W ∈ D s } ⊂ D g . We say that F s is the standard boundary component of degree s . If there exists an element γ ∈ Sp ( g, Q ) (equivalently γ ∈ Γ g ) with F = γ ( F s ) ⊂ D g , then F is said to be a rationalboundary component of degree s . The Siegel upper half plane H s is attached to H g as alimit of matrices in C ( g,g ) byΩ z→ lim Y Ð → ∞ ( Ω i Y ) , Ω ∈ H s , Y ∈ P g − s , meaning that all the eigenvalues of Y converge to ∞ . For a rational boundary component F ⊂ D g , we let P ( F ) = { α ∈ Sp ( g, Q ) ∣ α ( F ) = F } be the normalizer in Sp ( g, Q ) of F (or the parabolic subgroup of Sp ( g, Q ) associated to F ) and let Z ( F ) = { α ∈ Sp ( g, Q ) ∣ α ( W ) = W for all W ∈ F } be the centralizer of F . We put G ( F ) ∶ = P ( F )/ Z ( F ) ≅ Sp ( s, Q ) . Obviously G ( F ) acts on F . We choose the standard boundary component F = F s . Anelement γ of P ( F ) is of the form(5.9) γ = ⎛⎜⎜⎜⎝ A B ∗∗ u ∗ ∗ C D ∗ t u − ⎞⎟⎟⎟⎠ ∈ Sp ( g, Q ) , where γ = ( A B C D ) ∈ Sp ( s, Q ) and u ∈ GL ( g − s, Q ) . The unipotent radical U ( F ) of P ( F ) is given by(5.10) U ( F ) = ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ ⎛⎜⎜⎜⎝ I s t µλ I g − s µ κ I s − t λ I g − s ⎞⎟⎟⎟⎠ RRRRRRRRRRR λ, µ ∈ Q ( g − s,s ) , κ ∈ Q ( g − s,g − s ) ⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭ and the centralizer Z U ( F ) of U ( F ) is given by(5.11) Z U ( F ) = ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ ⎛⎜⎜⎜⎝ I s I g − s κ I s
00 0 0 I g − s ⎞⎟⎟⎟⎠ RRRRRRRRRRR κ ∈ Q ( g − s,g − s ) ⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭ . We have inclusions of normal subgroups Z U ( F ) ⊂ U ( F ) ⊂ P ( F ) . The Levi factor L ( F ) of P ( F ) is given by(5.12) L ( F ) = G h ( F ) G l ( F ) with(5.13) G h ( F ) = ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ ⎛⎜⎜⎜⎝ A B I g − s C D
00 0 0 I g − s ⎞⎟⎟⎟⎠ ∈ P ( F )RRRRRRRRRRR ( A B C D ) ∈ Sp ( s, Q ) ⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭ . and(5.14) G l ( F ) = ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ ⎛⎜⎜⎜⎝ I s S I s
00 0 0 t S − ⎞⎟⎟⎟⎠ ∈ P ( F )RRRRRRRRRRR S ∈ GL ( g − s, Q ) ⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭ . OLARIZED REAL TORI 17
The subgroup U ( F ) G h ( F ) is normal in P ( F ) . The map P ( F ) Ð→ Sp ( s, Q ) , γ ↦ γ is surjective and induces the isomorphism G h ( F s ) ≅ Sp ( s, Q ) . We note that the map f ∶ P ( F s ) ∩ Sp ( g, Z ) Ð→ Sp ( s, Z ) , γ ↦ γ is obtained via ( W I g − s ) z→ W , in the sense thatif γ ∈ P ( F s ) , then γ ⋅ ( W I g − s ) = ( γ ( W ) I g − s ) . We define D stg ∶ = ∐ ≤ s ≤ g F s and D ∗ g ∶ = ∐ F ∶ rational F , where F runs over all rational boundary components. Via the Cayley transform Φ g (cf. (5.2)), we identify D stg = H stg = ∐ ≤ s ≤ g H s . Definition 5.1.
Let u > . We denote by W g ( u ) the set of all matrices Ω = X + i Y in H g with X = ( x ij ) ∈ R ( g,g ) satisfying the conditions ( Ω1 ) and ( Ω2 ) : ( Ω1 ) ∣ x ij ∣ < u ; ( Ω2 ) if Y = t W DW is the Jacobi decomposition of Y with W = ( w ij ) strictly upper trian-gular and D = diag ( d , ⋯ , d g ) diagonal, then we have ∣ w ij ∣ < u, < u d , d i < u d i + , i = , ⋯ , g − . It is well known that for sufficiently large u >
0, the set W g ( u ) is a fundamental set forthe action of Γ g on H g , that is, Γ g ⋅ W g ( u ) = H g , and { γ ∈ Γ g ∣ γ ⋅ W g ( u ) ∩ W g ( u ) ≠ ∅ } is a finite set. We observe that if Ω = ( Ω Ω t Ω Ω ) ∈ W g ( u ) with Ω ∈ C ( s,s ) , then Ω ∈ W s ( u ) . Definition 5.2.
We can choose a sufficiently large u > such that for all ≤ s ≤ g, W s ( u ) is a fundamental set for the action of Γ s on H s . In this case we simply write W s = W s ( u ) with ≤ s ≤ g . We define W ∗ g ∶ = ∐ ≤ s ≤ g W s . For Ω ∗ ∈ W g − r , we let U be a neighborhood of Ω ∗ in W g − r and v a positive real number.For ≤ s ≤ r , we let W s ( U, v ) be the set of all Ω = ( Ω Ω t Ω Ω ) ∈ W g − s with Ω ∈ C ( g − r,g − r ) satisfying the conditions ( W ) and ( W ) : ( W ) Ω ∈ U ; ( W ) if Y = t W DW is the Jacobi decomposition of Y with W strictly upper triangularand D = diag ( d , ⋯ , d g ) diagonal, then we have d g − r + > v. A fundamental set of neighborhoods of Ω ∗ ∈ W g − r for the Satake topology on W ∗ g is givenby the collection { ⋃ ≤ s ≤ r W s ( U, v ) } ’s, where U runs through neighborhoods of Ω ∗ in W g − r and v ranges in R + . We regard W ∗ g ⊂ H stg ≅ D stg as a subset of D ∗ g . The Satake topology on D ∗ g is characterized as the unique topology T extending theordinary matrix topology on D g and satisfying the following properties (ST1)–(ST4) :(ST1) T induces on W ∗ g the topology defined in Definition 5.2(ST2) Sp ( g, Q ) acts continuously on D ∗ g ;(ST3) A ∗ g = Γ g / D ∗ g is a compact Hausdorff space ;(ST4) For any Ω ∈ D ∗ g , there exists a fundamental set of neighborhoods { U } of Ω such that γ ⋅ U = U if γ ∈ Γ g ( Ω ) ∶ = { γ ∈ Γ g ∣ γ ⋅ Ω = Ω } , and γ ⋅ U ∩ U = ∅ if γ ∉ Γ g ( Ω ) . For a proof of these above facts we refer to [4].Now we are ready to investigate the compactification of the moduli space X g R of realprincipally polarized abelian varieties of dimension g obtained by R. Silhol. Definition 5.3.
Let u > . We let F g ( u ) be the set of all Ω = X + i Y ∈ H g with X = Re Ω = ( x ij ) satisfying the following conditions (a) and (b) :(a) x ij = or ;(b) if Y = t W DW is the Jacobi decomposition of Y with W strictly upper triangular and D = diag ( d , ⋯ , d g ) diagonal, then we have ∣ w ij ∣ ≤ u and 0 < d i ≤ u d i + . We define F ′ g ( u ) to be the set of matrices in H g satisfying the condition ∣ x ij ∣ ≤ and theabove condition (b). Let u > be as in Definition 5.2. We put F g ∶ = F g ( u ) . It is well known that F g is a fundamental set for the action of Γ ⋆ g on H g . For two nonnegative integers s and t , we define two subsets F s,t and F s,t of D ∗ g as follows.(5.15) F s,t ∶ = ⎧⎪⎪⎪⎨⎪⎪⎪⎩ ⎛⎜⎝ − I s W
00 0 I t ⎞⎟⎠ ∈ D ∗ g ∣ W ∈ D g −( s + t ) ⎫⎪⎪⎪⎬⎪⎪⎪⎭ and(5.16) F s,t ∶ = ⎧⎪⎪⎪⎨⎪⎪⎪⎩ ⎛⎜⎝ − I s W
00 0 I t ⎞⎟⎠ ∈ F s,t ∣ W ∈ F g −( s + t ) ⎫⎪⎪⎪⎬⎪⎪⎪⎭ . For M ∈ Z ( g,g ) , we set F M ∶ = { Ω ∈ F g ∣ = M } . In particular, F = { Ω ∈ F g ∣ Re Ω = } , where denotes the g × g zero matrix. We let M ∶ = { M = ( m ij ) ∈ Z ( g,g ) ∣ M = t M, m ij = or } . OLARIZED REAL TORI 19
For any M ∈ M , we set B M ∶ = ( I g M I g ) ∈ Sp ( g, Q ) . By the definition we have F g = ⋃ M ∈ M B M ( F ) and F g = ⋃ M ∈ M B M ( F ) . We can show that H g = Γ ⋆ g ⋅ F g . Now we embed H g into D ∗ g via the Cayley transform (5.3). We let H g be the closure of H g in D ∗ g . Then the action of Γ ⋆ g extends to an action of Γ ⋆ g on H g (see (3.16), (3.17), (ST2)).R. Silhol proved that the quotient space Γ ⋆ g / H g is a connected, compact Hausdorff space(cf. [26, pp. 173-177]). Let π ∶ H g Ð→ Γ ⋆ g / H g be the canonical projection. For M ∈ M , wedefine H M = { M + i Y ∈ H g } . We let H M be the closure of H M in H g . Then without difficulty we can see that(5.17) Γ ⋆ g / H g = ⋃ ≤ s + t ≤ g ⋃ M ∈ M ( π ( B M ( F s,t ) ∪ H )) , Let { X i ∣ ≤ i ≤ N } with N = g + + [ g ] be the connected components of X g R ⊂ Γ ⋆ g / H g and let Σ i be the restriction to X i of the fundamental involution Σ (cf. Proposition 4.1).We note that Σ does not extend to a global involution of Γ ⋆ g / H g . But Σ i extends to aninvolution of the closure X i of X i in Γ ⋆ g / H g . We observe that for each 1 ≤ i ≤ N , we have X i = Γ ⋆ g ( M i )/ H M i for some M i ∈ M . Here Γ ⋆ g ( M i ) = { γ ∈ Γ ⋆ g ∣ γ ( H M i ) = H M i } . Definition 5.4.
Let z ∈ X i and z ∈ X j . We say that z and z are Σ -equivalent andwrite z ∼ z if Σ i ( z ) = Σ j ( z ) . Silhol [26, p. 185] showed that ∼ defines an equivalence relation in Γ ⋆ g / H g .By a direct computation, we obtain P ( F s,t ) = ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩⎛⎜⎜⎜⎜⎜⎜⎜⎝ v v ∗ A B ∗∗ ∗ u − u ∗ ∗∗ ∗ − u u ∗ ∗∗ C D ∗ v v ⎞⎟⎟⎟⎟⎟⎟⎟⎠ ∈ Sp ( g, Q ) ⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭ , where γ = ( A B C D ) ∈ Sp ( g − r, Q ) with r = s + t and U = ( u u u u ) ∈ GL ( r, Q ) , V = ( v v v v ) = t U − . Now we define(5.18) X g R ( s, t ) ∶ = ( Γ ⋆ g ∩ P ( F s,t ))/( F s,t ∩ H g ) . It is easily checked thatΓ ⋆ g ∩ P ( F s,t ) ≅ Γ ⋆ g −( s + t ) and F s,t ∩ H g ≅ H g −( s + t ) . We define(5.19) X g R ∶ = Γ ⋆ g / H g / ∼ . Silhol [26, Theorem 8.17] proved the following theorem.
Theorem 5.1. X g R is a connected compact Hausdorff space containing X g R as a dense opensubset. As a set, X g R = ∐ ≤ s + t ≤ g X g R ( s, t ) . We recall that H ∗ g denotes the Satake partial compactification of H g that is obtainedby attaching all rational boundary components with the Satake topology. We know that Sp ( g, Q ) acts on H ∗ g , the involution τ ∶ H g Ð→ H g (cf. (2.9)) extends to H ∗ g and τ ( α ⋅ x ) = τ ( α ) τ ( x ) for all α ∈ Sp ( g, Q ) and x ∈ H ∗ g . Let N = m with m a positive integer. We write X ( N ) ∶ = Γ g ( N )/ H g and V ( N ) ∶ = Γ g ( N )/ H ∗ g . We let(5.20) π BB ∶ H ∗ g Ð→ V ( N ) = Γ g ( N )/ H ∗ g be the canonical projection of H ∗ g to the Baily-Borel compactification of X ( N ) . The invo-lution τ passes to complex conjugation τ ∶ V ( N ) Ð→ V ( N ) , whose fixed points we denoteby V ( N ) R . Obviously the τ -fixed set X ( N ) R ∶ = { x ∈ X ( N ) ∣ τ ( x ) = x } is a subset of V ( N ) R . We let X ( N ) R denote the closure of X ( N ) R in V ( N ) R . Theorem 5.2.
There exists a natural rational structure on V ( N ) which is compatible withthe real structure defined by τ .Proof. It follows from Shimura’s result [22] that the Γ g ( N ) -automorphic forms on H g aregenerated by those automorphic forms with rational Fourier coefficients. ◻ If γ ∈ Γ g ( N ) and F is a rational boundary component of H ∗ g such that τ ( F ) = F , wedefine the set of γ -real points of F to be(5.21) F τγ ∶ = { x ∈ F ∣ τ ( x ) = γ ⋅ x } . Then π BB ( F τγ ) ⊂ V ( N ) R . Definition 5.5.
Let N = m . A Γ g ( N ) - real boundary pair ( F , γ ) of degree s consists of arational boundary component F of degree s and an element γ ∈ Γ g ( N ) such that F τγ ≠ ∅ . We say that two Γ g ( N ) -real boundary components ( F , γ ) and ( F ∗ , γ ∗ ) are equivalent if theresulting loci of real points π BB ( F τγ ) = π BB ( F τγ ∗ ) coincide. OLARIZED REAL TORI 21
We observe that if ( F , γ ) is a Γ g ( N ) -real boundary pair and if α ∈ Γ g ( N ) , we see that τ ( F ) = γ ( F ) and ( α ( F ) , τ ( α ) γ α − ) is an equivalent Γ g ( N ) -real boundary pair.Fix a positive integer s with 1 ≤ s ≤ g. We define the map Φ ∶ H s Ð→ H ∗ g by(5.22) Φ ( Ω ) = lim Y Ð→ ∞ ( Ω i Y ) , Ω ∈ H s , Y ∈ P g − s . Obviously Φ ( H s ) = F s is the standard boundary component of degree s (cf. (5.8)).Let ν s ∶ P ( F s ) Ð→ G h ( F s ) be the projection to the quotient. It is easily seen that ν s commutes with τ . Therefore F s is preserved by τ . The set F τs = { Φ ( i Y ) ∣ Y ∈ P s } is the set of τ -fixed points in F s and may be canonically identified with P s . We denoteby i I s its canonical base point. Then F s is attached to H g so that the cone Φ ( i P s ) iscontained in the closure of the cone i P g . Proposition 5.1.
