Siegel Moduli Space of Principally Polarized Abelian Manifolds

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1 Siegel Moduli Space of Principally Polarized Abelian Manifolds Andrea Anselli 8 february Recall Definition 11 A complex torus T of dimension d is given by V/Λ, where V is a C-vector space of dimension d and Λ is a lattice in V with rank 2d Let T V/Λ be a complex torus of dimension d Definition 12 A hermitian form on V is given by H : V V C such that H(z, ) is C-linear for every z V and H(w, z) H(z, w) for every z, w V Remark 13 If H is a hermitian form on V then, for every z, w V : H(z, w) E(iz, w) + ie(z, w), where E : V V R is R-bilinear, antisymmetric (E(w, z) E(z, w) for every z, w V ) and E(iz, iw) E(z, w) for every z, w V Moreover H is determined by E Definition 14 A Riemann form on T is given by a hermitian form H on V such that E(Λ, Λ) Z, where E : ImH It is called positive if it is positive definite as bilinear form, that is H(z, z) 0 for every z V and the equality holds if and only if z 0 Definition 15 An abelian manifold (am) A is a complex torus which posses a positive Riemann form We have seen the following theorem Theorem 16 The torus T is the manifold of complex points of an abelian variety if and only if it is an abelian manifold 2 Polarization Let A V/Λ be an abelian manifold of dimension d Definition 21 Let H 1, H 2 be two Riemann forms on A They are equivalent if there exists n 1, n 2 N such that n 1 H 1 n 2 H 2 We denote by H the equivalence class of a Riemann form H on A Definition 22 A polarized abelian manifold (pam) (A, H) is given by an abelian manifold A together with an equivalence class of Riemann forms H that contains a positive Riemann form (we can suppose that H is positive definite) The class H is called (homogeneous) polarization of A Remark 23 Let (A, H) be a polarized abelian manifold, we have seen that every D Div(A) whose associated Riemann form is H is an ample divisor Thus for every such D the following map is a polarization (as defined in the notes): ϕ D : A Ǎ : P ic0 (A), a τ ad D (1) 1

2 We have seen that ϕ D is an isogeny of degree det E, where E : ImH Moreover ϕ D depends only on D Div(A)/Div 0 (A) In facts D and D defines the same element in Div(A)/Div 0 (A) if and only if they have the same Riemann form associated and for every a A the divisor τ a D D correspond to the character λ e 2πiE(w,λ), where w V represents a and E ImH where H is the Riemann form associated to D, thus depends only on the class of D Definition 24 A morphism of polarized abelian manifold φ : (A 1 H1 ) (A 2 H2 ) is given by a morphism of complex abelian variety φ : A 1 V 1 /Λ 1 A 2 V 2 /Λ 2 (a holomorphic map which is a homomorphism of groups) such that φ H 2 is equivalent to H 1, where φ : V 1 V 2 is the C-linear map that lifts such that φ(λ 1 ) Λ 2 One can prove the following theorem Theorem 25 For every polarized abelian manifold (A, H) the automorphism group Aut(A, H) is a finite group Definition 26 A polarized abelian manifold (A, H) is called principally polarized abelian manifold (ppam) if there is an element in the homogeneous polarization class of A (we can suppose that this element is H) such that P f(e) 1, where E : ImH It means that with respect to a symplectic basis {λ 1,, λ 2d } of Λ the bilinear form E is given by the following matrix: J : 0 I d I d 0 Proposition 27 Up to isogenies every polarized abelian manifold is a principally polarized abelian manifold Sketch of proof Suppose that (A V/Λ, H) is a polarized abelian manifold of dimension d, with E : ImH and {λ 1,, λ 2d } a symplectic basis of Λ Thus E is given by the following matrix, where E diag(e 1,, e d ) and e j : E(λ j, λ j+d ) N for every j 1,, d: 0 E E 0 We define the following lattice in V : Λ : λ 1 e 1 Z + + λ d e d Z + λ d+1 Z + + λ 2d Z It is a lattice of rank 2d contained in Λ, the form E is alternating on Λ, by definition E(Λ, Λ ) Z and det E 1, thus A : V/Λ is a principally polarized abelian manifold The map A V/Λ V/Λ A defines an isogeny Remark 28 If (A, H) is a principally polarized abelian manifold then A is autodual It sufficies to consider an ample divisor D Div(A) whose associated Riemann form is H and the associated polarization φ D : A Ǎ The map φ D is an isogeny of degree det E 1, hence it is an isomorphism of complex abelian variety Example 29 The major example of principally polarized abelian manifold is the Jacobian of a compact Riemann surface C of genus g > 0 (even of non singular algebraic curve) In fact one can show there is a positive Riemann form H on C g with respect to the lattice Λ C g, where Λ is the additive subgroup of C g generated by: {( ) ( ) } ω 1,, ω g, ω 1,, ω g : i 1,, g, a i a i b i b i where H 1 (C, Z) π 1 (C) a 1,, a g, b 1,, b g and {ω 1,, ω g } is a C-basis of the g-dimensional C-vector space of holomorphic differential forms on C The Riemann form H is induced by the intersection pairing on H 1 (C, Z) Λ, the determinant of the intersection pairing is 1 hence H determines a principal polarization on Jac(C) : C g /Λ, the Jacobian of C 2

