Complex tori, theta groups and their Jordan properties
aa r X i v : . [ m a t h . AG ] F e b COMPLEX TORI, THETA GROUPS AND THEIR JORDANPROPERTIES
YURI G. ZARHIN
Abstract.
We prove that an analogue of Jordan’s theorem on finite subgroupsof general linear groups does not hold for the group of bimeromorphic auto-morphisms of a product of the complex projective line and a complex torus of positive algebraic dimension. Introduction
As usual, CP denotes the complex projective line . Recall that a group G iscalled Jordan (V.L. Popov [8]) if there exists a positive integer J that enjoys thefollowing properties. If B is a finite subgroup of G then there exists an abeliannormal subgroup A of B such that the index [ B : A ] ≤ J . If this is the case thensuch a smallest J is called the Jordan constant of G and denoted by J G ; otherwise,we say that G is not Jordan and its Jordan constant is ∞ . V.L. Popov [9] provedthat every complex or real Lie group with finitely many connected components isJordan. (His result also covers the case when the group of connected componentsis bounded [9].)If Z is a connected complex manifold then we write Bim( Z ) for its group ofbimeromorphic automorphisms and Aut( Z ) for its subgroup of all biholomorphicautomorphisms of Z . Jordan properties of Aut( Z ) and Bim( Z ) when Z is a compactcomplex manifolds have been studied recently by Sh. Meng and D.-Q. Zhang [6]and Yu. Prokhorov and C. Shramov [10, 11]. In particular, Prokhorov and Shramovhave classified all the surfaces with non-Jordan Bim. (The case of projective surfaceswas done earlier by V.L. Popov and the author in [8, 14]. See also [9] where Jordanproperties of the groups of biholomorphic automorphisms for certain compact andnon-compact complex manifolds have been studied.)The aim of this paper is to study Jordan properties of Aut( Y ) and Bim( Y ) where Y are certain CP -bundles over complex tori . Recall [7] that a complex compactmanifold X is a complex torus if it is (biholomorphic to) a connected compactcomplex Lie group . (Such a group is always commutative [7].) It is known [3, Ch.2, Sect. 6] that the algebraic dimension dim a ( X ) of X is positive if and only if X admits as a quotient-torus a positive-dimensional complex abelian variety. If x ∈ X then we write T x ∈ Aut( X ) for the translation map (1) T x : X → X, z z + x ∀ z ∈ X. Mathematics Subject Classification.
Key words and phrases. complex tori, theta groups, Jordan properties.The author was partially supported by Simons Foundation Collaboration grant
Clearly, all T x ’s constitute a commutative subgroup in Aut( X ), because T x ◦ T y = T x + y ∀ x, y ∈ X. If V a holomorphic vector bundle over X then for for each λ ∈ C ∗ we writemult( λ ) = mult V ( λ ) ∈ Aut( V )for the holomorphic automorphism of the total space of V that acts as multiplicationby λ in every fiber. The map(2) mult = mult V : C ∗ → Aut( V ) , λ mult V ( λ )is an injective group homomorphism . We write Aut ( V ) for the centralizer ofmult V ( C ∗ ) in Aut( V ). Clearly, Aut ( V ) is a subgroup of Aut( V ) that containsmult V ( C ∗ ).We write X for the trivial holomorphic line bundle X × C on X . If L is aholomorphic line bundle over X then we write L x for its fiber over x ∈ X and Y L for the CP -bundle over X that is the projectivization P ( L ⊕ X ) of the rank 2vector bundle L ⊕ X over X . Example 1.2. If L = X then Y L = P ( X ⊕ X ) = X × CP . Let L be a holomorphic line bundle over X . We write K ( L ) for the set of all x ∈ X such that L is isomorphic to the induced holomorphic line bundle T ∗ x L on X . It is known [5, pp. 7-8] that K ( L ) is a subgroup of X that coincides with thekernel of a certain holomorphic Lie group homomorphism from X to the dual torusof X . This implies that K ( L ) is a closed (hence compact) complex commutativeLie subgroup in X and therefore has finitely many connected components. Wewrite K ( L ) for the identity component of K ( L ); by definition, K ( L ) is a complexsubtorus in X , K ( L ) ⊂ K ( L ) ⊂ X ;the compactness of K ( L ) implies that the quotient K ( L ) /K ( L ) is a finite commu-tative group.Let us consider the subgroup S ( L ) ⊂ Aut( L ) of all holomorphic automorphisms u of the total space of L that enjoy the following properties.(i) There exists x ∈ X such that u : L → L is a lifting of T x : X → X , i.e., thefollowing diagram is commutative.(3) L u −−−−→ L y y X T x −−−−→ X (ii) For each z ∈ X the map between the fibers of L over z and z + x inducedby u is a linear isomorphism of one-dimensional C -vector spaces.By definition,(4) mult L ( C ∗ ) ⊂ S ( L ) ⊂ Aut ( L ) . There is a natural group homomorphism ρ = ρ L : S ( L ) → X OMPLEX TORI 3 that sends u to x if u is a lifting of T x . Clearly, ker( ρ L ) = mult L ( C ∗ ) ∼ = C ∗ . Thismeans that S ( L ) is included in an exact sequence of groups(5) 1 → C ∗ mult L −→ S ( L ) ρ L −→ X. Remark 1.4.