Let ( F , γ ) be a Γ g ( N ) -real boundary pair of degree s . Then there exists γ ∗ ∈ Γ g such that γ ∗ ( F s ) = F and τ ( γ ∗ ) − γ γ ∗ = ( A B t A − ) ∈ ker ( ν s ) . Moreover, we may take B = , i.e., there exist γ ′ ∈ Γ g ( m ) and γ ∈ Γ g so that F τγ ′ = F τγ , γ ( F s ) = F , and so that τ ( γ ) − γ ′ γ = ( A t A − ) ∈ ker ( ν s ) . Proof.
The proof can be found in [9, pp. 19-21]. ◻ As an application of Proposition 5.1, we get the following theorem.
Theorem 5.3.
Let m ≥ be a positive integer. Let F be a proper rational boundarycomponent of H g of degree g − . Let γ ∈ Γ g ( m ) such that F τγ = { x ∈ F ∣ τ ( x ) = γ ⋅ x } ≠ ∅ . Then F τγ is contained in the closure of H τ Γ g ( m ) g in H ∗ g , where H τ Γ g ( m ) g = { Ω ∈ H g ∣ τ ( Ω ) = − Ω = γ ⋅ Ω for some γ ∈ Γ g ( m ) } denotes the set of Γ g ( m ) -real points of H g . Proof.
The proof can be found in [9, pp. 23]. ◻ Theorem 5.4.
Let k be a positive integer with k ≥ . Let ( F , γ ) be a Γ g ( k ) -real boundarypair. Then there exists γ ∈ Γ g ( k ) such that F τγ = F τγ and F τγ is contained in theclosure H τγ g of H τγ g in H ∗ g . Proof.
The proof can be found in [9, pp. 23-26]. ◻ We may summarize the above results as follows. The Baily-Borel compactification V ( N ) = Γ g ( N )/ H ∗ g with N = m is stratified by finitely many strata of the form π BB ( F ) , where F is a rational boundary component. Each such strata is isomorphic to the standardrational boundary component F s ≅ H s . The stratum π BB ( F ) is called a boundary stratumof degree s . Let V ( N ) r denote the union of all boundary strata of rank g − r . We define V ( N ) r R ∶ = V ( N ) r ∩ V ( N ) R . According to Theorem 5.4, we have V ( N ) R ∪ V ( N ) R ⊂ X ( N ) R ⊂ V ( N ) R , where X ( N ) R denotes the closure of X ( N ) R in V ( N ) .6. Polarized Real Tori
In this section we introduce the notion of polarized real tori.First we review the properties of real tori briefly. We fix a positive integer g in thissection. Let T = R g / Λ be a real torus of dimension g , where Λ is a lattice in R g . T has aunique structure of a smooth (or real analytic) manifold such that the canonical projection p ∶ R g Ð→ T is smooth (or real analytic). We fix the standard basis { e , ⋯ , e g } for R g . Wesee that Λ = Π Z g for some Π ∈ GL ( g, R ) . A matrix Π is called a period matrix for T . Let C ∗ = { z ∈ C ∣ ∣ z ∣ = } be a circle. Since T is homeomorphic to C ∗ × ⋯ × C ∗ ( g -times), thefundamental group is π ( T ) ≅ π ( C ∗ ) × ⋯ × π ( C ∗ ) ≅ Z g . We see that H k ( T, Z ) ≅ Z g C k ≅ H k ( T, Z ) , k = , , ⋯ , g and H ∗ ( T, Z ) ≅ ⋀ H ( T, Z ) ≅ ⋀ Z g . Thus the Euler characteristic of T is zero. The mapping class group M CG ( T ) is M CG ( T ) = Aut ( π ( T )) = Aut ( Z g ) = GL ( g, Z ) . It is known that any connected compact real manifold can be embedded into the Euclideanspace R d with large d . Thus a torus T can be embedded in a real projective space P d ( R ) . Any connected compact abelian real Lie group is a real torus. Any two real tori of dimension g are isomorphic as real Lie groups. We easily see that if S is a connected closed subgroupof a real torus T , then S and T / S are real tori and T ≅ S × T / S. Let T = V / Λ and T ′ = V ′ / Λ ′ be two real tori. A homomorphism φ ∶ T Ð→ T ′ is a realanalytic map compatible with the group structures. It is easily seen that a homomorphism φ ∶ T Ð→ T ′ can be lifted to a uniquely determined R -linear map Φ ∶ V Ð→ V ′ . This yieldsan injective homomorphism of abelian groups τ a ∶ Hom ( T, T ′ ) Ð→ Hom R ( V, V ′ ) , φ z→ Φ , where Hom ( T, T ′ ) is the abelian group of all homomorphisms of T into T ′ and Hom R ( V, V ′ ) is the abelian group of all R -linear maps of V into V ′ . The above τ a is called a real analytic OLARIZED REAL TORI 23 representation of Hom ( T, T ′ ) . The restriction Φ Λ of Φ to Λ is Z -linear. Φ Λ determines Φand φ completely. Thus we get an injective homomorphism τ r ∶ Hom ( T, T ′ ) Ð→ Hom Z ( Λ , Λ ′ ) , φ z→ Φ Λ , called the rational representation of Hom ( T, T ′ ) . Lemma 6.1.
Let φ ∶ T Ð→ T ′ be a homomorphism of real tori. Then(1) the image Im φ is a real subtorus of T ′ ;(2) the kernel ker φ of φ is a closed subgroup of T and the identity component ( ker φ ) ofker φ is a real subtorus of T of finite index in ker φ .Proof. It follows from the fact that a connected compact abelian real Lie group is a realtorus. Since ker φ is compact, ker φ has only a finite number of connected components. ◻ A surjective homomorphism φ ∶ T Ð→ T ′ of real tori with finite kernel is called a realisogeny or simply an isogeny . The exponent e ( φ ) of an isogeny φ is defined to be theexponent of the finite group ker φ , that is, the smallest positive integer e such that e ⋅ x = x ∈ ker φ . Two real tori are said to be isogenous if there is an isogeny between them.It is clear that a homomorphism φ ∶ T Ð→ T ′ is an isogeny if and only if it is surjective anddim T = dim T ′ . We can see that if Γ ⊂ T is a finite subgroup, the quotient space T / Γ is areal torus and the natural projection p Γ ∶ T Ð→ T / Γ is an isogeny.For a homomorphism φ ∶ T Ð→ T ′ of real tori, we define the degree of φ to bedeg φ ∶ = ⎧⎪⎪⎨⎪⎪⎩ ord (ker φ ) if ker φ is f inite ;0 otherwise. Let T = V / Λ be a real torus of dimension g . For any nonzero integer n ∈ Z , we definethe isogeny n T ∶ T Ð→ T by n T ( x ) ∶ = n ⋅ x for all x ∈ T . The kernel T ( n ) of n T iscalled the group of n - division points of T . It is easily seen that T ( n ) ≅ ( Z / n Z ) g becauseker n T = n Λ / Λ ≅ Λ / n Λ ≅ ( Z / n Z ) g . So deg n T = n g . We put Hom Q ( T, T ′ ) ∶ = Hom ( T, T ′ ) ⊗ Z Q and End ( T ) ∶ = Hom ( T, T ) , End Q ( T ) ∶ = End ( T ) ⊗ Z Q . For any α ∈ Q and φ ∈ Hom ( T, T ′ ) , we define the degree of α φ ∈ Hom Q ( T, T ′ ) bydeg ( α φ ) ∶ = α g deg φ. Lemma 6.2.
For any isogeny φ ∶ T Ð→ T ′ of real tori with exponent e , there exists anisogeny ψ ∶ T ′ Ð→ T , unique up to isomorphisms, such that ψ ○ φ = e T and φ ○ ψ = e T ′ . Proof.
Since ker φ ⊆ ker e T , there exists a unique map ψ ∶ T ′ Ð→ T such that ψ ○ φ = e T . Itis easy to see that ψ is also an isogeny and that ker ψ ⊆ ker e T ′ . Therefore there is a uniqueisogeny φ ′ ∶ T ′ Ð→ T such that φ ′ ○ ψ = e T ′ . Since φ ′ ○ e T = φ ′ ○ ψ ○ φ = e T ′ ○ φ = φ ○ e T and e T is surjective, we have φ ′ = φ. Hence we obtain ψ ○ φ = e T and φ ○ ψ = e T ′ . ◻ According to Lemma 6.2, we see that isogenies define an equivalence relation on the setof real tori, and that an element in End ( T ) is an isogeny if and only if it is invertible inEnd Q ( T ) .For a real torus T = V / Λ of dimension g , we put V ∗ ∶ = Hom R ( V, R ) . Then the followingcanonical R -bilinear form ⟨ , ⟩ T ∶ V ∗ × V Ð→ R , ⟨ ℓ, v ⟩ T ∶ = ℓ ( v ) , ℓ ∈ V ∗ , v ∈ V is non-degenerate. Thus the set ̂ Λ ∶ = { ℓ ∈ V ∗ ∣ ⟨ ℓ, Λ ⟩ T ⊆ Z } is a lattice in V ∗ . The quotient ̂ T ∶ = V ∗ /̂ Λis a real torus of dimension g which is called the dual real torus of T . Identifying V with thespace of R -linear forms V ∗ Ð→ R by double duality, the non-degeneracy of ⟨ , ⟩ T impliesthat Λ is the lattice in V dual to ̂ Λ. Therefore we get ̂̂ T = T. Let φ ∶ T Ð→ T be a homomorphism of real tori with T i = V i / Λ i ( i = , ) and withreal analytic representation Φ ∶ V Ð→ V . Since the dual map Φ ∗ ∶ V ∗ Ð→ V ∗ satisfies thecondition Φ ∗ (̂ Λ ) ⊆ ̂ Λ , Φ ∗ induces a homomorphism, called the dual map ̂ φ ∶ ̂ T Ð→ ̂ T . If ψ ∶ T Ð→ T is another homomorphism of real tori, then we get ̂ ψ ○ φ = ̂ φ ○ ̂ ψ. If φ ∶ T Ð→ T is an isogeny of real tori, then dual map ̂ φ ∶ ̂ T Ð→ ̂ T is also an isogeny. Definition 6.1.
A real torus T = R g / Λ with a lattice Λ in R g is said to be polarized if thethe associated complex torus A = C g / L is a polarized real abelian variety, where L = Z g + i Λ is a lattice in C g . Moreover if A is a principally polarized real abelian variety, T is said tobe principally polarized . Let Φ ∶ T Ð→ A be the smooth embedding of T into A defined by (6.1) Φ ( v + Λ ) ∶ = i v + L, v ∈ R g . Let L be a polarization of A , that is, an ample line bundle over A . The pullback Φ ∗ L iscalled a polarization of T . We say that a pair ( T, Φ ∗ L ) is a polarized real torus . Example 6.1.
Let Y ∈ P g be a g × g positive definite symmetric real matrix. ThenΛ Y = Y Z g is a lattice in R g . Then the g -dimensional torus T Y = R g / Λ Y is a principallypolarized real torus. Indeed, A Y = C g / L Y , L Y = Z g + i Λ Y is a princially polarized real abelian variety. Its corresponding hermitian form H Y is givenby H Y ( x, y ) = E Y ( i x, y ) + i E Y ( x, y ) = t x Y − y, x, y ∈ C g , where E Y denotes the imaginary part of H Y . It is easily checked that H Y is positive definiteand E Y ( L Y × L Y ) ⊂ Z (cf. [17, pp. 29–30]). The real structure σ Y on A Y is a complexconjugation. OLARIZED REAL TORI 25
Example 6.2.
Let Q = (√ √ √ − √ ) be a 2 × ( , ) .Then Λ Q = Q Z is a lattice in R . Then the real torus T Q = R / Λ Q is not polarized becausethe associated complex torus A Q = C / L Q is not an abelian variety, where L Q = Z + i Λ Q is a lattice in C . Definition 6.2.
Two polarized tori T = R g / Λ and T = R g / Λ are said to be isomorphic ifthe associated polarized real abelian varieties A = C g / L and A = C g / L are isomorphic,where L i = Z g + i Λ i ( i = , ) , more precisely, if there exists a linear isomorphism ϕ ∶ C g Ð→ C g such that ϕ ( L ) = L , (6.2) ϕ ∗ ( E ) = E , (6.3) ϕ ∗ ( σ ) = ϕ ○ σ ○ ϕ − = σ , (6.4) where E and E are polarizations of A and A respectively, and σ and σ denotes thereal structures (in fact complex conjugations) on A and A respectively. Example 6.3.
Let Y and Y be two g × g positive definite symmetric real matrices. ThenΛ i ∶ = Y i Z g is a lattice in R g ( i = , ) . We let T i ∶ = R g / Λ i , i = , g . Then according to Example 6.1, T and T are principallypolarized real tori. We see that T is isomorphic to T as polarized real tori if and only ifthere is an element A ∈ GL ( g, Z ) such that Y = A Y t A. Example 6.4.