3 The following theorem is a deep result and encourages us to proceed in the study of the moduli space of principally polarized abelian manifolds Theorem 210 (Torelli) Let C 1 and C 2 be two compact Riemann surfaces and let H 1 and H 2 be the Riemann forms induced by the intersection pairings If (Jac(C 1 ), H 1 ) (Jac(C 2 ), H 2 ) as (principally) polarized abelian manifolds, then C 1 C 2 as complex manifolds 3 Siegel Moduli Space For every d 1 we define: A d : {isomorphism classes of principally polarized abelian manifold of dimension d}, we want to give to A d the structure of a complex analytic space Example 31 Case d 1 We have seen that every abelian manifold of dimension 1 (ie every elliptic curve) is principally polarized by the Riemann form associated to the divisor (0) and all the Riemann forms are equivalent (they are all integers multiple of a Riemann form, it s Neron-Severi group is isomorphic to Z) Let H : {τ C : Imτ > 0} be the Poincaré upper half plane and let: { } a b SL 2 (Z) : : a, b, c, d Z, ad bc 1 c d The group SL 2 (Z) acts on H in the following way: ( ) a b, τ c d One has: A 1 SL 2 (Z)\H, aτ + b cτ + d this bijection is defined as follows Any element in A 1 can be represented by the elliptic curve C/(Zω 1 +Zω 2 ), where {ω 1, ω 2 } is a R-basis of C such that Im(ω 1 /ω 2 ) > 0 To this element of A 1 correspond the class of τ : ω 1 /ω 2 Moreover, by the j-invariant, A 1 inherits from C the structure of a complex manifold This material will be seen as a special case of our further considerations Let (A V/Λ, H) a principal polarized abelian manifold By choosing a C-basis {v 1,, v d } for V and a symplectic basis {ω 1,, ω 2d } of Λ one can represent the polarized abelian manifold in the following way: (C d,, J), where Mat d 2d (C) is called the period matrix and has as columns the components of the elements of the symplectic basis {ω 1,, ω 2d } in the basis {v 1,, v d } of V and J is the matrix that represents the imaginary part E of the Riemann form H in the symplectic basis The following two lemmas prove that the period matrices are characterize by the following two conditions, called Riemann s relations: (RI) J T 0, (RII) 2i(J 1 T ) 1 > 0 Remark 32 Every element of C d can be written in the form x, for some x R 2d We have for every x, y R 2d : E(x, y) x T Jy Let C Mat 2d (R) be the matrix such that i C (iω j c 1,j ω 1 + c 2d,j ω d for every j 1,, 2d), as E(i, ) is a symmetric bilinear form and E(i, i ) E(, ) the following equation hold: C T J JC 3

4 Lemma 33 The matrix C T J is symmetric if and only if (RI) holds Proof Clearly J T J, thus: (C T J) T J T C JC If C T J is symmetric then C T J JC, thus CJ 1 J 1 C T, thus CJ 1 T J 1 C T T By definition of C we obtain: ij 1 T CJ 1 T J 1 C T T ij 1 T, therefore 2iJ 1 T 0 By applying the same argument we can obtain the converse Lemma 34 The matrix associated to H with respect the canonical basis of C d is M : 2i(J 1 T ) 1 Proof The matrix M is hermitian, in fact: M T 2i((J T ) 1 T ) 1 2i( J 1 T ) 1 M, thus it sufficies to show that H(u, u) u T Mu for every u C d (and then we conclude by the polarization identity) We denote by i d : ii d, clearly we have: id 0 C Recalling that H(u, u) E(iu, u) for every u C d we obtain, for x R 2d such that u x: Clearly: H(u, u) xt JCx 1 x T id 0 J x T T, T T, T 1 J x T T, T u T, u T, x 1 id 0 u x u Moreover, by using and Lemma 33 and the following relation: 1 0 X 0 Y 1 Y 0 X 1 0 we obtain: T, T 1 J 1 id 0, 1 J 1 T, T id 0 J 1 T J 1 T J 1 T J 1 T 0 J 1 T J 1 T 0 0 ij 1 T ij 1 T 0 x 1 id id i(j 1 T ) 1 i(j 1 T ) 1 0 4