Let ψ : L ∼ = L be an isomorphism of holomorphic line bundles L and L over X . Then K ( L ) = K ( L ), and all the isomorphisms between L and L are of the form mult L ( c ) ψ = ψ mult L ( c ) where c runs through C ∗ . Thisimplies that the induced by ψ group isomorphism(6) ψ S : S ( L ) ∼ = S ( L ) , u ψ u ψ − does not depend on a choice of ψ . In addition,(7) ψ S (mult L ( c )) = mult L ( c ) ∀ c ∈ C ∗ and ψ S may be extended to the commutative diagram(8) S ( L ) ψ S −→ S ( L ) ρ L ց ւ ρ L X .
In what follows we write C ) for the number of elements of a finite set C .By a short exact sequence of complex (resp. real) Lie groups we mean a shortexact sequence of groups, each of which is a complex (resp. real) Lie group and allthe homomorphisms involved are homomorphisms of corresponding complex (resp.real) Lie groups. We do not assume these groups to be connected or to have finitelymany connected components.The following assertion was inspired by results of D. Mumford [7, Sect. 23], whodealt with abelian varieties. Theorem 1.5. If L is any holomorphic line bundle over X then the group S ( L ) carries the natural structure of a complex Lie group that enjoys the following prop-erties. (0) The action map S ( L ) × L → L , ( u , l ) u ( l ) ∀ u ∈ S ( L ) , l ∈ L of S ( L ) onthe total space of L is holomorphic. (i) ρ L ( S ( L )) = K ( L ) and the short exact sequence of groups induced by (5)(9) 1 → C ∗ mult L → S ( L ) ρ L −→ K ( L ) → is a short exact sequence of complex Lie groups. (ii) Let us consider the preimage S ( L ) := ρ − L ( K ( L ) ) ⊂ S ( L ) , which is anormal clopen complex Lie subgroup of finite index (cid:0) K ( L ) /K ( L ) (cid:1) in S ( L ) . Then S ( L ) is the identity component and the center of S ( L ) . Inparticular, S ( L ) is commutative if and only if K ( L ) is connected. (iii) If ψ : L → L ′ is an isomorphism of holomorphic vector bundles over X then ψ S : S ( L ) ∼ = S ( L ′ ) defined in (6) is an isomorphism of complex Liegroups. Corollary 1.6. If L ∈
Pic ( X ) then S ( L ) is commutative.Proof of Corollary 1.6. It is known [5, Corollary 1.9 on p. 7] that if
L ∈
Pic ( X )then K ( L ) = X and therefore is connected. Now the desired result follows fromTheorem 1.5(ii). (cid:3) YURI G. ZARHIN
The following assertion was actually proven in [15] in the case when dim( X ) = 1.(See also [1, Cor. 3.6].) Theorem 1.7.
Let L be a holomorphic line bundle over X . Then there is a groupembedding Υ L : S ( L ) ֒ → Aut( Y L ) of S ( L ) into the group Aut( Y L ) of holomorphic automorphisms of Y L = P ( L ⊕ X ) such that the action map S ( L ) × Y L → Y L , ( u , y ) Υ L ( u )( y ) ∀ u ∈ S ( L ) , y ∈ Y L is holomorphic. In addition, the action of every u ∈ S ( L ) on Y L is a lifting of T x : X → X where x = ρ L ( u ) . In other words, the following diagram is commutative. (10) Y L Υ L ( u ) −−−−→ Y L y y X T x −−−−→ X The following assertion was actually proven in [14] in the case when X is anabelian variety and L is ample. Theorem 1.8.
The Jordan constant of S ( L ) is p K ( L ) /K ( L ) ) . We use Theorem 1.7 and ideas related to Theorem 1.8 in the proof of the followingmain result of this paper.
Theorem 1.9.
Let X be a complex torus of positive algebraic dimension. Thenthe group Bim( X × CP ) is not Jordan. The special case of Theorem 1.9 when X is a complex abelian variety was donein [14]. We also prove the following generalizations of Theorem 1.9. Theorem 1.10.
Let ψ : X → A be a surjective holomorphic group homomorphismfrom a complex torus X onto a complex abelian variety A of positive dimension.Let F be a holomorphic line bundle on X that enjoys the following property:there exist a holomorphic line bundle M on A and a holomorphic line bundle F ∈ Pic ( X ) such that F is isomorphic to the tensor product ψ ∗ M ⊗ F .Then the group Bim( Y F ) is not Jordan. Example 1.11.