Let Y = (√ √ √ √ ) . Let T Y = R / Λ Y be a two dimensional principallypolarized torus, where Λ Y = Y Z is a lattice in R . Let T Q be the torus in Example 6.2.Then T Y is diffeomorphic to T Q . But T Q is not polarized. T Y admits a differentiableembedding into a complex projective space but T Q does not.Let Y ∈ P g be a g × g positive definite symmetric real matrix. Then Λ Y = Y Z g is alattice in R g . We already showed that the g -dimensional torus T Y = R g / Λ Y is a principallypolarized real torus (cf. Example 6.1). We know that the following complex torus A Y = C g / L Y , L Y = Z g + i Λ Y is a princially polarized real abelian variety. We define a map Φ Y ∶ T Y Ð→ A Y byΦ Y ( a + Λ Y ) ∶ = i a + L Y , a ∈ R g . Then Φ Y is well defined and is an injective smooth map. Therefore T Y is smoothly embeddedinto a complex projective space and hence into a real projective space because A Y can beholomorphically embedded into a complex projective space (cf. [17, pp. 29–30]).Let A = C g / L and A ′ = C g ′ / L ′ be two abelian complex tori of dimension g and dimension g ′ respectively, where L (resp. L ′ ) is a lattice in C g (resp. C g ′ ). A homomorphism f ∶ A Ð→ A ′ lifts to a uniquely determined C -linear map F ∶ C g Ð→ C g ′ . This yields an injectivehomomorphism ρ a ∶ Hom ( A , A ′ ) Ð→ Hom C ( C g , C g ′ ) = C ( g ′ ,g ) , f z→ F = ρ a ( f ) . Its restriction F ∣ L to the lattice L is Z -linear and determines F and f completely. Thereforewe get an injective homomorphism ρ r ∶ Hom ( A , A ′ ) Ð→ Hom Z ( L, L ′ ) , f z→ F ∣ L . Let ̃ Π ∈ C ( g, g ) and ̃ Π ′ ∈ C ( g ′ , g ′ ) be period matrices for A and A ′ respectively. With respectto the chosen bases, ρ a ( f ) (resp. ρ r ( f ) ) can be considered as a matrix in C ( g ′ ,g ) ( resp. Z ( g ′ , g ) ) . We have the following diagram : Z g ̃ Π −−−− Ð→ C g ×××Ö ρ r ( f ) ×××Ö ρ a ( f ) Z g ′ ̃ Π ′ −−−− Ð→ C g ′ , that is, by the equation ρ a ( f ) ̃ Π = ̃ Π ′ ρ r ( f ) . Conversely any two matrices A ∈ C ( g ′ ,g ) and R ∈ Z ( g ′ , g ) satisfying the equation A ̃ Π = ̃ Π ′ R define a homomorphism A Ð→ A ′ .For two real tori T and T of dimension g and dimension g respectively, we letExt ( T , T ) be the set of all isomorphism classes of extensions of T by T up to real analyticisomorphism. Since any two real tori of dimension g + g are isomorphic as real analyticreal Lie groups, Ext ( T , T ) is trivial. This leads us to consider polarized real tori T and T with T i = R g i / Λ i ( i = , ) . Here Λ i is a lattice in R g i for i = , . Let A and A be thepolarized real abelian varieties associated to T and T respectively, that is, A i = C g i / L i , L i = Z g i + Λ i Z g i , i = , . Let Ext ( T , T ) pt be the set of all isomorphism classes of extensions of A by A . We canshow that a homomorphism φ ∶ A ′ Ð→ A such that A ′ is the real abelian variety associatedto a polarized real torus T ′ induces a map(6.5) φ ∗ ∶ Ext ( T , T ) pt Ð→ Ext ( T ′ , T ) pt and that a homomorphism ψ ∶ A Ð→ A ′ such that A ′ is the real abelian variety associatedto a polarized real torus T ′ induces a map(6.6) ψ ∗ ∶ Ext ( T , T ) pt Ð→ Ext ( T , T ′ ) pt . Indeed, if(6.7) e ∶ Ð→ A ι Ð→ A p Ð→ A Ð→ ( T , T ) pt , the image φ ∗ ( e ) is defined to be the identity component ofthe kernel of the homomorphism C p,φ ∶ A × A ′ Ð→ A defined by C p,φ ( x, y ) ∶ = p ( x ) − φ ( y ) , x ∈ A , y ∈ A ′ . The dualization of the exact sequence (6.7) gives an element ˆ e ∈ Ext (̂ A , ̂ A ) . We define(6.8) ψ ∗ ( e ) ∶ = ̂̂ ψ ∗ ( ˆ e ) ∈ Ext ( T , T ′ ) pt = Ext ( A , A ′ ) . Therefore Ext ( , ) pt is a functor which is contravariant in the first and covariant in thesecond argument. OLARIZED REAL TORI 27
We can equip the set Ext ( T , T ) pt with the canonical group structure as follows : Let e and e ◇ be the extensions in Ext ( T , T ) pt which are represented by the exact sequence (6.7)and the following exact sequence e ◇ ∶ Ð→ A Ð→ A ◇ Ð→ A Ð→ . The product e × e ◇ is represented by the exact sequence e × e ◇ ∶ Ð→ A × A Ð→ A × A ◇ Ð→ A × A Ð→ . If ∆ ∶ A Ð→ A × A is the diagonal map, x z→ ( x, x ) , x ∈ A and µ ∶ T × T Ð→ T is theaddition map, ( s, t ) z→ s + t, s, t ∈ A , the sum e + e ◇ is defined to be the image of e × e ◇ under the compositionExt ( T × T , T × T ) pt ∆ ∗ Ð→ Ext ( T , T × T ) pt µ ∗ Ð→ Ext ( T , T ) pt , that is,(6.9) e + e ◇ ∶ = µ ∗ ∆ ∗ ( e × e ◇ ) . We can show that Ext ( T , T ) pt is an abelian group with respect to the addition (6.9)(cf. [5]).Now we describe the group Ext ( T , T ) pt in terms of period matrices. First we fix periodmatrices Π and Π for T and T respectively, that is, Λ i = Π i Z g i for i = , . We know thatΠ i ∈ GL ( g i , R ) for i = , . To each extension e ∶ Ð→ A Ð→ A Ð→ A Ð→ ( T , T ) pt , there is associated a period matrix for A of the form(6.10) (̃ Π σ ̃ Π ) , ̃ Π i = ( I g i , Π i ) for i = , , σ ∈ C ( g , g ) . Conversely it is obvious that for any σ ∈ C ( g , g ) , the matrix of the form (6.10) is a periodmatrix defining an extension of A by A in Ext ( T , T ) pt . Lemma 6.3.
Let σ and σ ′ be elements in C ( g , g ) . Suppose that Π and Π are periodmatrices for polarized real tori T and T respectively. Then the period matrices ̃ Π σ = (̃ Π σ ̃ Π ) and ̃ Π σ ′ = (̃ Π σ ′ ̃ Π ) , ̃ Π i = ( I g i , Π i ) for i = , define isomorphic extensions of A by A in Ext ( T , T ) pt if and only if (6.11) σ ′ = σ + ̃ Π M + A ̃ Π with some M ∈ Z ( g , g ) and A ∈ C ( g ,g ) . Proof.
Let ̃ Π σ and ̃ Π σ ′ define isomorphic extensions e and e ′ of A by A : e ∶ Ð→ A Ð→ A Ð→ A Ð→ ∣∣ ××Ö f ∣∣ e ′ ∶ Ð→ A Ð→ A ′ Ð→ A Ð→ Then we have the following commutative diagram : Z g + g ̃ Π σ −−−− Ð→ C g + g ×××Ö ρ r ( f ) ×××Ö ρ a ( f ) Z g + g ̃ Π σ ′ −−−− Ð→ C g + g Therefore there are A ∈ C ( g ,g ) and M ∈ Z ( g , g ) that satisfy the following equation(6.12) ( I g A I g ) ̃ Π σ = ̃ Π σ ′ ( I g − M I g ) . We obtain the equation (6.11) from the equation (6.12).Conversely if we have σ and σ ′ in C ( g ,g ) satisfying the equation (6.11), then we see easilythat ̃ Π σ and ̃ Π σ ′ define isomorphic extensions of A by A . ◻ Proposition 6.1.
Let σ and σ ′ be elements in C ( g , g ) . Suppose that Π and Π are periodmatrices for real tori T and T respectively. Assume that the following period matrices ̃ Π σ = (̃ Π σ ̃ Π ) and ̃ Π σ ′ = (̃ Π σ ′ ̃ Π ) , ̃ Π i = ( I g i , Π i ) for i = , define extensions e and e ′ of A by A in Ext ( T , T ) pt . Then the period matrix ̃ Π σ + σ ′ = (̃ Π σ + σ ′ ̃ Π ) defines the extension e + e ′ in Ext ( T , T ) pt .Proof. We denote A = C g + g /̃ Π σ Z g + g and A ′ = C g + g /̃ Π σ ′ Z g + g . Then we have the extensions e ∶ Ð→ A Ð→ A Ð→ A Ð→ e ′ ∶ Ð→ A Ð→ A ′ Ð→ A Ð→ ( T , T ) pt . The complex torus A × A ′ defined by the extension e × e ′ in Ext ( A × A , A × A ) is given by the period matrix ◻ = ⎛⎜⎜⎜⎜⎝̃ Π σ ̃ Π σ ′ ̃ Π
00 0 0 ̃ Π ⎞⎟⎟⎟⎟⎠ . Let ∆ ∶ A Ð→ A × A be the diagonal map. Then we have the induced map ∆ ∗ ∶ Ext ( A × A , A × A ) Ð→ Ext ( A , A × A ) . If∆ ∗ ( e × e ′ ) ∶ Ð→ A × A Ð→ S Ð→ A Ð→ OLARIZED REAL TORI 29 is given, the complex torus S is given by a period matrix of the form ◻ = ⎛⎜⎝̃ Π α ̃ Π β ̃ Π ⎞⎟⎠ with α ∈ C ( g , g ) and β ∈ C ( g , g ) . The homomorphismExt ( A × A , A × A ) Ð→ Ext ( A , A × A ) , e × e ′ z→ ∆ ∗ ( e × e ′ ) corresponds to a homomorphism S z→ A × A ′ of real tori given by the equation ◻ ⋅ ⎛⎜⎜⎜⎝ I g M I g M I g I g ⎞⎟⎟⎟⎠ = ⎛⎜⎜⎜⎝ I g A I g A I g I g ⎞⎟⎟⎟⎠ ⋅ ◻ . Thus we have the equations α = σ + ̃ Π M − A ̃ Π and β = σ ′ + ̃ Π M − A ̃ Π . According to Lemma 6.3, ⎛⎜⎝̃ Π σ ̃ Π σ ′ ̃ Π ⎞⎟⎠ is also a period matrix for S respectively ∆ ∗ ( e × e ′ ) . We denote e + e ′ ∶ Ð→ A Ð→ B Ð→ A Ð→ . A period matrix for B is of the formΠ B ∶ = (̃ Π τ ̃ Π ) , τ ∈ R ( g ,g ) . The homomorphism µ ∗ ∶ ∆ ∗ ( e × e ′ ) z→ e + e ′ defines a homomorphism S z→ B which isgiven by the equation(6.13) ( I g I g A I g ) ⎛⎜⎝̃ Π σ ̃ Π σ ′ ̃ Π ⎞⎟⎠ = (̃ Π τ ̃ Π ) ( I g I g M I g ) with A ∈ C ( g ,g ) and M ∈ Z ( g , g ) . Comparing both sides in the equation (6.13), we obtain τ = σ + σ ′ − ̃ Π M + A ̃ Π . According to Lemma 6.3, we see that ̃ Π σ + σ ′ = (̃ Π σ + σ ′ ̃ Π ) is a period matrix for B , respectively e + e ′ . ◻ Let T , T , Π , Π , ̃ Π , ̃ Π , σ, σ ′ , ̃ Π σ and ̃ Π σ ′ be as above in Proposition 6.1. We note thatthe assignment σ z→ C g + g /̃ Π σ Z g + g , σ ∈ C ( g , g ) induces a surjective homomorphism of abelian groups(6.14) Φ Π , Π ∶ C ( g , g ) Ð→ Ext ( T , T ) pt . According to Lemma 6.3, we see that the kernel of Φ Π , Π is given byker Φ Π , Π = ̃ Π Z ( g , g ) + C ( g ,g ) ̃ Π . Obviously the homomorphism Φ Π , Π depends on the choice of the period matrices Π andΠ . Proposition 6.2.
Let T and T ′ be polarized real tori of dimension g and dimension g ′ with period matrices Π and Π ′ respectively. Let T and T ′ be polarized real tori ofdimension g and dimension g ′ with period matrices Π and Π ′ respectively. Then(a) for a homomorphism f ∶ A ′ Ð→ A such that A ′ is the polarized real abelian varietyassociated to a polarized real torus T ′ , the following diagram C ( g , g ) Φ Π1 , Π2 −−−− Ð→ Ext ( T , T ) pt ×××Ö ⋅ ρ r ( f ) ×××Ö f ∗ C ( g , g ′ ) Φ Π1 , Π ′ −−−− Ð→ Ext ( T ′ , T ) pt commutes and(b) for a homomorphism h ∶ A Ð→ A ′ such that A ′ is the polarized real abelian varietyassociated to a polarized real torus T ′ , the following diagram C ( g , g ) Φ Π1 , Π2 −−−− Ð→ Ext ( T , T ) pt ρ a ( h ) ⋅ ×××Ö ×××Ö h ∗ C ( g ′ , g ) Φ Π ′ , Π2 −−−− Ð→ Ext ( T , T ′ ) pt commutes.Proof. (a) For an extension e ∈ Ext ( T , T ) pt we choose σ ∈ C ( g , g ) with Φ Π , Π ( σ ) = e and σ ′ ∈ C ( g , g ′ ) with Φ Π , Π ′ ( σ ′ ) = f ∗ ( e ) . We see that the following diagram with exact rows f ∗ ( e ) ∶ Ð→ A Ð→ A ′ Ð→ A ′ Ð→ ∣∣ ××Ö f ∗ ××Ö fe ∶ Ð→ A Ð→ A Ð→ A Ð→ σ and σ ′ are related by the equation(6.15) ( I g A ρ a ( f )) (̃ Π σ ′ ̃ Π ′ ) = (̃ Π σ ̃ Π ) ( I g M ρ r ( f )) with A ∈ C ( g ,g ′ ) and M ∈ Z ( g , g ′ ) . Comparing both sides in the equation (6.15). we get σ ′ = σ ⋅ ρ r ( f ) + ̃ Π M − A ̃ Π ′ . According to Lemma 6.3, we haveΦ Π , Π ′ ( σ ′ ) = Φ Π , Π ′ ( σ ⋅ ρ r ( f )) = f ∗ ( e ) . This completes the proof of (a).
OLARIZED REAL TORI 31 (b) For an extension e ◇ ∈ Ext ( T , T ) pt we choose σ ◇ ∈ C ( g , g ) with Φ Π , Π ( σ ◇ ) = e ◇ and σ ′◇ ∈ C ( g ′ , g ) with Φ Π ′ , Π ( σ ′◇ ) = h ∗ ( e ◇ ) . We see that the following diagram with exact rows e ◇ ∶ Ð→ A Ð→ A ◇ Ð→ A Ð→ ××Ö h ××Ö h ∗ ∣∣ e ∶ Ð→ A ′ Ð→ A ′◇ Ð→ A Ð→ σ ◇ and σ ′◇ are related by the equation(6.16) (̃ Π ′ σ ′◇ ̃ Π ) ( ρ r ( h ) M ◇ I g ) = ( ρ a ( h ) A ◇ I g ) (̃ Π σ ◇ ̃ Π ) . with A ◇ ∈ C ( g ′ ,g ′ ) and M ◇ ∈ Z ( g ′ , g ′ ) . Comparing both sides in the equation (6.16). we get σ ′◇ = ρ a ( h ) ⋅ σ ◇ + A ◇ ̃ Π − ̃ Π ′ M ◇ . According to Lemma 6.3, we get h ∗ ( e ◇ ) = h ∗ ( Φ Π , Π ( σ ◇ )) = Φ Π ′ , Π ( ρ a ( h ) ⋅ σ ◇ ) . This completes the proof of (b). ◻ Corollary 6.1.
For e ∈ Ext ( T , T ) pt and n ∈ Z , we have n ∗ A ( e ) = n ⋅ e = ( n A ) ∗ ( e ) . Proof.
We consider the following commutative diagram : C ( g , g ) Φ Π1 , Π2 −−−−−−− Ð→ Ext ( T , T ) pt ⋅ ρ r ( n A )×××Ö ×××Ö( n A ) ∗ C ( g , g ) Φ Π1 , Π2 −−−−−−− Ð→ Ext ( T , T ) pt Since ρ r ( n A ) = n I g , we get ( n A ) ∗ ( e ) = Φ Π , Π ( nσ ) = n ⋅ Φ Π , Π ( σ ) = n ⋅ e. By a similar argument, we get ( n A ) ∗ ( e ) = n ⋅ e. ◻ Proposition 6.3.