5 Thus: H(u, u) u T, u T 0 i(j 1 T ) 1 i(j 1 T ) 1 0 u u iu T (J 1 T ) 1 u iu T (J 1 T ) 1 u 2iu T Mu If we write 1, 2, for suitable 1, 2 Mat d (C), the Riemann s conditions can be rewritten in the following way (one has J 1 J): (RI ) 2 T 1 1 T 2 0, (RII ) 2i( 2 T 1 1 T 2 ) > 0 We define the following: R : { 1, 2 : 1, 2 Mat d (C) and the conditions (RI ), (RII ) hold} and recall the definition of the symplectic group: SP 2d (R) : {M Mat 2d (R) : MJM T J} The property of the symplectic group are collected in the following proposition Proposition 35 The symplectic group SP 2d (R) is a subgroup of GL 2d (R) and it is closed by transposition (if M SP 2d (R), then M T SP 2d (R)) Proof Clearly det J 0, thus if M SP 2d (R) then (det M) 2 1, thus M GL 2d (R) Moreover from MJM T J we obtain J M 1 J(M 1 ) T, thus M 1 SP 2d (R) For the product and for the identity element analogous arguments hols, therefore SP 2d (R) is a subgroup of GL 2d (R) Moreover let M SP 2d (R), taking the inverse of the relation J M 1 J(M 1 ) T, we obtain: thus M T SP 2d (R) J J 1 (M 1 J(M 1 ) T ) 1 M T J 1 M M T J(M T ) T, We define the Siegel upper half space: H d : {τ Mat d (C) : τ is symmetric and Imτ > 0} It is an open of the complex manifold of the symmetric d d matrices in complex coefficients, that has dimension d(d + 1)/2 Proposition 36 Let 1, 2 R, the following hold: (i) If g GL d (C), then g : g 1, g 2 R A B (ii) If M SP C D 2d (R), then M : 1 A + 2 C, 1 B + 2 D R (iii) 1, 2 GL d (C) (iv) H d Proof (i) We have to prove that the conditions (RI ), (RII ) hold for g g 1, g 2 5

6 (RI ) g 2 (g 1 ) T g 1 (g 2 ) T g( 2 T 1 1 T 2 )g T 0, (RII ) 2i(g 2 g 1 T g1 g 2 T ) g2i(2 T 1 1 T 2 )g T > 0 (ii) As M SP 2d (R) we have: BA T AB T 0, BC T AD T I d, DA T CB T I d, DC T CD T 0 To prove (RI ) we observe that: ( 1 B + 2 D)(A T T 1 + C T T 2 ) ( 1 A + 2 C)(B T T 1 + D T T 2 ) 1 T T 1 An analogous argument hold for proving (RII ) (iii) Suppose that there exists v C d such that v T 1 0 Thus: v T ( 1 T 2 2 T 1 )v 0 For (RII ) the matrix 2i( 1 T 2 2 T 1 ) is positive definite, therefore v 0 Thus 1 GL d (C) The matrix J SP 2d (R), thus 2, 1 J R for (ii) Hence also 2 GL d (C) (iv) For (iii) and (i) the element 1 2 1, I d 1 2 R The condition (RI ) means that is symmetric, the condition (RII ) means that Im( 1 2 1) > 0 Lemma 37 Every element of A d contains a representative of the form: for some τ H d (C d, τ, I d, J), Proof We have alredy seen that each element can be represented by a triple (C d,, J), where 1, 2 R For Proposition 36 (i) we have , I d R and for (iv) the matrix τ : H d The multiplication by 1 2 corresponds to a change of basis of C d, thus it doesn t change the isomorphism class and the matrix that represent the imaginary part of the positive Riemann form (in fact the symplectic basis remains unchanged, it only change the period matrix, that is the description of the symplectic basis in the basis of C d ) To get uniqueness we have to factor out by an appropriate group of automorphisms Remark 38 The group SP 2d (R) acts on H d in the following way: ( ) A B M, τ M(τ) : (Aτ + B)(Cτ + D) 1 C D To prove that the action is well defined note that : τ, I d R, thus by Proposition 36 (ii): τa + C, τb + D M R Thus, by Proposition 36 (iv), (τb + D) 1 (τa + C) H d Clearly H d is closed by transposition and τ T τ, thus: M T (τ) (A T τ + C T )(B T τ + D T ) 1 (τb + D) 1 (τa + C) T H d To conclude it sufficies to consider M T instead of M (SP 2d (R) is closed under transposition) 6