Taking X = A and ψ the identity map, we obtain from Theorem1.10 that if X is a positive-dimensional complex abelian variety then the groupBim( Y F ) is not Jordan for every holomorphic line bundle over X . (Actually, thisassertion follows from results of [14].) Theorem 1.12.
Let X be a complex torus and F be a holomorphic line bundle on X . Let X be a complex subtorus in X that enjoys the following properties. (i) X has positive dimension. (ii) The quotient A := X/X is a complex abelian variety of positive dimension. (iii) The restriction of F to X lies in Pic ( X ) . (iv) Hom( X , A ) = { } .Then the group Bim( Y F ) is not Jordan. OMPLEX TORI 5
Example 1.13.
Suppose that X is a two-dimensional complex torus that containsa one-dimensional subtorus X . Then X is a one-dimensional abelian variety(elliptic curve) and the quotient X = X/X is also a one-dimensional torus andtherefore is also an elliptic curve. Now the condition Hom( X , X ) = { } meansthat X and X are not isogenous. It follows from Theorem 1.12 that if X and X are not isogenous and F is a holomorphic line bundle on X , whose restrictionto X lies in Pic ( X ) (i.e., has degree 0) then Bim( Y F ) is not Jordan.The paper is organized as follows. In Section 2 we discuss certain natural non-linear transformation groups that act in complex vector spaces. Sections 3 dealsmostly with linear algebra (Hermitian forms, lattices, discriminant groups) relatedto holomorphic bundles on complex tori via
Appel - Humbert theorem . In Section 4we discuss in detail theta groups , which are pretty well known in the case of abelianvarieties [7, 5]. Theorems 1.5 and 1.7 are proven in Section 5. Jordan propertiesof theta groups are discussed in Section 6; they are used in the proof of Theorem1.8 in Section 7. Theorem 1.9 is proven in Section 8. Section 9 deals with pencils (one-dimensional families) of Hermitian forms; its results are used in Section 10 inthe proofs of Theorems 1.12 and 1.10. In Section 11 we discuss theta groups thatcorrespond to line bundles from Pic and identify them with the complement of thetotal space of the line bundle to the zero section. (In particular, we give anotherproof of their commutativity) Acknowledgements . This paper is a result of an attempt to answer questionsof Constantin Shramov. I am grateful to him for interesting stimulating questionsand discussions. My special thanks go to Vladimir L. Popov, whose thoughtfulcomments helped to improve the exposition.2.
Preliminaries
Throughout the paper we will freely use the following well known commuta-tor pairing (11) e : C × C → A that arises from a short exact sequence of groups ( central extension of C by A )(12) 1 → A → B q → C → A is a central subgroup of B and C is a commutative group. Recall that inorder to find e ( c , c ) ∈ A for c , c ∈ C one has to choose preimages b , b ∈ B with respect to surjection q : B → C , i.e., q ( b ) = c , q ( b ) = c , and put(13) e ( c , c ) := b b b − b − ∈ A ; e ( c , c ) does not depend on a choice of b , b . It is well known that e is bimulti-plicative and alternating. It follows from the very definition of e that a subgroup K ⊂ B is commutative if and only if its image q ( K ) is an isotropic subgroup of C with respect to e . Let V ∼ = C g be a finite-dimensional complex vector space of finite positivedimension g . Let L ∼ = C be a one-dimensional complex vector space and V L := V × L viewed as a complex manifold. We write Aut( V L ) for the group of holomorphic YURI G. ZARHIN automorphisms of V L . For each λ ∈ C ∗ we write mult ( λ ) for the holomorphicautomorphism of V L defined by mult ( λ ) : ( v, c ) ( v, λc ) ∀ v ∈ V, c ∈ L . The map mult : C ∗ → Aut( V L ) , λ mult ( λ )is an injective group homomorphism with image mult ( C ∗ ). We write Aut ( V L )for the centralizer of mult ( C ∗ ) in Aut( V L ). Clearly, Aut ( V L ) is a subgroup ofAut( V L ) containing mult ( C ∗ ). In what follows we discuss certain subgroups ofAut ( V L ) related to line bundles on complex tori of the form V /L where L is a lattice of maximal rank in V , i.e., a discrete subgroup of rank 2dim C ( V ) = 2 g . Let H : V × V → C be a Hermitian form on V and E : V × V → R , u, v Im( H ( u, v ))its imaginary part. Then E is an alternating R -bilinear form such that(14) E ( i u, i v ) = E ( u, v ) , H ( u, v ) = E ( u, i v ) + i E ( u, v ) ∀ u, v ∈ V } . As usual, consider the kernel of H (15) ker( H ) := { u ∈ V | H ( u, v ) = 0 ∀ v ∈ V } , which is a C -vector subspace in V .For each u ∈ V let us consider B H,u ∈ Aut ( V L ) defined as follows. B H,u (( v, c )) = ( v + u, e πH ( v,u ) c ) ∀ v ∈ V, c ∈ L . Clearly, B H, is the identity automorphism of V L . If u , u ∈ V then B H,u ◦ B H,u (( v, c )) = (cid:16) v + u + u , e πH ( v + u ,u ) e πH ( v,u ) c (cid:17) =(16) (cid:16) v + u + u , e πH ( u ,u ) e πH ( v,u + u ) c (cid:17) = mult (cid:16) e πH ( u ,u ) (cid:17) ◦ B H,u + u (( v, c )) . This implies that in Aut ( V L ) we have B H,u B H,u = mult (cid:16) e πH ( u ,u ) (cid:17) ◦ B H,u + u and therefore B H,u B H,u B − H,u ◦B − H,u = mult (cid:16) e πH ( u ,u ) (cid:17) (cid:16) mult (e πH ( u ,u ) ) (cid:17) − = mult (cid:16) e π i E ( u ,u ) (cid:17) . In particular, B H,u and B H,u commute if and only if E ( u , u ) ∈ Z . In addition,it follows from (16) applied to u = u, u = − u that(17) B − H,u = mult (e − πH ( u,u ) ) ◦B H, − u , ( mult ( λ ) ◦ B H,u ) − = mult (cid:18) e − πH ( u,u ) λ (cid:19) ◦B H, − u . We write ˜ G ( H, V ) for the subset { mult ( λ ) ◦ B H,u | λ ∈ C ∗ , u ∈ V } ⊂ Aut ( V L ) . It follows from (16) that ˜ G ( H, V ) is the subgroup of Aut ( V L ) generated by mult ( C ∗ )and all B H,u ( u ∈ V ). Clearly ˜ G ( H ) is included in the short exact sequence(18) 1 → C ∗ mult −→ ˜ G ( H, V ) κ → V → OMPLEX TORI 7 where mult ( C ∗ ) is a central subgroup of ˜ G ( H, V ) and the surjective group homo-morphism κ : ˜ G ( H, V ) → V kills mult ( C ∗ ) while κ ( B H,u ) = u ∀ u ∈ V. In other words, each u ∈ V lifts to B H,u ∈ ˜ G ( H, V ). This implies that the commu-tator pairing V × V → mult ( C ∗ )attached to central extension (18) coincides with u , u mult (cid:16) e π i E ( u ,u ) (cid:17) ∀ u , u ∈ V. In particular, if H = 0 then E = 0 and ˜ G ( H, V ) is a commutative group. Moregenerally, if Π ⊂ V is an additive subgroup in V then we may define ˜ G ( H, Π) asthe subgroup of Aut ( V L ) generated by mult ( C ∗ ) and all B H,u ( u ∈ Π). Clearly˜ G ( H, Π) = κ − (Π) is included in the short exact sequence(19) 1 → C ∗ mult −→ ˜ G ( H, Π) κ → Π → mult ( C ∗ ) is a central subgroup of ˜ G ( H, Π). Each u ∈ Π lifts to B H,u ∈ ˜ G ( H, Π) and the commutator pairing Π × Π → mult ( C ∗ )attached to central extension (19) coincides with u , u mult (cid:16) e π i E ( u ,u ) (cid:17) ∀ u , u ∈ Π . In particular, if the restriction of H to Π is identically then ˜ G ( H, Π) is a commu-tative group. It follows from (16) that ˜ G ( H, ker( H )) is a central subgroup in ˜ G ( H, V ) The aim of this subsection and Subsection 2.7 is to endow ˜ G ( H, V ) with thenatural structure of a connected real
Lie group and its certain subgroups ˜ G ( H, Π)(including Π = ker( H )) with the natural structure of a complex Lie group. Let usconsider the complex manifold V × H C ∗ := V × C ∗ endowed with the compositionlaw (cid:0) V × H C ∗ (cid:1) × (cid:0) V × H C ∗ (cid:1) → V × H C ∗ , (20) ( u, λ ) , ( v, µ ) ( u, λ ) ◦ ( v, µ ) := (cid:16) u + v, λµ e πH ( u,v ) (cid:17) . The bijectivity of the mapΨ H : V × H C ∗ → ˜ G ( H, V ) , ( u, λ ) mult ( λ ) ◦ B u combined with (16) and (17) proves that the composition law (20) makes V × H C ∗ a group with identity element (0 ,
1) and the operation of taking the inverse definedby the map(21) V × H C ∗ → V × H C ∗ , ( v, λ ) ( v, λ ) − := ( − u, e − πH ( u,u ) /λ ) . In addition, Ψ H is a group isomorphism. Formulas (16) and (17) tell us that thegroup V × H C ∗ is actually a real Lie group with real structure induced by thenatural complex structure on V × H C ∗ . (However, if H = 0 then the real Lie YURI G. ZARHIN group V × H C ∗ is actually a commutative complex Lie group.) Clearly, V × H C ∗ is included in the short exact sequence of real Lie groups(22) 1 → C ∗ λ (0 ,λ ) −→ V × H C ∗ ( v,λ ) v −→ V → . Applying Ψ H to (22), we obtain that (18) is actually a short exact sequence of realLie groups.Let Π be a closed additive subgroup of V . The third theorem of Cartan tells usthat Π is a real Lie subgroup of V . Clearly,Π × H C ∗ := Π × C ∗ ⊂ V × C ∗ = V × H C ∗ is a closed subgroup of V × H C ∗ and therefore is its real Lie subgroup. Notice thatΨ H (Π × H C ∗ ) = ˜ G ( H, Π), which implies that Ψ H : Π × H C ∗ → ˜ G ( H, Π) is a groupisomorphism that provides ˜ G ( H, Π) with the structure of a real Lie group. Clearly,Π × H C ∗ is included in the short exact sequence of real Lie groups(23) 1 → C ∗ λ (0 ,λ ) −→ Π × H C ∗ ( v,λ ) v −→ Π → . Applying Ψ H to (23), we obtain that (19) is actually a short exact sequence of realLie groups. Remark 2.5.