We have an isomorphism of abelian groups C ( g ,g ) /( I g , Π ) Z ( g , g ) ( Π I g ) ≅ Ext ( T , T ) pt . Proof.
Let σ = ( σ , σ ) ∈ C ( g , g ) with σ , σ ∈ C ( g ,g ) corresponding to the extension e = Φ Π , Π ( σ ) ∈ Ext ( T , T ) pt . By Lemma 6.3, the matrix σ − σ ̃ Π = ( σ , σ ) − σ ( I g , Π ) = ( , σ − σ Π ) corresponds to the same extension e . This shows that every extension in Ext ( T , T ) pt can be represented by a matrix σ = ( , α ) with α ∈ C ( g ,g ) . Hence we get a surjectivehomomorphism of abelian groups C ( g ,g ) Ð→ Ext ( T , T ) pt . According to Lemma 6.3, the matrices α and α ′ ∈ C ( g ,g ) define the same extension if andonly if(6.17) ( , α − α ′ ) = ̃ Π ( M M M M ) + A ̃ Π with ( M M M M ) ∈ Z ( g , g ) and A ∈ C ( g ,g ) . From the equation (6.17) we get A = − M − Π M . Thus we have α − α ′ = Π M − Π M Π + M − M Π = ( I g , Π ) ( − M M − M M ) ( Π I g ) . This completes the proof of the above proposition. ◻ Line Bundles over a Polarized Real Torus
Before we investigate complex line bundles over a real torus, we need a knowledge ofholomorphic line bundles on a complex torus. We briefly review some results on holomorphicline bundles on a complex torus (cf. [13], [17]).Let X = C g / L be a complex torus, where L is a lattice in C g . The exponential sequence0 Ð→ Z Ð→ O X Ð→ O ∗ X Ð→ ⋯ Ð→ H ( X, Z ) Ð→ H ( X, O X ) Ð→ H ( X, O ∗ X ) c Ð→ H ( X, Z ) Ð→ ⋯ We recall that the N´eron-Severi group
N S ( X ) (resp. P ic ( X ) ) is defined to be the imageof c (resp. the kernel of c ). For a hermitian form H on C g whose imaginary part E H ∶ = Im ( H ) is integral on L × L , a semi-character for H is defined to be a map α ∶ L Ð→ C ∗ isdefined to be a map such that α ( ℓ + ℓ ) = α ( ℓ ) α ( ℓ ) e π i E H ( ℓ ,ℓ ) , ℓ , ℓ ∈ L. We let Her ( L ) be the set of all hermitian forms on C g whose imaginary parts are integralon L × L . For any H ∈ Her ( L ) , we denote by SC ( H ) the set of all semi-characters for H .To each pair ( H, α ) with H ∈ Her ( L ) and α ∈ SC ( H ) , we associate the automorphic factor J H,α ∶ L × C g Ð→ C ∗ defined by(7.1) J H,α ( ℓ, z ) ∶ = α ( ℓ ) e π H ( ℓ,ℓ ) + π H ( z,ℓ ) , ℓ ∈ L, z ∈ C g . A lattice L acts on the trivial line bundle C g × C on C g freely by(7.2) ℓ ⋅ ( z, ξ ) = ( z + ℓ, J H,α ( ℓ, z ) ξ ) , ℓ ∈ L, z ∈ C g , ξ ∈ C . The quotient(7.3) L ( H, α ) ∶ = ( C g × C )/ L obtained by the action (7.2) of L has a natural structure of a holomorphic line bundle over X . We note that for each such pairs ( H , α ) and ( H , α ) , we have J H ,α ⋅ J H ,α = J H + H ,α α and L ( H , α ) ⊗ L ( H , α ) = L ( H + H , α α ) . OLARIZED REAL TORI 33
Let B ( L ) be the set of all pairs ( H, α ) with H ∈ Her ( L ) and α ∈ SC ( H ) . Then B ( L ) hasa group structure equipped with multiplication law ( H , α ) ⋅ ( H , α ) = ( H + H , α α ) , H i ∈ Her ( L ) , α i ∈ SC ( H i ) , i = , . Appell-Humbert Theorem says that we have the following canonical isomorphism of exactsequences 0 Ð→ Hom ( L, C ∗ ) Ð→ B ( L ) Ð→ N S ( X ) Ð→ ×××Ö c L ×××Ö β L ×××Ö id0 Ð→ P ic ( X ) Ð→ P ic ( X ) Ð→ N S ( X ) Ð→ β L ∶ B ( L ) Ð→ P ic ( X ) = H ( X, O ∗ X ) is the group isomorphism defined by β L (( H, α )) ∶ = L ( H, α ) , ( H, α ) ∈ B ( L ) and c L is the isomorphism induced by β L . It is known that N S ( X ) is a free abelian groupof rank ρ ( X ) ≤ g , where ρ ( X ) is the Picard number of X . By Appell-Humbert Theorem, N S ( X ) is realized in several ways as follows : N S ( X ) = P ic ( X )/ P ic ( X ) = c ( H ( X, O ∗ X )) = { H ∶ C g × C g Ð→ C hermitian , Im ( H )( L × L ) ⊆ Z } = { E ∶ C g × C g Ð→ R alternating , E ( L × L ) ⊆ Z , E ( i ⋅ , ⋅ ) symmetric } . Let ̂ X = P ic ( X ) be the dual complex torus of X . There exists the holomorphic linebundle P over X × ̂ X uniquely determined up to isomorphism, the so-called Poincar´ebundle satisfying the following properties (PB1) and (PB2) :(PB1) P ∣ X × L ≅ L for all L ∈ ̂ X , and(PB2) P ∣ { }× ̂ X is trivial on ̂ X. We can see that H g ( X, P ) = C and H q ( X, P ) = q ≠ g. Let T Λ = V / Λ be a real torus of dimension g , where V ≅ R g is a real vector space ofdimension g and Λ is a lattice in V . Let ρ ∶ Λ Ð→ C ∗ be a character of Λ. Let B ∶ V × V Ð→ R be a real valued symmetric bilinear form on V . We define the map I B,ρ ∶ Λ × V Ð→ C ∗ by(7.4) I B,ρ ( λ, v ) = ρ ( λ ) e π B ( λ,λ ) + π B ( v,λ ) , λ ∈ Λ , v ∈ V, η ∈ C . It is easily checked that I B,ρ satisfies the following equation I B,ρ ( λ + λ , v ) = I B,ρ ( λ , λ + v ) I B,ρ ( λ , v ) , λ , λ ∈ Λ , v ∈ V. Then Λ acts on the trivial line bundle V × C over V freely by(7.5) λ ⋅ ( v, η ) = ( v + λ, I B,ρ ( λ, v ) η ) , λ ∈ Λ , v ∈ V, η ∈ C . Thus the quotient space(7.6) L ( B, ρ ) = ( V × C )/ Λhas a natural structure of a smooth (or real analytic) line bundle over a real torus T Λ . Lemma 7.1.
Suppose B ∶ V × V Ð→ R is a positive definite bilinear form on V . We definethe function θ B,ρ ∶ V Ð→ C by (7.7) θ B,ρ ( v ) = ∑ λ ∈ Λ ρ ( λ ) − e − π B ( λ,λ ) − π B ( v,λ ) , v ∈ V. Then map Θ B,ρ ∶ V Ð→ V × C defined by (7.8) Θ B,ρ ( v ) = ( v, θ B,ρ ( v )) , v ∈ V defines a smooth (or real analytic) global section of the line bundle L ( B, ρ ) .Proof. For any λ ∈ Λ and v ∈ V , we have θ B,ρ ( λ + v ) = ∑ µ ∈ Λ ρ ( µ ) − e − π B ( µ,µ ) − π B ( λ + v,µ ) = ρ ( λ ) e π B ( λ,λ ) + π B ( v,λ ) ∑ µ ∈ Λ ρ ( λ + µ ) − e − π B ( λ + µ,λ + µ ) − π B ( v,λ + µ ) = I B,ρ ( λ, v ) θ B,ρ ( v ) . Therefore Θ
B,ρ is a smooth global section of L ( B, ρ ) . ◻ Lemma 7.2.
Suppose B ∶ V × V Ð→ R is a positive definite bilinear form on V . Assume B is integral on Λ × Λ , that is, B ( Λ × Λ ) ⊂ Z . Then for any character ρ ∶ Λ Ð→ C , the function f B,ρ ∶ V Ð→ C defined by (7.9) f B,ρ ( v ) = ∑ λ ∈ Λ ρ ( λ ) e − π B ( λ,λ ) + π i B ( v,λ ) , v ∈ V is invariant under the action of Λ . Therefore f B,ρ may be regarded as a function on T Λ . Proof.
It follows immediately from the definition. ◻ We see that L Λ = Z g + i Λ ⊂ C g is a lattice in C g . We consider the complex torus T Λ = C g / L Λ . We define the R -linear map S B ∶ C g × C g Ð→ R and E B ∶ C g × C g Ð→ R by(7.10) S B ( x, y ) = B ( x , y ) + B ( x , y ) and(7.11) E B ( x, y ) = B ( x , y ) − B ( x , y ) , where x = x + i x ∈ C g and y = y + i y ∈ C g with x , x , y , y ∈ R g . It is easily seen that S B is symmetric and E B is alternating. We note that S B ( x, y ) = E B ( i x, y ) for all x, y ∈ C g . We define the hermitian form H B ∶ C g × C g Ð→ C by(7.12) H B ( x, y ) ∶ = S B ( x, y ) + i E B ( x, y ) , x, y ∈ C g . Moreover we assume that E B is integral on L Λ × L Λ . Let α ∶ L Λ Ð→ C ∗ be a semi-characterof L Λ for H B such that α ( ℓ + ℓ ) = α ( ℓ ) α ( ℓ ) e π i E B ( ℓ ,ℓ ) , ℓ , ℓ ∈ L Λ . OLARIZED REAL TORI 35
Then the mapping J B,α ∶ L Λ × C g Ð→ C ∗ defined by(7.13) J B,α ( ℓ, z ) = α ( ℓ ) e π H B ( ℓ,ℓ ) + π H B ( z,ℓ ) , ℓ ∈ L Λ , z ∈ C g is an automorphic factor for L Λ on C g . Clearly L Λ acts on the trivial line bundle C g × C over C g freely by(7.14) ℓ ⋅ ( z, ξ ) = ( ℓ + z, J B,α ( ℓ, z ) ξ ) , ℓ ∈ L Λ , z ∈ C g , ξ ∈ C . The quotient L ( B, α ) ∶ = ( C g × C )/ L Λ of C g × C by L Λ has a natural structure of a holomorphic line bundle over a complex torus T Λ .In summary, to each pair ( B, α ) with s symmetric R -bilinear form B on V such that E B is integral on L Λ × L Λ and a semi-character α for H B there is associated the holomorphicline bundle L ( B, α ) over T Λ .We assume that B is non-degenerate of signature ( r, s ) with r + s = g. Then the hermitianform H B is also non-degenerate of signature ( r, s ) . Moreover we assume that E B is integralon L Λ × L Λ . Under these assumptions, Matsushima [15] proved that the cohomology group H q ( T Λ , L ( B, α )) = q ≠ s and that H s ( T Λ , L ( B, α )) is identified with the complexvector space of all C ∞ functions f on C g satisfying the following conditions :(a) f is a differentiable theta functions for the automorphic factor J B,α ; namely we have f ( ℓ + z ) = J B,α ( ℓ, z ) f ( z ) , ℓ ∈ L Λ , z ∈ C g , (b) ∂f∂z i = i ∈ { , , ⋯ , r } and ∂f∂z i + π z i f = i ∈ { r + , ⋯ , g } , where ( z , ⋯ , z g ) is the coordinate of C g determined by a privileged basis of C g for thehermitian form H B . We can show that the cohomology group H s ( T Λ , L ( B, α ) ⊗ ) defines asmooth embedding of T Λ into the projective space P d ( C ) with d + = dim H s ( T Λ , L ( B, α ) ⊗ ) which is holomorphic in z , ⋯ , z r and anti-holomorphic in z r + , ⋯ , z g (cf. [15] and [17]).We consider the canonical semi-character γ Λ ,B of L Λ defined by(7.15) γ Λ ,B ( κ + i λ ) ∶ = e π i E B ( κ, i λ ) , κ ∈ Z g , λ ∈ Λ . Then γ Λ ,B defines the holomorphic line bundle L ( B, γ Λ ,B ) over a complex torus T Λ . Forany z ∈ T Λ we denote by T z the translation of T Λ by z . Let π Λ ∶ C g Ð→ T Λ be the naturalprojection. Then there exists an element c Λ ,B,α of C g such that(7.16) L ( B, α ) = T ∗ π Λ ( c Λ ,B,α ) L ( B, γ Λ ,B ) .c Λ ,B,α is called a characteristic of the holomorphic line bundle L ( B, α ) . We refer to [13] fordetail.Now we let T Λ = V / Λ be a polarized real torus of dimension g . Its associated polarizedreal abelian variety A Λ = C g / L Λ , L Λ = Z g + i Λ admits a positive definite hermitian form H Λ on C g whose imaginary part Im ( H Λ ) is integralon Λ × Λ (cf. [17, p. 35]). We write H Λ ( x, y ) = S Λ ( x, y ) + i E Λ ( x, y ) , x, y ∈ C g , where S Λ and E Λ are the real part (resp. imaginary part) of H Λ respectively. We knowthat S Λ is a real valued symmetric bilinear form on V and E Λ is a real valued alternatingbilinear form on V . Let α Λ ∶ L Λ Ð→ C ∗ be a canonical semi-character of L Λ defined by(7.17) α Λ ( κ + i λ ) ∶ = e π i E Λ ( κ, i λ ) , κ ∈ Z g , λ ∈ Λ . We let J H Λ ,α Λ ∶ L Λ × C g Ð→ C ∗ be the automorphic factor for Λ on V that is canonicallygiven by(7.18) J H Λ ,α Λ ( ℓ, z ) = α Λ ( ℓ ) e π H Λ ( ℓ,ℓ ) + π H Λ ( z,ℓ ) , ℓ ∈ L Λ , z ∈ C g . Obviously L Λ acts on C g × C freely by ℓ ⋅ ( z, ξ ) = ( ℓ + z, J H Λ ,α Λ ( ℓ, z ) ξ ) , ℓ ∈ L Λ , z ∈ C g , ξ ∈ C . So the quotient space(7.19) L ( H Λ , α Λ ) ∶ = ( C g × C )/ L Λ has a natural structure of a holomorphic line bundle over an abelian variety A Λ .Now we define the map Φ Λ ∶ T Λ Ð→ A Λ by(7.20) Φ Λ ( v + Λ ) ∶ = i v + L Λ , v ∈ R g . Φ Λ is a well defined injective mapping. It is well known that H q ( A Λ , L ( H Λ , α Λ )) = q ≠ L ( H Λ , α Λ ) ⊗ n for any positiveinteger n ≥ A Λ as a closed complex manifold in aprojective complex manifold P d ( C ) (cf. [17, pp. 29–33]). Therefore we have a differentiableembedding of T Λ into a complex projective space P d ( C ) and hence into a real projectivespace P N ( R ) with large enough N > . We will characterize the pullback L ( α Λ ) ∶ = Φ ∗ Λ L ( H Λ , α Λ ) . We first define the automorphicfactor I α Λ ∶ Λ × R g Ð→ C ∗ by(7.21) I α Λ ( λ, v ) ∶ = α Λ ( i λ ) e π H Λ ( λ,λ ) + πH Λ ( v,λ ) , λ ∈ Λ , v ∈ R g . This automorphic factor I α Λ yields the smooth (or real analytic) line bundle over T Λ whichis nothing but the pullback L ( α Λ ) . We observe that if θ is a holomorphic theta functionfor L ( H Λ , α Λ ) , then the function f θ ∶ R g Ð→ C defined by f θ ( v ) ∶ = θ ( i v ) , v ∈ R g defines aglobal smooth (or real analytic) section of L ( α Λ ) . Now we will show that a holomorphic line bundle L ( H Λ , α Λ ) over A Λ naturally yields asmooth line bundle over a polarized torus T Λ . Let B Λ be the restriction of S Λ to R g × R g . First we define the automorphic factor I B Λ ,α Λ ∶ Λ × R g Ð→ C ∗ by(7.22) I B Λ ,α Λ ( λ, v ) ∶ = α Λ ( i λ ) e πB Λ ( λ,λ ) + πB Λ ( v,λ ) , λ ∈ Λ , v ∈ R g . This automorphic factor I B Λ ,α Λ ( λ, v ) yields a smooth line bundle(7.23) L ( B Λ , α Λ ) ∶ = ( R g × C )/ Λover a polarized real torus T Λ . Since B Λ is positive definite, according to Lemma 4.1, thespace Γ ( T Λ , L ( B Λ , α Λ )) of smooth (or real analytic) global sections of L ( B Λ , α Λ ) is not zero. OLARIZED REAL TORI 37 If B Λ is integral on Λ × Λ, according to Lemma 4.2, we see that the function f Λ ,α Λ ∶ R g Ð→ C defined by f Λ ,α Λ ( v ) = ∑ λ ∈ Λ α Λ ( i λ ) e − π B Λ ( λ,λ ) + π i B Λ ( v,λ ) , v ∈ R g is a function on T Λ . So far we have proved the following.