7 Remark 39 Let 1, 2 be another period matrix, defined by a different symplectic basis {ω 1,, ω 2d } for Λ Clearly there exists M GL 2d (Z) such that M T The form E, with matrix J with respect to the basis {ω 1,, ω 2d }, has the matrix MJM T with respect to the basis {ω 1,, ω 2d } Therefore the description of the form E (and thus H) remains unchanged if and only if M SP 2d (R) We define: Γ : Aut(J) GL 2d (Z) SP 2d (R) SP 2d (Z) Any two representative of an element in A d of the form (C d, τ, I d, J) and (C d, τ, I d, J) respectively, with τ, τ H d, differ by a change of symplectic basis preserving J (as above) followed by a change of basis in C d transforming the second half of the lattice vectors into the unit vectors By this two remarks we obtain the following Theorem Theorem 310 The following is a bijection: A d H d /Γ SP 2d (Z)\H d It is defined as follows Every element of A d admits a representant of the form (C d, τ, I d, J), where τ H d To this element correspond the class of τ We recall the following facts and an important theorem Let X be a complex analytic space and let G be a subgroup of Aut(X) acting on X The quotient X/G, endowed with the quotient topology, admits the structure of a ringed space For every U open of X/G: O X/G (U) : {f : U C : f π O X (π 1 U)} The group G acts properly and discontinuously on X if for any K 1, K 2 compact subsets of X the following set is finite: {g G : gk 1 K 2 } Theorem 311 Let X be a complex analytic space and let G be a group acting properly and discontinuously on X The quotient X/G is a complex analityc space Moreover, if X is normal (in particular if X is a complex manifold), so is X/G One can prove the following proposition Proposition 312 Any discrete subgroup of SP 2d (R) (in particular Γ) acts properly and discontinuously on H d Thus we can apply the Theorem 311 obtaining the following Theorem 313 The quotient H d /Γ is a normal complex analytic space of dimension d(d + 1/2) Therefore we have reach the goal of this section: give to A d the structure of a complex analytic space Moreover we have recover the familiar situation when d 1 4 Application 1 There are complex tori which are not the manifold of complex points of an abelian variety Consider the following example for d 2 In C 2 consider the following period matrix, for α, β, γ, δ R algebraically indipendent over Q: : α + i β 1 0 γ δ + i 0 1 Let T : C 2 /Λ, where Λ is the lattice generated by the columns of the period matrix One can prove that M(T ) C The following theorem due to Siegel: 7

8 Theorem 41 Let M be a compact, connected, complex manifold of dimension d Then M(M) has trascendence degree over C at most d If d is atteined then M(M) is a finitely generated field over C If M X(C), the complex points on a nonsingular algebraic variety X, then M(M) C(X), the field of rational functions on X Thus, in this case, M(M) is a finitely generated field of trascendence degree d 2 The generic abelian variety has Z as endomorphism ring Let A be the representant of an element in A d and let τ H d the corresponding matrix As we have seen before the multiplication by g M d (C) gives an element of End(A) if and only if there exists M SP 2d (Z) such that gτ, I d τ, I d M Therefore: gτ τb + D, gτ τa + C, thus τbτ + Dτ τa C 0 If B C 0 and A D ni d for some n Z there are no conditions imposed on τ Otherwise the coefficients of τ must satisfy certain non trivial quadratic polynomials with coefficients in Z, but in general this cannot happen References 1 G Cornell, JH Silverman Arithmetic Geometry, Springer-Verlag New York Inc, C Birkenhake, H Lange, Complex Abelian Varieties, Springer-Verlag Berlin Heidelberg, P Griffiths, J Harris, Principle of Algebraic Geometry, John Wiley & Sons, Inc, S Lang, Introduction to Algebraic and Abelian Functions, Springer-Verlag New York,