Let Π be the identity component of Π, which is a connected real Liesubgroup of V , i.e., is an R -vector subspace of V . Then Π × H C ∗ is the connectedcomponent of Π × H C ∗ that contains the identity element (0 ,
1) of the group law,i.e., the identity component of Π × H C ∗ . It follows that ˜ G ( H, Π ) is the identitycomponent of ˜ G ( H, Π). This implies that Π / Π is canonically isomorphic to thegroup ˜ G ( H, Π) / ˜ G ( H, Π ) of connected components of ˜ G ( H, Π).
Example 2.6.
If Π = ker( H ) then the group law on ker( H ) × H C ∗ is( u, λ ) , ( v, µ ) ( u, λ ) ◦ ( v, µ ) := (cid:16) u + v, λµ e πH ( u,v ) (cid:17) = (cid:0) u + v, λµ e π · (cid:1) = ( u + v, λµ ) , since H ( u, v ) = 0 for all u, v ∈ ker( H ). This means that ker( H ) × H C ∗ is actuallythe direct product ker( H ) × C ∗ of complex Lie groups ker( H ) and C ∗ ; in particular,it is connected commutative . Now and till the rest of this section let us assume that Π = ker( H ) . Sinceker( H ) is a complex vector subspace in V , it is a complex Lie subgroup in V andtherefore Π is also a closed complex Lie subgroup in V . This implies that Π × H C ∗ is a closed complex submanifold of V × H C ∗ . Recall that Π × H C ∗ is a closed realLie subgroup of the real Lie group V × H C ∗ . We claim that Π × H C ∗ is actually a complex Lie group.
Lemma 2.8. If Π = ker( H ) then the real Lie group Π × H C ∗ is the complex Liegroup with respect to the structure of the complex manifold on Π × H C ∗ describedabove. In addition, (23) is a short exact sequence of complex Lie groups.Proof. We need to check that the maps (16) and (17), being restricted to (cid:0) Π × H C ∗ (cid:1) × (cid:0) Π × H C ∗ (cid:1) and Π × H C ∗ respectively are holomorphic (not just real analytic). Thecomplex analyticity condition could be checked locally, in the open neighborhoods( u + ker( H )) × C ∗ , ( v + ker( H )) × C ∗ ⊆ Π × H C ∗ × Π × H C ∗ OMPLEX TORI 9 of points ( u , λ ) , ( v , µ ) ∈ Π × H C ∗ × Π × H C ∗ . Then (16) gives us the compositionlaw ( u + u, λ ) ◦ ( v + v, µ ) = ( u + v + u + v, e πH ( u + u,v + w ) λµ ) =( u + v + u + v, e π ( H ( u ,v ) λµ ) , which is obviously holomorphic in u, v, ∈ ker( H ), λ, µ ∈ C ∗ . Similarly, (17) givesus the operation of taking the inverse( u + u, λ ) ( u + u, λ ) − = ( − u − u, e − πH ( u + u,u + u ) /λ ) = ( − u − u, e − πH ( u ,u ) /λ ) , which is obviously holomorphic in u ∈ ker( H ), λ ∈ C ∗ . The second assertion ofLemma is also obvious. (cid:3) Remark 2.9.