Theorem 7.1.
Let T Λ = V / Λ be a polarized real torus of dimension g . Then there is asmooth line bundle L ( B Λ , α Λ ) over T Λ which is constructed canonically by (7.23). Example 7.1.
Let Y ∈ P g be a g × g positive definite symmetric real matrix. ThenΛ Y = Y Z g is a lattice in R g . Then the g -dimensional torus T Y = R g / Λ Y is a principallypolarized real torus. Indeed, A Y = C g / L Y , L Y = Z g + i Λ Y is a princially polarized real abelian variety (cf. Example 6.1). Its corresponding hermitianform H Y is given by H Y ( x, y ) = S Y ( x, y ) + i E Y ( x, y ) = t x Y − y, x, y ∈ C g , where S Y and E Y denote the real part and the imaginary part of H Y respectively. Let α ∶ L Y Ð→ C ∗ be a semi-character of L Y . To a pair ( H Y , α ) the canonical automorphicfactor J Y,α ∶ L Y × C g Ð→ C is associated by J Y,α ( ℓ, z ) = α ( ℓ ) e π i t ℓ Y − ℓ + π i t zY − ℓ , ℓ ∈ L Y , z ∈ C g . The associated automorphic factor I Y,α ∶ Λ Y × R g Ð→ C ∗ is given by I Y,α ( λ, v ) = α ( i λ ) e π t λ Y − λ + π t v Y − λ , λ ∈ Λ Y , v ∈ R g . We get the associated line bundle L ( B Y , α ) = ( R g × C )/ Λ Y given by I Y,α , where B Y is the restriction of S Y to R g × R g . Then the function θ Y,α ∶ R g Ð→ C defined by θ Y,α ( v ) = ∑ λ ∈ Λ Y α ( i λ ) e − π t λ Y − λ − π t v Y − λ , v ∈ R g yields a smooth global section of L ( B Y , α ) over a real torus T Y . The canonical semi-character α Y of L Y is given by α Y ( κ + i λ ) = e − π i t κY − λ , κ ∈ Z g , λ ∈ Λ Y . Moduli Space for Principally Polarized Real Tori
We have the natural action of GL ( g, R ) on P g given by(8.1) A ∗ Y = AY t A, A ∈ GL ( g, R ) , Y ∈ P g . We put G g = GL ( g, Z ) (see Notations in the introduction). The fundamental domain R g for G g / P g which was found by H. Minkowski [16] is defined as a subset of P g consisting of Y = ( y ij ) ∈ P g satisfying the following conditions (M.1)–(M.2) (cf. [10, p. 191] or [14,p. 123]):(M.1) aY t a ≥ y kk for every a = ( a i ) ∈ Z g in which a k , ⋯ , a g are relatively prime for k = , , ⋯ , g .(M.2) y k,k + ≥ k = , ⋯ , g − . We say that a point of R g is Minkowski reduced or simply M - reduced . R g has the followingproperties (R1)-(R6):(R1) For any Y ∈ P g , there exist a matrix A ∈ GL ( g, Z ) and R ∈ R g such that Y = R [ A ] (cf. [10, p. 191] or [14, p. 139]). That is, GL ( g, Z ) ○ R g = P g . (R2) R g is a convex cone through the origin bounded by a finite number of hyperplanes. R g is closed in P g (cf. [14, p. 139]).(R3) If Y and Y [ A ] lie in R g for A ∈ GL ( g, Z ) with A ≠ ± I g , then Y lies on the boundary ∂ R g of R g . Moreover R g ∩ ( R g [ A ]) ≠ ∅ for only finitely many A ∈ GL ( g, Z ) (cf. [14, p. 139]).(R4) If Y = ( y ij ) is an element of R g , then y ≤ y ≤ ⋯ ≤ y gg and ∣ y ij ∣ < y ii for 1 ≤ i < j ≤ g. We refer to [10, p. 192] or [14, pp. 123-124].For Y = ( y ij ) ∈ P g , we put dY = ( dy ij ) and ∂∂Y = ( + δ ij ∂∂y ij ) . For a fixed element A ∈ GL ( g, R ) , we put Y ∗ = A ⋆ Y = AY t A, Y ∈ P g . Then(8.2) dY ∗ = A dY t A and ∂∂Y ∗ = t A − ∂∂Y A − . We consider the following differential operators(8.3) D k = σ (( Y ∂∂Y ) k ) , k = , , ⋯ , g, where σ ( M ) denotes the trace of a square matrix M . By Formula (8.2), we get ( Y ∗ ∂∂Y ∗ ) i = A ( Y ∂∂Y ) i A − for any A ∈ GL ( g, R ) . So each D i is invariant under the action (8.1) of GL ( g, R ) .Selberg [20] proved the following. Theorem 8.1.
The algebra D ( P g ) of all differential operators on P g invariant under theaction (8.1) of GL ( g, R ) is generated by D , D , ⋯ , D g . Furthermore D , D , ⋯ , D g are al-gebraically independent and D ( P g ) is isomorphic to the commutative ring C [ x , x , ⋯ , x g ] with g indeterminates x , x , ⋯ , x g . OLARIZED REAL TORI 39
Proof.
The proof can be found in [14, pp. 64-66]. (cid:3)
We can see easily that ds = σ (( Y − dY ) ) is a GL ( g, R ) -invariant Riemannian metric on P g and its Laplacian is given by∆ = σ (( Y ∂∂Y ) ) . We also can see that dµ g ( Y ) = ( det Y ) − g + ∏ i ≤ j dy ij is a GL ( g, R ) -invariant volume element on P g . The metric ds on P g induces the metric ds R on R g . Minkowski [16] calculated the volume of R g explicitly. P g parameterizes principally polarized real tori of dimension g . The Minkowski domain R g is the moduli space for isomorphism classes of principally polarized real tori of dimension g . According to (R2) we see that R g is a semi-algebraic set with real analytic structure.Unfortunately R g does not admit the structure of a real algebraic variety and does notadmit a compactification which is defined over the rational number field Q . We see that R g is real analytically isomorphic to the semi-algebraic subset X g ( , ) of X g R . We define theembedding Φ g ∶ P g Ð→ H g by(8.4) Φ g ( Y ) = i Y, Y ∈ P g . We have the following inclusions P g Φ g Ð→ i P g ↪ H g ↪ H g ⊂ H ∗ g . G g acts on P g and i P g , Γ ⋆ g acts on H g , and Γ g acts on H g and H ∗ g . It might be interestingto characterize the boundary points of the closure of i P g (or P g ) in H ∗ g explicitly. In Section5 we reviewed Silhol’s compactification X g R of X g R which is analogous to the Satake-Baily-Borel compactification. The theory of automorphic forms on R g has been developed bySelberg [20], Maass [14] et al past a half century. According to Theorem 5.1, X g R is aconnected compact Hausdorff space containing X g R as an open dense subset of X g R . But X g R does not admit an algebraic structure.For any positive integer h ∈ Z + , we let(8.5) GL g,h ∶ = GL ( g, R ) ⋉ R ( h,g ) be the semi-direct product of GL ( g, R ) and R ( h,g ) with the multiplication law(8.6) ( A, a ) ⋅ ( B, b ) = ( AB, a t B − + b ) , A, B ∈ GL ( g, R ) , a, b ∈ R ( h,g ) . Then we have the natural action of GL g,h on the Minkowski-Euclid space P g × R ( h,g ) definedby(8.7) ( A, a ) ⋅ ( Y, ζ ) = ( AY t A, ( ζ + a ) t A ) , ( A, a ) ∈ GL g,h , Y ∈ P g , ζ ∈ R ( h,g ) . For a variable ( Y, V ) ∈ P g × R ( h,g ) with Y ∈ P g and V ∈ R ( h,g ) , we put Y = ( y µν ) with y µν = y νµ , V = ( v kl ) ,dY = ( dy µν ) , dV = ( dv kl ) , [ dY ] = ∏ µ ≤ ν dy µν , [ dV ] = ∏ k,l dv kl , and ∂∂Y = ( + δ µν ∂∂y µν ) , ∂∂V = ( ∂∂v kl ) , where 1 ≤ µ, ν, l ≤ g and 1 ≤ k ≤ h. Lemma 8.1.
For all two positive real numbers A and B , the following metric ds g,h ; A,B on P g × R ( h,g ) defined by (8.8) ds g,h ; A,B = A σ ( Y − dY Y − dY ) + B σ ( Y − t ( dV ) dV ) is a Riemannian metric on P g × R ( h,g ) which is invariant under the action (8.7) of GL g,h .The Laplacian ∆ g,h ; A,B of ( P g × R ( h,g ) , ds g,h ; A,B ) is given by ∆ g,h ; A,B = A σ (( Y ∂∂Y ) ) − h A σ ( Y ∂∂Y ) + B ∑ k ≤ p (( ∂∂V ) Y t ( ∂∂V )) kp . Moreover ∆ g,h ; A,B is a differential operator of order 2 which is invariant under the action(8.7) of GL g,h . Proof.
For a fixed element ( A, a ) ∈ GL g,h , we set ( Y ∗ , V ∗ ) = ( A, a ) ⋅ ( Y, V ) . Then Y ∗ = A Y t A, V ∗ = ( V + a ) t A. The first statement follows immediately from the fact that dY ∗ = A dY t A and dV ∗ = dV t A. Using the formula (13) in [8, p. 245], we can compute the Laplacian ∆ g,h ; A,B of ( P g × R ( h,g ) , ds g,h ; A,B ) . The last statement follows from the fact that ∂∂Y ∗ = t A − ∂∂Y A − , ∂∂V ∗ = ∂∂V ⋅ A − . (cid:3) Lemma 8.2.
The following volume element dv g,h ( Y, V ) on P g × R ( h,g ) defined by (8.9) dv g,h ( Y, V ) = ( det Y ) − g + h + [ dY ][ dV ] is invariant under the action (8.7) of GL g,h . OLARIZED REAL TORI 41
Proof.
For a fixed element ( A, a ) ∈ GL g,h , we set ( Y ∗ , V ∗ ) = ( A, a ) ⋅ ( Y, V ) = ( AY t A, ( V + a ) t A ) . Let ∂ ( Y ∗ ,V ∗ ) ∂ ( Y,V ) be the Jacobian determinant of the action (8.7) of GL g,h on P g × R ( h,g ) . It isknown that the Jacobian determinant of the action Y z→ Y ∗ is given by ∂ ( Y ∗ ) ∂ ( Y ) = ( det A ) g + . Take the diagonal matrix g = ( d , ⋯ , d g ) with distinct real numbers d i . Obviously if a = ( a kl ) , V = ( v kl ) and V ∗ = ( v ∗ kl ) , then v ∗ kl = ( v kl + a kl ) d l for all k, l. Thus we have(8.10) ∂ ( V ∗ ) ∂ ( V ) = ( d ⋯ d g ) h = ( det A ) h . Since the set of all g × g real matrices whose eigenvalues are all distinct is everywhere densein GL ( g, R ) , and ∂ ( V ∗ ) ∂ ( V ) is a rational function, the relation (8.10) holds for any A ∈ GL ( g, R ) . It is easy to see that ∂ ( Y ∗ , V ∗ ) ∂ ( Y, V ) = ∂ ( Y ∗ ) ∂ ( Y ) ⋅ ∂ ( V ∗ ) ∂ ( V ) . Thus we obtain [ dY ∗ ][ dV ∗ ] = ∣ det A ∣ g + h + [ dY ][ dV ] . Since det Y ∗ = ( det A ) det Y, we have ( det Y ∗ ) − g + h + [ dY ∗ ][ dV ∗ ] = ( det Y ) − g + h + [ dY ][ dV ] . Hence the volume element (8.9) is invariant under the action (8.7). (cid:3)
It is known that dµ g ( Y ) ∶ = ( det Y ) − g + [ dY ] is a volume element on P g invariant under the action (8.1) of GL ( g, R ) (cf. [14, p. 23]). Let r be a positive integer with 0 < r < g. We define a bijective transformation P g Ð→ P r × P s × R ( s,r ) , r + s = g, Y z→ ( F, G, H ) by(8.11) Y = ( F G ) [( I r H I s )] , Y ∈ P n , F ∈ P r , G ∈ P s , H ∈ R ( s,r ) . According to [14, pp. 24-26], we obtain(8.12) [ dY ] = ( det G ) r [ dF ][ dH ][ dG ] , equivalently(8.13) dµ g ( Y ) = ( det F ) − s ( det G ) r dµ r ( F ) dµ s ( G ) [ dH ] . Therefore we get(8.14) dv g,h ( Y, V ) = ( det F ) − h + s ( det G ) r − h dµ r ( F ) dµ s ( G ) [ dH ] [ dV ] . Similarly if Y ∈ P g , g = r + s with 0 < r < g , we write(8.15) Y = ( P Q ) [( I r R I s )] , Y ∈ P g , P ∈ P r , Q ∈ P s , R ∈ R ( r,s ) . According to [14, pp. 27], we obtain(8.16) [ dY ] = ( det P ) s [ dP ][ dQ ][ dR ] , equivalently(8.17) dµ g ( Y ) = ( det P ) s ( det G ) − r dµ r ( P ) dµ s ( Q ) [ dR ] . Therefore we get(8.18) dv g,h ( Y, V ) = ( det P ) s − h ( det G ) − r + h dµ r ( P ) dµ s ( Q ) [ dR ] [ dV ] . The coordinates ( F, G, H ) or ( P, Q, R ) are called the partial Iwasawa coordinates on P g . Theorem 8.2.