Let Π = ker( H ).(i) Lemma 2.8 and the bijectivity of Ψ H allow us to endow the real Lie group˜ G ( H, Π) = Ψ H (Π × H C ∗ ) with the compatible natural structure of a com-plex Lie group, whose identity component ˜ G ( H, Π ) = ˜ G ( H, ker( H )) = Ψ H (ker( H ) × H C ∗ )is a central subgroup of ˜ G ( H, Π) (and even of ˜ G ( H, V )).(ii) The action map ˜ G ( H, Π) × V L → V L , Ψ H ( u, λ ) , ( v, c ) mult( λ ) ◦ B u (( v, c )) = ( v + u, λ e πH ( v,u ) c )is holomorphic . Indeed, it suffices to check that it is holomorphic at allΨ H ( u, c ) from the open subgroup ˜ G ( H, ker( H )). In this case H ( v, u ) = 0and we get the map (ker( H ) × C ∗ ) × V L → V L , ( u, λ ) , ( v, c ) ( v + u, λc ),which is obviously holomorphic. Clearly, (19) is a short exact sequence of complex Lie groups.3.
Hermitian forms, lattices, line bundles
In what follows, a lattice is an additive discrete subgroup in a finite-dimensionalcomplex (or real) vector space.Let X be a positive-dimensional complex torus, i.e., X = V /L where V ∼ = C g afinite-dimensional complex vector space of positive dimension g and L ⊂ V a latticeof maximal possible rank 2 g . We view X as a connected complex commutativecompact Lie group. By Appel-Humbert theorem [7, 5], holomorphic line bundles L on X are classified (up to an isomorphism) by A.-H. data ( H, α ) where H : V × V → C is an Hermitian form on V and α : L → U(1) is a map from L to the unit circle U(1) that enjoy the following properties:(24) E ( l , l ) := Im( H ( l , l )) ∈ Z , α ( l + l ) = ( − E ( l ,l ) α ( l ) α ( l ) ∀ l , l ∈ L. We denote by L ( H, α ) the corresponding line bundle on X , whose definition isrecalled in Subsection 3.2. Let us consider the following discrete action of the group L on V L by holomor-phic automorphisms. An element l ∈ L acts as A H,l : V L → V L , ( v, c ) (cid:16) v + l, α ( l )e πH ( v,l )+ πH ( l,l ) · c (cid:17) ∀ v ∈ V, c ∈ C . In other words, A H,l = mult (cid:16) α ( l )e πH ( l,l ) (cid:17) B H,l ∈ Aut ( V L ) . In particular, A H,l ∈ ˜ G ( H, L ) ∀ l ∈ L. It is well known (and could be easily checked by direct computations) that A H,l ◦ A H,l = A H,l + l ∀ l , l ∈ L. In particular, the map(25) A L : L → Aut ( V L ) , l
7→ A
H,l in an injective group homomorphism, whose image we denote by(26) ˜ L = ˜ L ( H, α ) := A L ( L ) ⊂ ˜ G ( H, V ) ⊂ Aut ( V L ) . Clearly, ˜ L is a subgroup of Aut ( V L ) that meets mult ( C ∗ ) only at the identityelement. In addition,(27) ˜ L = ˜ L ( H, α ) ⊂ ˜ G ( H, L ) ⊂ Aut ( V L ) . It is also clear that for each additive subgroup Π ⊂ V we have(28) ˜ L \ ˜ G ( H, Π) = A L (Π \ L ) = {A H,l | l ∈ Π \ L } . The holomorphic line bundle L ( H, α ) → X is defined [7, Sect. 2] as thequotient V L / ˜ L = V L /L → V /L = X. In particular, V L / ˜ L is the total space of the holomorphic line bundle L ( H, α ).In the obvious notation,(29) L ( H, α ) ⊗ L ( H ′ , α ′ ) ∼ = L ( H + H ′ , αα ′ ) . In particular, X is isomorphic to L (0 , α ) where(30) α : L → { } ⊂ U(1)is the trivial character of L . Definition 3.3.
One says [7, 5, 3] that a holomorphic line bundle on X lies inPic ( X ) if it is isomorphic to L (0 , α ) for some α , i.e., the corresponding Hermitianform is 0. We keep the notation and assumptions of Section 3. Since L spans the R -vectorspace V , we have ker( H ) = { ˜ x ∈ V | H (˜ x, l ) = 0 ∀ l ∈ L } . Let us put(31) L ⊥ E := { ˜ x ∈ V | E (˜ x, l ) ∈ Z ∀ l ∈ L } . Clearly, L ⊥ E is a closed (not necessarily connected) real Lie subgroup of V thatcontains L as a discrete subgroup. In particular, the identity component (cid:0) L ⊥ E (cid:1) of L ⊥ E is an R -vector subspace of V . Lemma 3.5. (i) (cid:0) L ⊥ E (cid:1) = ker( H ) . In particular, (cid:0) L ⊥ E (cid:1) is a C -vector subspaceof V and L ⊥ E is a closed complex Lie subgroup of V . OMPLEX TORI 11 (ii) ˜ G ( H, L ⊥ E ) is a complex Lie group that is included in the short exact sequenceof complex Lie groups (32) 1 → C ∗ mult → ˜ G ( H, L ⊥ E ) κ → L ⊥ E → defined in (19) for Π = L ⊥ E . (iii) ˜ G ( H, ker( H )) = κ − (ker( H )) is the identity component of ˜ G ( H, L ⊥ E ) , whichis a central clopen subgroup of ˜ G ( H, L ⊥ E ) containing mult ( C ∗ ) and includedin the short exact sequence of complex Lie groups (33) 1 → C ∗ mult → ˜ G ( H, ker( H )) κ → ker( H ) → defined in (19) for Π = ker( H ) that is induced from (32) by ker( H ) ⊂ L ⊥ E . (iv) The action map ˜ G ( H, L ⊥ E ) × V L → V L is holomorphic.Proof. Clearly,(34) (cid:0) L ⊥ E (cid:1) ⊂ { v ∈ V | E ( l, v ) = 0 ∀ l ∈ L } =: V . Since E is R -bilinear, V is a real vector subspace of V . In light of first formula of(14), V = i V , i.e., V is a complex vector subspace of V . Since L spans V over R and E is R -bilinear,(35) V = { v ∈ V | E ( u, v ) = 0 ∀ u ∈ V } . Since V is a C -vector subspace, (35) and second formula of (14) imply that(36) V = { v ∈ V | H ( u, v ) = 0 ∀ u ∈ V } = ker( H ) . Now (35) and (36) imply that V = ker( H ) ⊂ (cid:0) L ⊥ E (cid:1) , because ker( H ) is connected. In light of (34),ker( H ) = V ⊃ (cid:0) L ⊥ E (cid:1) . This implies that ker( H ) = (cid:0) L ⊥ E (cid:1) , which proves (i). Now assertions (ii), (iii) and(iv) follow from Remark (2.9). (cid:3) Let us put L := L \ ker( H ) = { l ∈ L | E ( l, v ) = 0 ∀ v ∈ V } = { l ∈ L | E ( l, m ) = 0 ∀ m ∈ L } . Then L is a free saturated Z -submodule of L and E induces a nondegenerate alternating bilinear form on L/L . In particular, the rank of the free Z -module L/L is even. Since the rank of L is even, the rank of L is also even. Let 2 g bethe rank of L . Then the rank of L/L is 2( g − g ). Notice also that since L is alattice in ker( H ), 2 g ≤ dim R (ker( H )) . Since L is saturated in L , there exists a saturated free Z -submodule L ⊂ L ofrank 2 g − g such that L = L ⊕ L . This implies that V = L ⊗ R = ( L ⊗ R ) ⊕ ( L ⊗ R ) . Clearly, the restriction E (cid:12)(cid:12) L : L × L → Z of E to L is a nondegenerate alternating bilinear form. This implies that therestriction E (cid:12)(cid:12) L ⊗ R : ( L ⊗ R ) × ( L ⊗ R ) → R is a nondegenerate alternating R -bilinear form. It follows that( L ⊗ R ) \ ker( H ) = { } and therefore 2 g = dim R ( V ) ≥ dim R (( L ⊗ R )) + dim R (ker( H )) =(2 g − g ) + dim R (ker( H )) ≥ (2 g − g ) + 2 g = 2 g. This implies that dim R (ker( H )) = 2 g , i.e., L is a lattice of maximal rank in thereal vector space V and ker( H ) = L ⊗ R . Remark 3.7.
It follows from (24) that the restriction of α to L is a group homo-morphism, i.e.,(37) α ( l + l ) = α ( l ) α ( l ) ∀ l , l ∈ L . Theta groups
We keep the notation and assumptions of Section 3.