Any geodesic through the origin ( I g , ) is of the form γ ( t ) = ( λ ( t )[ k ] , Z ( ∫ t λ ( t − s ) ds ) [ k ]) , where k is a fixed element of O ( g ) , Z is a fixed h × g real matrix, t is a real variable, λ , λ , ⋯ , λ g are fixed real numbers but not all zero and λ ( t ) ∶ = diag ( e λ t , ⋯ , e λ g t ) . Furthermore, the tangent vector γ ′ ( ) of the geodesic γ ( t ) at ( I g , ) is ( D [ k ] , Z ) , where D = diag ( λ , ⋯ , λ g ) . Proof.
Let W = ( X, Z ) be an element of p with X ≠ . Then the curve α ( t ) = exp tW = ( e tX , Z ( ∫ t e − sX ds )) , t ∈ R is a geodesic in GL g,h with α ′ ( ) = W passing through the identity of GL g,h . Thus the curve γ ( t ) = α ( t ) ⋅ ( I g , ) = ( e tX , Z ( ∫ t e − sX ds ) e tX ) is a geodesic in P g × R ( h,g ) passing through the origin ( I g , ) . Since X is a symmetric realmatrix, there is a diagonal matrix Λ = diag ( λ , ⋯ , λ g ) with λ , ⋯ , λ g ∈ R such that X = t k Λ k for some k ∈ O ( g ) , where λ , ⋯ , λ n are real numbers and not all zero. Thus we may write γ ( t ) = ( ( δ kl e λ k t )[ k ] , Z ( ∫ t e ( t − s ) Λ ds ) [ k ] ) . Hence this completes the proof. (cid:3)
Theorem 8.3.
Let ( Y , V ) and ( Y , V ) be two points in P g × R ( h,g ) . Let g be an element in GL ( g, R ) such that Y [ t g ] = I g and Y [ t g ] is diagonal. Then the length s (( Y , V ) , ( Y , V )) OLARIZED REAL TORI 43 of the geodesic joining ( Y , V ) and ( Y , V ) for the GL g,h -invariant Riemannian metric ds g,h ; A,B is given by (8.19) s (( Y , V ) , ( Y , V )) = A ⎧⎪⎪⎨⎪⎪⎩ g ∑ j = ( ln t j ) ⎫⎪⎪⎬⎪⎪⎭ / + B ∫ ⎛⎝ g ∑ j = ∆ j e −( ln t j ) t ⎞⎠ / dt, where ∆ j = ∑ hk = ̃ v kj ( ≤ j ≤ g ) with ( V − V ) t g = (̃ v kj ) and t , ⋯ , t g denotes the zeros of det ( t Y − Y ) .Proof. Without loss of generality we may assume that ( Y , V ) = ( I g , ) and ( Y , V ) = ( T, ̃ V ) with T = diag ( t , ⋯ , t n ) diagoanl because the element ( g, − V ) ∈ GL g,h can be re-garded as an isometry of P g × R ( h,g ) for the Riemannian metric ds g,h ; A,B (cf. Lemma 8.1).Let γ ( t ) = ( α ( t ) , β ( t )) with 0 ≤ t ≤ P g × R ( h,g ) joining two points γ ( ) = ( Y , V ) and γ ( ) = ( Y , V ) , where α ( t ) is the uniquely determined curve in P g and β ( t ) is the uniquely determined curve in R ( h,g ) . We now use the partial Iwasawa coordinates in P g . Then if Y ∈ P g , we write for anypositive integer r with 0 < r < g, r + s = g,Y = ( F G ) [( I r H I s )] , F ∈ P r , G ∈ P s , H ∈ R ( h,g ) . For V ∈ R ( h,g ) , we write V = ( R, S ) , R ∈ R ( h,r ) , S ∈ R ( h,s ) . Now we express ds g,h ; A,B in terms of
F, G, H, R and S. Lemma 8.3. ds g,h ; A,B = A ⋅ { σ (( F − dF ) ) + σ (( G − dG ) ) + σ ( F − t ( dH ) G dH )} + B ⋅ { σ ( F − t ( dR ) dR ) + σ (( G − + F − [ t H ]) t ( dS ) dS )} − B ⋅ σ ( F − t H t ( dS ) dR ) . Proof of Lemma 8.3.
First we see that if Y ∈ P g , then Y − = ( F − G − ) [( I r − t H I s )] = ( F − − F − t H − HF − G − + F − [ t H ]) ,dY = ( dF + dG [ H ] + t ( dH ) ⋅ GH + t HG ⋅ dH t ( dH ) ⋅ G + t H ⋅ dGdG ⋅ H + G ⋅ dH dG ) and dV = ( dR, dS ) . For brevity, we put dY ⋅ Y − = ( L L L L ) and Y − t ( dV ) dV = ( M M M M ) . Here L , L , L and L denote the r × r , r × s , s × r and s × s matrix valued differential oneforms respectively, and M , M , M and M denote the r × r , r × s , s × r and s × s matrixvalued differential two forms respectively.By an easy computation, we get L = dF ⋅ F − + t HG ⋅ dH ⋅ F − ,L = − dF ⋅ F − t H − t HG ⋅ dH ⋅ F − t H + t ( dH ) + t H ⋅ dG ⋅ G − ,L = G ⋅ dH ⋅ F − ,L = dG ⋅ G − − G ⋅ dH ⋅ F − t H,M = F − t ( dR ) dR − F − t H t ( dS ) dR,M = − HF − t ( dR ) dS + ( G − + F − [ t H ]) t ( dS ) dS. Therefore we have ds g,h ; A,B = A ⋅ σ (( dY ⋅ Y − ) ) + B ⋅ σ ( Y − t ( dV ) dV ) = A ⋅ { σ ( L + L L ) + σ ( L L + L )} + B ⋅ { σ ( M ) + σ ( M ) } = A ⋅ { σ (( F − dF ) ) + σ (( G − dG ) ) + σ ( F − t ( dH ) G dH )} + B ⋅ { σ ( F − t ( dR ) dR ) + σ (( G − + F − [ t H ]) t ( dS ) dS )} − B ⋅ σ ( F − t H t ( dS ) dR ) . ◻ Let s (( Y , V ) , ( Y , V )) be the length of the geodesic γ ( t ) = ( α ( t ) , β ( t )) with 0 ≤ t ≤ . We put α ( t ) = ( F ( t ) G ( t )) [( I r H ( t ) I s )] , β ( t ) = ( R ( t ) , S ( t )) , ≤ t ≤ , where F ( t ) , G ( t ) , H ( t ) , R ( t ) and S ( t ) are the uniquely determined curves in P r , P s , R ( s,r ) , R ( h,r ) and R ( h,s ) respectively.then we have s (( Y , V ) , ( Y , V ))= A ⋅ ∫ { σ (( F − dFdt ) ) + σ (( G − dGdt ) ) + σ ( F − t ( dHdt ) G dHdt )} / dt + B ⋅ ∫ { σ ( γ ( t ) − t ( dVdt ) dVdt )} / dt ≥ A ⋅ ∫ { σ (( F − dFdt ) ) + σ (( G − dGdt ) )} / dt + B ⋅ ∫ { σ ( F − t ( dRdt ) dRdt ) + σ ( G − t ( dSdt ) dSdt ) } / dt. OLARIZED REAL TORI 45
The reason is that the quadratic form σ ( F − t ( dH ) G dH ) is positive definite. Indeed, if M, N ∈ GL ( g, R ) such that F = t M M and G = t N N, then σ ( F − t ( dH ) G dH ) = σ ( t W W ) , W ∶ = N ⋅ dH ⋅ M − . If σ ( F − t ( dHdt ) G dHdt ) = , then dHdt = H ( t ) is constant in the interval [ , ] . Since H ( ) = , H ( t ) = ( ≤ t ≤ ) . Moreover the curve α ( t ) must be diagonal, that is, α ( t ) = ( δ µν e χ ν ( t ) ) , χ ν ( ) = , χ ν ( ) = ln t ν , ≤ ν ≤ g, where g ν ( t ) ( ≤ ν ≤ g ) are continuously differentiable in [0,1]. Thus we have dαdt = ( δ µν e χ ν ( t ) dχ ν dt ) and hence α ( t ) − dαdt = ( δ µν dχ ν dt ) . Therefore we have ∫ { σ (( α ( t ) − dαdt ) )} / dt = ∫ { σ (( F − dFdt ) ) + σ (( G − dGdt ) )} / dt = ∫ ⎧⎪⎪⎨⎪⎪⎩ n ∑ j = ( dχ j dt ) ⎫⎪⎪⎬⎪⎪⎭ / dt. The minimum value of ∑ gj = ( dχ j dt ) is obtained if the curve α ( t ) is the straight line, i.e., χ j ( t ) = t ln t j ( ≤ j ≤ g ) , ≤ t ≤ ( χ , ⋯ , χ g ) -space. Thus we get ∫ { σ (( α ( t ) − dαdt ) )} / dt = ⎧⎪⎪⎨⎪⎪⎩ n ∑ j = ( ln t j ) ⎫⎪⎪⎬⎪⎪⎭ / . We put β ( t ) = ( β kj ( t ) ) with 0 ≤ t ≤ , ≤ k ≤ h, ≤ j ≤ g. Then we obtain ∫ { σ ( α ( t ) − t ( dβdt ) dβdt )} / dt = ∫ { σ ( F − t ( dRdt ) dRdt ) + σ ( G − t ( dSdt ) dSdt )} / dt = ∫ ⎧⎪⎪⎨⎪⎪⎩ g ∑ j = h ∑ k = e − t ln t j ( dβ kj dt ) ⎫⎪⎪⎬⎪⎪⎭ / dt. Each curve β kj ( t ) ( ≤ t ≤ ) is a curve in R such that β k j ( ) = β kj ( ) = ̃ v kj . Thuseach curve β kj ( t ) must be a straight line, that is, for all k, j with 1 ≤ k ≤ h and 1 ≤ j ≤ g , β kj ( t ) = ̃ v kj t, ≤ t ≤ . Therefore we have ∫ { σ ( α ( t ) − t ( dβdt ) dβdt )} / dt = ∫ ⎧⎪⎪⎨⎪⎪⎩ n ∑ j = e − t ln t j ( h ∑ k = ̃ v kj )⎫⎪⎪⎬⎪⎪⎭ / dt = ∫ ⎛⎝ g ∑ j = ∆ j e − t ln t j ⎞⎠ / dt. Finally we obtain(8.20) s (( Y , V ) , ( Y , V )) = A ⎧⎪⎪⎨⎪⎪⎩ g ∑ j = ( ln t j ) ⎫⎪⎪⎬⎪⎪⎭ / + B ∫ ⎛⎝ g ∑ j = ∆ j e −( ln t j ) t ⎞⎠ / dt. Hence we complete the proof. (cid:3)
For a fixed element ( A, a ) ∈ GL g,h , we let Θ A,a ∶ P g × R ( h,g ) Ð→ P g × R ( h,g ) be the mappingdefined by Θ A,a ( Y, V ) ∶ = ( A, a ) ⋅ ( Y, V ) , ( Y, V ) ∈ P g × R ( h,g ) . We consider the behaviour of the differential map d Θ A,a of Θ
A,a at ( I g , ) . Then d Θ A,a isgiven by d Θ A,a ( u, v ) = ( Au t A, v t A ) , where ( u, v ) is a tangent vector of P g × R ( h,g ) at ( I g , ) . We let ˜ θ be the involution of GL g,h defined by˜ θ (( A, a )) ∶ = ( t A − , − a ) , ( A, a ) ∈ GL g,h . Then the differential map of ˜ θ at ( I g , ) , denoted by the same notation ˜ θ is given by˜ θ ∶ g Ð→ g , ˜ θ ( X, Z ) = ( − t X, − Z ) , OLARIZED REAL TORI 47 where X ∈ R ( g,g ) and Z ∈ R ( h,g ) . We note that k is the (+1)-eigenspace of ˜ θ and p = {( X, Z ) ∣ X ∈ R ( g,g ) , X = t X, Z ∈ R ( h,g ) } is the (-1)-eigenspace of ˜ θ .Now we consider some differential forms on P g × R ( h,g ) which are invariant under theaction of GL ( g, Z ) ⋉ Z ( h,g ) . We let G g,h ∶ = GL ( g, Z ) ⋉ Z ( h,g ) be the discrete subgroup of GL g,h . Let α ∗ = ∑ µ ≤ ν f µν ( Y, V ) dy µν + h ∑ k = g ∑ l = φ kl ( Y, V ) dv kl be a differential 1-form on P g × R ( h,g ) that is invariant under the action of G g,h . We put e µν = ⎧⎪⎪⎨⎪⎪⎩ µ = ν otherwise . We let f ( Y, V ) = ( e µν f µν ( Y, V )) and φ ( Y, V ) = t ( φ kl ( Y, V )) , where f ( Y, V ) is a g × g matrix with entries f µν ( Y, V ) and φ ( Y, V ) is a g × h matrix withentries φ kl ( Y, V ) . Then α ∗ = σ ( f dY + φ dV ) . If ˜ γ = ( γ, α ) ∈ G g,h with γ ∈ GL ( g, Z ) and α ∈ Z ( h,g ) , then we have the following transfor-mation relation(8.21) f ( γY t γ, ( V + α ) t γ ) = t γ − f ( Y, V ) γ − and(8.22) φ ( γY t γ, ( V + α ) t γ ) = t γ − φ ( Y, V ) . We let ω = dy ∧ dy ∧ ⋯ ∧ dy nn ∧ dv ∧ ⋯ ∧ dv hg be a differential form on P g × R ( h,g ) of degree ̃ N ∶ = g ( g + ) + gh. If ω = h ( Y, V ) ω is a differentialform on P g × R ( h,g ) of degree ˜ N that is invariant under the action of G g,h . Then the function h ( Y, V ) satisfies the transformation relation(8.23) h ( γY t γ, ( V + α ) t γ ) = ( det γ ) −( g + h + ) h ( Y, V ) for all γ ∈ GL ( g, Z ) and α ∈ Z ( h,g ) . We write ω = dy ∧ dy ∧ ⋯ ∧ dy gg and ω = dv ∧ ⋯ ∧ dv hg . Now we define ω ab = ǫ ab ⋀ ≤ µ ≤ ν ≤ g ( µ,ν )≠( a,b ) dy µν ∧ ω , ≤ a ≤ b ≤ g and ˜ ω cd = ˜ ǫ cd ω ∧ ⋀ ≤ k ≤ h, ≤ l ≤ g ( k,l )≠( c,d ) dv kl , ≤ c ≤ h, ≤ d ≤ g. Here the signs ǫ ab and ˜ ǫ cd are determined by the relations ǫ ab ω ab ∧ dy ab = ω and ˜ ǫ cd ω cd ∧ dv cd = ω . We let β ∗ = ∑ µ ≤ ν s µν ( Y, V ) ω µν + h ∑ k = g ∑ l = ϕ kl ( Y, V ) ˜ ω kl be a differential form on P g × R ( h,g ) of degree ̃ N − G g,h , where s µν ( Y, V ) and ϕ kl are smooth functions on P g × R ( h,g ) . We set s = ( ǫ µν s µν ) , ǫ µν = ǫ νµ , s µν = s νµ and ϕ = ( ˜ ǫ kl ϕ kl ) . If we write Ω ( Y, V ) = ( s ( Y, V ) ϕ ( Y, V )) , then we obtain β ∗ ∧ ( dYdV ) = Ω ω . If ˜ γ = ( γ, α ) ∈ G g,h , then we have the following transformation relations :(8.24) s ( γY t γ, ( V + α ) t γ ) = ( det γ ) −( g + h + ) γ s ( Y, V ) t γ and(8.25) ϕ ( γY t γ, ( V + α ) t γ ) = ( det γ ) −( g + h + ) ϕ ( Y, V ) t γ. G g,h acts on P g × R ( h,g ) properly discontinuously. The quotient space(8.26) G g,h /( P g × R ( h,g ) ) may be regarded as a family of principally polarized real tori of dimension gh . To eachequivalence class [ Y ] ∈ G g / P g with Y ∈ P g we associate a principally polarized real torus T [ h ] Y = T Y × ⋯ × T Y with T = R g / Λ Y , where Λ Y = Y Z g is a lattice in R g .Let Y and Y be two elements in P g with [ Y ] ≠ [ Y ] , that is, Y ≠ A Y t A for all A ∈ G g . We put Λ i = Y i Z g for i = , . Then a torus T = R g / Λ is diffeomorphic to T = R g / Λ assmooth manifolds but T is not isomorphic to T as polarized tori. Lemma 8.4.
The following set (8.27) R g,h ∶ = { ( Y, V ) ∣ Y ∈ R g , ∣ v kj ∣ ≤ , V = ( v kj ) ∈ R ( h,g ) } is a fundamental set for G g,h / P g × R ( h,g ) . Proof.
It is easy to see that R g,h is a fundamental set for G g,h / P g × R ( h,g ) . We leave thedetail to the reader. ◻ For two positive integers g and h , we consider the Heisenberg group H ( g,h ) R = { ( λ, µ ; κ ) ∣ λ, µ ∈ R ( h,g ) , κ ∈ R ( h,h ) , κ + µ t λ symmetric } endowed with the following multiplication law ( λ, µ ; κ ) ○ ( λ ′ , µ ′ ; κ ′ ) = ( λ + λ ′ , µ + µ ′ ; κ + κ ′ + λ t µ ′ − µ t λ ′ ) . We define the semidirect product of Sp ( g, R ) and H ( g,h ) R G J = Sp ( g, R ) ⋉ H ( g,h ) R OLARIZED REAL TORI 49 endowed with the following multiplication law ( M, ( λ, µ ; κ )) ⋅ ( M ′ , ( λ ′ , µ ′ ; κ ′ )) = ( M M ′ , ( ˜ λ + λ ′ , ˜ µ + µ ′ ; κ + κ ′ + ˜ λ t µ ′ − ˜ µ t λ ′ )) with M, M ′ ∈ Sp ( g, R ) , ( λ, µ ; κ ) , ( λ ′ , µ ′ ; κ ′ ) ∈ H ( g,h ) R and ( ˜ λ, ˜ µ ) = ( λ, µ ) M ′ . Then G J actson the Siegel-Jacobi space H g × C ( h,g ) transitively by(8.28) ( M, ( λ, µ ; κ )) ⋅ ( Ω , Z ) = ( M ⋅ Ω , ( Z + λ Ω + µ )( C Ω + D ) − ) , where M = ( A BC D ) ∈ Sp ( g, R ) , ( λ, µ ; κ ) ∈ H ( g,h ) R and ( Ω , Z ) ∈ H g × C ( h,g ) . We note thatthe Jacobi group G J is not a reductive Lie group and also that the space H g × C ( h,g ) is nota symmetric space. We refer to [28, 29, 30, 31, 32, 33] for more detail on the Siegel-Jacobispace H g × C ( h,g ) .We let Γ g,h ∶ = Γ g ⋉ H ( g,h ) Z be the discrete subgroup of G J , where H ( g,h ) Z = { ( λ, µ ; κ ) ∈ H ( g,h ) R ∣ λ, µ ∈ Z ( h,g ) , κ ∈ Z ( h,h ) } . We define the map Φ g,h ∶ P g × R ( h,g ) Ð→ H g × C ( h,g ) by(8.29) Φ g,h ( Y, ζ ) ∶ = ( i Y, ζ ) , ( Y, ζ ) ∈ P g × R ( h,g ) . We have the following inclusions P g × R ( h,g ) Φ g,h Ð→ H g × C ( h,g ) ↪ H g × C ( h,g ) ↪ H ∗ g × C ( h,g ) . G g,h acts on P g × R ( h,g ) , Γ ⋆ g ⋉ H ( g,h ) Z acts on H g × C ( h,g ) and Γ g,h acts on H g × C ( h,g ) and H ∗ g × C ( h,g ) . It might be interesting to characterize the boundary points of the closure of theimage of Φ g,h in H ∗ g × C ( h,g ) . Real Semi-Abelian Varieties
In this section we review the work of Silhol on semi-abelian varieties [26] which is neededin the next section.
Definition 9.1.
A complex semi - abelian variety A is the extension of an abelian variety ̃ A by a group of multiplicative type. A semi-abelian variety is said to be real if it admits ananti-holomorphic involution which is a group homomorphism. Let T be a group of multiplicative type. We consider the exponential map exp ∶ t Ð→ T .The real structure S on T lifts to a real structure S t on t . Then L t ∶ = ker exp is a free Z -module and S t induces an involution on L t . By standard results (cf. [25, I. (3.5.1)]), we can find a basis of L t with respect to which the matrix for S t is of the form ⎛⎜⎜⎜⎜⎜⎜⎜⎝ I s ⋯ B ⋯ B ⋯ ⋱ ⋯ B
00 0 0 ⋯ − I t ⎞⎟⎟⎟⎟⎟⎟⎟⎠ , B ∶ = ( ) . Since fixing a basis of L t is equivalent to fixing an isomorphism T ≅ ( C ∗ ) r , we get T = T × T × T , r = s ′ + p + t ′ , where(i) T = C ∗ × ⋯ × C ∗ ( s ′ -times) and S induces on each factor the involution z z→ z. In thiscase we write T = G m × ⋯ × G m ;(ii) T = C ∗ × ⋯ × C ∗ ( t ′ -times) and S induces on each factor the involution z z→ z − . Inthis case we write T = G ∞ m × ⋯ × G ∞ m ;(iii) T = ( C ∗ × C ∗ ) × ⋯ × ( C ∗ × C ∗ ) ( p -times) and S induces on each factor ( C ∗ × C ∗ ) theinvolution ( z , z ) z→ ( z , z ) . In this case we write T = G ∗ m × ⋯ × G ∗ m .Let ∆ = { ζ ∈ C ∣ ∣ ζ ∣ < } be the unit disk and let ∆ ∗ = { ζ ∈ C ∣ < ∣ ζ ∣ < } be a puncturedunit disk. Let ϕ ∶ Z ∗ Ð→ ∆ ∗ be a holomorphic family of matrices ϕ − ( ζ ) = Z ( ζ ) in H g . Wehave the natural action of the lattice Z g on ∆ ∗ × C g defined by(9.1) ( λ, µ ) ⋅ ( ζ, z ) ∶ = ( ζ, z + λ + Z ( ζ ) µ ) , ζ ∈ ∆ ∗ , λ, µ ∈ Z g , z ∈ C g . Then the quotient space(9.2) A ∗ ∶ = ( ∆ ∗ × C g )/ Z g is a holomorphic family of principally polarized abelian varieties associated to a holomorphicfamily ϕ ∶ Z ∗ Ð→ ∆ ∗ .Now we write Z ( ζ ) = X ( ζ ) + i Y ( ζ ) ∈ H g and Y ( ζ ) = t W ( ζ ) D ( ζ ) W ( ζ ) ∈ P g ( the J acobi decomposition ) with diag ( d ( ζ ) , ⋯ , d g ( ζ )) ∈ R ( g,g ) is a diagonal matrix.Now we assume the following conditions (F1)–(F3) : for any ζ ∈ ∆ ∗ r ∶ = { ζ ∈ C ∣ < ∣ ζ ∣ < r } ,(F1) There exists a positive number r > ζ ∈ ∆ ∗ r , Z ( ζ ) ∈ W g ( u ) for some u > , where ∆ ∗ r ∶ = { ζ ∈ C ∣ < ∣ ζ ∣ < r } ;(F2) X ( ζ ) converges in R ( g,g ) and W ( ζ ) converges in GL ( g, R ) as ζ → ζ → d i ( ζ ) = d i converges for 1 ≤ i ≤ g − t , and lim ζ → d i ( ζ ) = ∞ for g − t < i ≤ g. OLARIZED REAL TORI 51
Let Z ( ) = ⎛⎜⎜⎜⎜⎜⎝ z ⋯ z ,g − t ⋯ ⋮ ⋱ ⋮ ⋱ ⋮ z g − t, ⋯ z g − t,g − t ⋯ ⋮ ⋱ ⋮ ⋮ ⋯ ⋮ z g, ⋯ z g,g − t ⋯ ⎞⎟⎟⎟⎟⎟⎠ , z ij = lim ζ → z ij ( ζ ) . The action (9.1) extends to the action of Z g on ∆ × C g by letting Z ( ) be the fibre at ζ = . We take the quotient space(9.3) A ∶ = ( ∆ × C g )/ Z g . Then we see that A is an analytic variety fibred holomorphically over ∆, and the fibre at0 is a semi-abelian variety(9.4) A = C g / L , L ∶ = Z g Z ( ) + Z g ⊂ C g of the abelian variety(9.5) ̃ A ∶ = C g − t / L ◇ , L ◇ ∶ = Z g − t Z ◇ ( ) + Z g − t ⊂ C g − t by ( C ∗ ) t , where Z ◇ ( ) = ⎛⎜⎝ z ⋯ z ,g − t ⋮ ⋱ ⋮ z g − t, ⋯ z g − t,g − t ⎞⎟⎠ ∈ H g − t . The extension 1 Ð→ ( C ∗ ) t Ð→ A Ð→ ̃ A Ð→ z ◇ g − k = ( z g − k, , ⋯ , z g − k,g − t ) ∈ C g − t , k = , ⋯ , t − C g − t Ð→ ̃ A Ð→ Pic ( ̃ A ) , where the last map is the isomorphism defined by the polarization.These above facts can be generalized as follows. Proposition 9.1.
Let ϕ ∶ Z ∗ Ð→ ∆ ∗ be a holomorphic family of matrices ϕ − ( ζ ) = Z ( ζ ) in H g such that ϕ − ( ζ ) = Z ( ζ ) converges in H ∗ g as ζ → . Then there exists an analytic variety A ( Z ∗ ) Ð→ ∆ such that(i) the fibre at ζ ( ≠ ) ∈ ∆ is the principally polarized abelian variety C g / L ζ with the lattice L ζ = Z g Z ( ζ ) + Z g ;(ii) the zero fibre A ( Z ∗ ) is a semi-abelian variety.Proof. The proof can be found in [26, p. 189]. ◻ Theorem 9.1.
Let ϕ ∶ Z ∗ Ð→ ∆ ∗ be a holomorphic family of matrices ϕ − ( ζ ) = Z ( ζ ) in H g such that ϕ − ( ζ ) = Z ( ζ ) converges in H ∗ g as ζ → . We assume that Z ( ζ ) = ϕ − ( ζ ) ∈ H g for ζ ∈ R ∩ ∆ ∗ . Let ( s, t ) be such that lim ζ → Z ( ζ ) ∈ γ B M ( F s,t ∩ H ) for some M ∈ Z ( g,g ) and some γ ∈ Γ ⋆ g . Then (a) A ( Z ∗ ) has a natural real structure extending the real structures of the A ( Z ∗ ) ζ ′ s for ζ ∈ R ∩ ∆ ∗ ;(b) As a real variety, A ( Z ∗ ) is the extension of a real abelian variety ̃ A ( Z ∗ ) by ( G m ) s ′ × ( G ∗ m ) p × ( G ∞ m ) t ′ , s = s ′ + p, t = t ′ + p ; (c) Let x ∈ X g R ( s, t ) ⊂ X g R be the image of lim ζ → Z ( ζ ) in X g R and let [ x ] be the image of x under the isomorphism X g R ( s, t ) ≅ X g − r R with r = s + t. Then [ x ] is the real isomorphismclass of ̃ A ( Z ∗ ) .Proof. The proof can be found in [26, pp. 191–192]. ◻ Corollary 9.1.
Let ϕ ∶ Z ∗ Ð→ ∆ ∗ be as in Theorem 9.1. Assume lim ζ Ð→ Z ( ζ ) ∈ F ,t ( resp. F s, ) . Then the class of the extension Ð→ ( G ∞ m ) t → A ( Z ∗ ) Ð→ ̃ A ( Z ∗ ) Ð→ ( resp. Ð→ ( G m ) s Ð→ A ( Z ∗ ) Ð→ ̃ A ( Z ∗ ) Ð→ ) is defined by t purely imaginary divisors on A ( Z ∗ ) ( resp. s real divisors A ( Z ∗ ) ) . Real Semi-Tori
A real semi-torus T of dimension g is defined to be an extension of a real torus ̃ T ofdimension g − t by a real group ( R ∗ ) t of multiplicative type, where R ∗ = R − { } . Let I = { ξ ∈ R ∣ − < ξ < } be the unit interval and I ∗ = I − { } be the punctured unitinterval. Let ̟ ∶ Y ∗ Ð→ I ∗ be a real analytic family of matrices ̟ − ( ξ ) = Y ( ξ ) ∈ P g . Wehave the natural action of the lattice Z g in R g on I ∗ × R g defined by(10.1) α ⋅ ( ξ, x ) = ( ξ, x + Y ( ξ ) α ) , α ∈ Z g , ξ ∈ I ∗ , x ∈ R g . The quotient space(10.2) T ∗ ∶ = ( I ∗ × R g )/ Z g is a real analytic family of real tori of dimension g associated to a real analytic family ̟ ∶ Y ∗ Ð→ I ∗ . We let Y ( ξ ) = t W ( ξ ) D ( ξ ) W ( ξ ) be the Jacobi decomposition of Y ( ξ ) , where D ( ξ ) = diag ( d ( ξ ) , ⋯ , d g ( ξ )) is a real diagonalmatrix and W ( ξ ) is a strictly upper triangular real matrix of degree g . Now we assume thefollowing conditions (T1)-(T4):(T1) There exists a positive number r with 0 < r, ξ ∈ I ∗ r , i Y ( ξ ) ∈ W g ( u ) for some u > , where I ∗ r ∶ = { ξ ∈ R ∣ − r < ξ < r } ;(T2) W ( ξ ) converges in GL ( g, R ) as ξ → ξ → d i ( ξ ) = d i converges for 1 ≤ i ≤ g − t , and lim ξ → d i ( ξ ) = ∞ for g − t < i ≤ g. OLARIZED REAL TORI 53
Let Y ( ) = ⎛⎜⎜⎜⎜⎜⎝ y ⋯ y ,g − t ⋯ ⋮ ⋱ ⋮ ⋱ ⋮ y g − t, ⋯ y g − t,g − t ⋯ ⋮ ⋱ ⋮ ⋮ ⋯ ⋮ y g, ⋯ y g,g − t ⋯ ⎞⎟⎟⎟⎟⎟⎠ , y ij = lim ξ → y ij ( ζ ) . The action (10.1) extends to the action of Z g on I × R g by letting Y ( ) be the fibre at ξ = . We take the quotient space(10.3) T ∶ = ( I × R g )/ Z g . Then we see that T is a real analytic variety fibred real analytically over I , and the fibreat 0 is a real semi-torus(10.4) T = R g / Λ , Λ ∶ = Z g Y ( ) ⊂ R g of the real torus ̃ T ∶ = R g − t / Λ ◇ , Λ ◇ ∶ = Z g − t Y ◇ ( ) is a lattice in R g − t by ( C ∗ ) t , where Y ◇ ( ) = ⎛⎜⎝ y ⋯ y ,g − t ⋮ ⋱ ⋮ y g − t, ⋯ y g − t,g − t ⎞⎟⎠ ∈ P g − t . Open Problems and Remarks
In this final section we give some open problems related to polarized real tori to be studiedin the future.
Problem 1.
Characterize the boundary points of the closure of i P g in H ∗ g explicitly. Problem 2.
Find the explicit generators of the ring D ( g, h ) of differential operators onthe Minkowski-Euclidean space P g × R ( h,g ) which are invariant under the action (8.7) of GL g,h = GL ( g, R ) ⋉ R ( h,g ) . Problem 3.
Find all the relations among a complete explicit list of generators of D ( g, h ) .The orthogonal group O ( g ) of degree g acts on the subspace p = { ( X, Z ) ∣ X = t X ∈ R ( g,g ) , Z ∈ R ( h,g ) } of the vector space R ( g,g ) × R ( h,g ) by(11.1) k ⋅ ( X, Z ) = ( kX t k, Z t k ) , k ∈ O ( g ) , ( X, V ) ∈ p . The action (11.1) induces the action of O ( g ) on the polynomial ring Pol ( p ) on p . We denoteby I ( p ) the subring of Pol ( p ) consisting of polynomials on p invariant under the action of O ( g ) . We see that there is a canonical linear bijectionΘ ∶ I ( p ) Ð→ D ( g, h ) of I ( p ) onto D ( g, h ) . We refer to [8] and [27] for more detail. Remark 11.1.
M. Itoh [11] proved that I ( p ) is generated by α j ( ≤ j ≤ g ) and β ( k ) pq ( ≤ k ≤ g − , ≤ p ≤ q ≤ h ) , where (11.2) α j ( X, Z ) = tr ( X j ) , ≤ j ≤ g and (11.3) β ( k ) pq ( X, Z ) = ( Z X k t Z ) pq , ≤ k ≤ g − , ≤ p ≤ q ≤ h. Here A pq denotes the ( p, q ) -entry of a matrix A of degree h . Remark 11.2.
M. Itoh [11] found all the relations among the above generators α j ( ≤ j ≤ g ) and β ( k ) pq ( ≤ k ≤ g − , ≤ p ≤ q ≤ h ) of I ( p ) . Problem 4.
Develop the theory of harmonic analysis on the Minkowski-Euclidean space P g × R ( h,g ) with respect to a discrete subgroup of GL ( g, Z ) ⋉ Z ( h,g ) . Problem 5.
Characterize the boundary points of the closure of the image of Φ g,h in H ∗ g × C ( h,g ) (cf. see (8.29)). Problem 6.
Find the explicit generators of the ring D ( H g × C ( h,g ) ) of differential operatorson the Siegel-Jacobi space H g × C ( h,g ) which are invariant under the action (8.28) of theJacobi group G J = Sp ( g, R ) ⋉ H ( g,h ) R . We refer to [28] for more detail. Problem 7.
Find all the relations among a complete list of generators of D ( H g × C ( h,g ) ) . Problem 8.
Develop the theory of harmonic analysis on the Siegel-Jacobi space H g × C ( h,g ) with respect to a congruent subgroup of Γ g,h = Sp ( g, Z ) ⋉ H ( g,h ) Z . We refer to [29] for moredetail. Appendix : Non-Abelian Cohomology
In this section we review some results on the first cohomology set H ( < τ > , Γ ) obtainedby Goresky and Tai [9], where < τ >= { , τ } is a group of order 2 and γ is a certain arithmeticsubgroup. These results are often used in this article.First of all we recall the basic definitions. Let S be a group. A group M is called a S - g roup if there exists an action of G on M, S × M Ð→ M, ( σ, a ) z→ σ ( a ) such that σ ( ab ) = σ ( a ) σ ( b ) for all σ ∈ S and a, b ∈ M . From now on we let 1 S (resp. 1 M ) be theidentity element of S (resp. M ). We observe that if M is a S -group, then σ ( M ) = M forall σ ∈ S. OLARIZED REAL TORI 55
Definition.
Let M be a S -group, where S is a group. We define H ( S, M ) ∶ = { a ∈ M ∣ σ ( a ) = a for all σ ∈ S } . A map f ∶ S Ð→ M is called a - cocycle with values in M if f ( στ ) = f ( σ ) σ ( f ( τ )) for all σ, τ ∈ S. We observe that if f is a -cocycle, then f ( S ) = M . We denote by Z ( S, M ) theset of all -cocycles of S with values in M . Let f and f be two -cocycles in Z ( S, M ) . We say that f is cohomologous to f , denoted f ∼ f , if there exists an element h ∈ M such that f ( σ ) = h − f ( σ ) σ ( h ) for all σ ∈ S. Let f ♭ ∶ S Ð→ M be the trivial map, i.e., f ♭ ( σ ) = M for all σ ∈ S. A map f ∶ S Ð→ M iscalled a - coboundary if f ∼ f ♭ , i.e., if there exists h ∈ M such that f ( σ ) = h − σ ( h ) for all σ ∈ S. Obviously a 1-coboundary is a 1-cocycle. It is easy to see that ∼ is an equivalence relationon Z ( S, M ) . So we define the first cohomology set H ( S, M ) ∶ = Z ( S, M )/ ∼ . Remark.
In general, H ( S, M ) does not admit a group structure. But H ( S, M ) has anidentity, that is, the cohomologous class containing the trivial 1-cocycle f ♭ . Example.
Let L be a Galois extension of a number field K with Galois group G . A linearalgebraic group defined over K has naturally the structure of G -group. It is known that H ( G, GL ( n, L )) is trivial for all n ≥ . Using the following exact sequence of G -groups1 Ð→ SL ( n, L ) Ð→ GL ( n, L ) Ð→ L ∗ Ð→ , L ∗ = L − { } , we can show that H ( G, SL ( n, L )) is trivial.We put G = Sp ( g, R ) and K = U ( g ) . Then D = G / K is biholomorphic to H g . Let S τ = { , τ } be a group of order 2 as before. We define the S τ -group structure on G via theaction (2.7) of S τ on G . Let Γ be an arithmetic subgroup of Sp ( g, Q ) . We let X Γ ∶ = Γ / G / K ≅ Γ / H g and let π Γ ∶ D Ð→ X Γ be the natural projection. For any γ ∈ Γ , we define the map f ∶ S τ Ð→ Γ by ( ) f γ ( ) = Γ and f γ ( τ ) = γ, where 1 Γ denotes the identity element of Γ . Lemma 1.
Let γ ∈ Γ . Then(a) f γ is a -cocycle if and only if γ τ ( γ ) = Γ , equivalently, τ ( γ ) γ = Γ . (b) A cocycle f γ is a -coboundary if and only if there exists h ∈ Γ such that γ = τ ( h ) h − . Proof.
The proof follows immediately form the definition. ◻ To each such a 1-cocycle f γ we associate the γ -twisted involution τ γ ∶ D Ð→ D and τ γ ∶ Γ Ð→ Γ. Indeed the involution τ γ ∶ D Ð→ D is defined by ( ) τ γ ( xK ) = τ ( γxK ) = τ ( γ ) τ ( x ) K, x ∈ G and the involution τ γ ∶ Γ Ð→ Γ is defined by ( ) τ γ ( γ ) = τ ( γγ γ − ) , γ ∈ Γ . Let D τγ ∶ = { x ∈ D ∣ ( τ γ )( x ) = x } and Γ τγ ∶ = { γ ∈ Γ ∣ ( τ γ )( γ ) = γ } be the fixed point sets. Lemma 2.
Let x ∈ D . Then π Γ ( x ) ∈ X τ Γ if and only if there exists an element γ ∈ Γ suchthat x ∈ D τγ .Proof. It is easy to prove this lemma. We leave the proof to the reader. ◻ Theorem A.
Assume Γ is torsion free. Let C Γ be the set of all connected components ofthe fixed point set X τ Γ . Then the map Φ Γ ∶ H ( S τ , Γ ) Ð→ C Γ defined by Φ Γ ([ f γ ]) ∶ = π Γ ( D τγ ) = Γ τγ / D τγ determines a one-to-one correspondence between H ( S τ , Γ ) and C Γ .Proof. The proof can be found in [9, pp. 3-4]. ◻ Theorem B.
Let S τ = { , τ } be a group of order . Then Sp ( g, R ) has a S τ -group structurevia the action (2.7) and hence U ( g ) also admits a S τ -group structure through the restrictionof the action (2.7) to U ( g ) . And H ( S τ , U ( g )) and H ( S τ , Sp ( g, R )) are trivial.Proof. The proof can be found in [9, pp. 8-9]. However I will give a sketchy proof for thereader. Assume f k is a 1-cocycle in Z ( S τ , U ( g )) with k = ( A B − B A ) ∈ U ( g ) . Using the fact τ ( k ) k = I g , we see that A = t A, B = t B, AB = BA and A + B = I g . Therefore we can find h ∈ O ( g ) such that h ( A + iB ) h − = D ∈ U ( g ) is diagonal. We take µ = √ D ∈ U ( g ) by choosing a square root of each diagonal entry. We set δ = h − µh. Then k = τ ( δ ) δ − . By Lemma 1, f k is a 1-coboundary. Hence H ( S τ , U ( g )) is trivial.Let G = Sp ( g, R ) as before. Suppose f M ∈ Z ( S τ , G ) with M = ( A BC D ) ∈ G. Then wesee that f M is a 1-coboundary with values in G if and only if H Mτg ≠ ∅ . We can find M ∈ G such that H M τg ≠ ∅ and f M ∼ f M ∼ f ♭ . Therefore f M ∼ f ♭ , that is, f M is a 1-coboundarywith values in G . Hence H ( S τ , G ) is trivial. ◻ Theorem C.
For all m ≥ , the mapping H ( S τ , Γ g ( m )) Ð→ H ( S τ , Γ g ( , m )) is trivial.Proof. The proof can be found in [9, pp. 7-10]. We will give a sketchy proof for the reader.In order to prove this theorem, we need the following lemma.
Lemma 3. If τ ( γ ) γ ∈ Γ g ( m ) with γ ∈ Γ g , then γ = βu for some β ∈ Γ g ( , m ) and forsome u ∈ GL ( g, Z ) . OLARIZED REAL TORI 57
Lemma 4.
Let γ ∈ Γ g ( ) and suppose Ω ∈ H g is not fixed by any element of Γ g other than ± I g . Suppose τ ( Ω ) = γ ⋅ Ω . Then there exists an element h ∈ Γ g such that γ = τ ( h ) h − . Lemma 4 is a consequence of the theorem of Silhol [26, Theorem 1.4] and Comessatti.Suppose f γ is a cocycle in Z ( S τ , Γ g ( m )) with γ ∈ Γ g ( m ) . According to Theorem B,its image in H ( S τ , G )) is a coboundary and so there exists h ∈ G with γ = τ ( h ) h − . Thus H τγ = h ⋅ ( i P g ) . By Lemma 2.2, there exists Ω ∈ H τγ which are not fixed by anyelement of Γ g other than ± I g and the set of such points is the complement of a countableunion of proper real algebraic subvarieties of H τγ . According to Lemma 4, γ = τ ( h ) h − for some h ∈ Γ g . By Lemma 3, we may write h = β u for some β ∈ Γ g ( , m ) and for some u ∈ GL ( g, Z ) . Then γ = τ ( h ) h − = τ ( β ) β − . By Lemma 1, the cohomology class [ f γ ] istrivial in H ( S τ , Γ g ( , m )) . ◻ Theorem D.
Let Γ = Γ g ( , m ) and Γ = Γ g ( m ) . Then we have the following results :(a) D τ = G τ / K τ ; (b) For each cohomology class [ f γ ] ∈ H ( S τ , Γ ) , there exists h ∈ Γ such that γ = τ ( h ) h − ,in which case D τγ = h D τ and Γ τγ = h Γ τ h − . (c) The association f γ Ð→ h (cf. see (2)) determines a one-to-one correspondence between H ( S τ , Γ ) and Γ / Γ / Γ τ .(d) X τ Γ ∶ = ∐ h ∈ Γ / Γ / Γ τ h Γ τ h − / h D τ . Proof. (a) follows from the fact that H ( S τ , U ( g )) is trivial (cf. Theorem B). (b) followsfrom Theorem C. (c) follows from Theorem C, and the facts that Γ is a normal subgroup ofΓ and that τ acts on Γ / Γ trivially. (d) follows from Theorem C. (c) follows from TheoremC, and the facts that Γ is a normal subgroup of Γ and that τ acts on Γ / Γ trivially togetherwith the fact that Γ s torsion free. ◻ Corollary.
Let m be a positive integer with m ≥ . Let S τ be as in Theorem A . Let Γ = Γ g ( m ) and X = Γ / H g . The set X R of real points of X is given by X R = ∐ h Γ [ h ] / h ⋅ ( i P g ) = Γ / H τ Γ , where h is indexed by elements h ∈ Γ g ( m )/ Γ g ( , m )/ Γ [ L ] g ( ) = H ( S τ , Γ g ( m )) and Γ [ h ] ∶ = h Γ [ L ] g ( m ) h − . Proof.
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