Suppose that L = L , i.e., E = 0, i.e., H = 0. This means that L is afree Z -module of positive rank 2( g − g ). Let us choose once and for all a ba-sis { ¯ l , . . . , ¯ l g − g } of L and consider the skew symmetric nondegenerate squarematrix ˜ E of E (cid:12)(cid:12) L attached to this basis, whose order is 2 g − g and entries are(38) ˜ E ij := E (cid:12)(cid:12) L (¯ l i , ¯ l j ) = E (¯ l i , ¯ l j ) ∈ Z The determinant det( E (cid:12)(cid:12) L ) of ˜ E is a nonzero integer that does not depend on achoice of the basis. Since ˜ E is skew symmetric, det( E (cid:12)(cid:12) L ) is the square of the pfaffian of ˜ E . Since all the entries of ˜ E are integers, its pfaffian is also an integerand therefore det( E (cid:12)(cid:12) L ) is a square in Z ; in particular, it is a positive integer .Its square root q det( E (cid:12)(cid:12) L ) will play a prominent role in Section 6. On the otherhand, det( E (cid:12)(cid:12) L ) admits the following well known interpretation. Let us put L ⊥ ,E = { ˜ x ∈ L ⊗ R | E (˜ x, l ) ∈ Z ∀ l ∈ L } . The nondegeneracy of E (cid:12)(cid:12) L implies that L ⊥ ,E is a free Z -module of rank 2 g − g that is contained in L ⊗ Q and contains L as a subgroup of finite index. It is wellknown that(39) [ L ⊥ ,E : L ] = L ⊥ ,E /L ) = det( E (cid:12)(cid:12) L ) . Let us also point out the following obvious but useful equality:(40) L ⊥ E = ker( H ) ⊕ L ⊥ ,E = ( L ⊗ R ) ⊕ L ⊥ ,E . In order to get an explicit description of the finite discriminant group L ⊥ ,E /L ,notice that the structure theorem for alternating nongenerate bilinear forms over Z implies the existence of a basis e , f , . . . , e g − g , f g − g ∈ L of the free Z -module L and positive integers d ( E ) , . . . , d g − g ( E ) that enjoy thefollowing properties. Each d i ( E ) divides d i +1 ( E ) (if 1 ≤ i < g ), L = ⊕ g − g i =1 ( Z · e i ⊕ Z · f i ) ; E ( e i , f j ) = 0 if i = j ; E ( e i , f i ) = d i ( E ) ∀ i. (See also [3, pp. 7-8].) It follows thatdet( ˜ E ) = g − g Y i =1 d i ( E ) ! ,L ⊥ ,E = ⊕ g − g i =1 d i ( E ) ( Z · e i ⊕ Z · f i ) . If we define free rank two Z -submodules U i := Z · e i ⊕ Z · f i ⊂ L then we get a direct orthogonal (with respect to E ) splittings(41) L = L ⊕ L , L = ⊕ g − g i =1 U i , L ⊥ ,E = ⊕ g − g i =1 d i ( E ) U i . In addition,(42) E (cid:18) d i ( E ) e i , d i ( E ) f i (cid:19) = 1 d i ( E ) ∀ i ; E (cid:18) d i ( E ) U i , d j ( E ) U j (cid:19) = { } ∀ i = j. It follows from (41) that(43) L ⊥ ,E /L = ⊕ g − g i =1 (cid:18) d i ( E ) U i (cid:19) /U i ∼ = ⊕ g − g i =1 ( Z /d i ( E ) Z ) , L ⊥ ,E /L ) = g − g Y i =1 d i ( E ) ! = det( ˜ E ) . It also follows from (41) that(44) X = V /L ⊃ L ⊥ E /L = (ker( H ) /L ) ⊕ (cid:0) L ⊥ ,E /L (cid:1) = (ker( H ) /L ) ⊕ (cid:18) ⊕ g − g i =1 (cid:18) d i ( E ) U i (cid:19) /U i (cid:19) . (See also [3, pp. 7-8].) Remark 4.2.
Suppose that H = 0. It follows from [5, pp. 7-8] that(45) K ( L ( H, α )) = L ⊥ E /L ⊂ X. Now (44) implies that ker( H ) /L is the identity component K ( L ( H, α )) of thecomplex Lie group K ( L ( H, α )) while the group K ( L ( H, α )) /K ( L ( H, α )) is iso-morphic to L ⊥ ,E /L and therefore(46) (cid:0) K ( L ( H, α )) /K ( L ( H, α )) (cid:1) = (cid:0) L ⊥ ,E /L (cid:1) = det( E (cid:12)(cid:12) L ) = g − g Y i =1 d i ( E ) ! . Let us consider the alternating bilinear pairing(47) e E : L ⊥ ,E /L × L ⊥ ,E /L → C ∗ , ( v + L , v + L ) e π i E ( v ,v ) ∀ v + L , v + L ∈ L ⊥ ,E /L . It follows from (42) and (41) that e E is a nondegenerate pairing . Lemma 4.4.
Suppose that H = 0 . Let n be a positive integer such that (48) E ( l , l ) ∈ n Z ∀ l , l ∈ L. Then L ⊥ ,E /L ) is divisible by n .Proof. Since H = 0, we have g < g , i.e. g − g ≥
1. It follows from (38) that all theentries of the order 2( g − g ) square matrix ˜ E are divisible by n in Z and thereforedet( ˜ E ) is divisible by n g − g ) in Z . This implies that L ⊥ ,E /L ) = det( ˜ E ) isdivisible by n g − g ) and therefore is divisible by n . (cid:3) Theorem 4.5. (i) ˜ L = ˜ L ( H, α ) is a central discrete subgroup of ˜ G ( H, L ⊥ E ) that meets mult ( C ∗ ) only at the identity. (ii) ˜ L := ˜ L \ ˜ G ( H, ker( H )) = A L ( L ) = {A H,l | l ∈ L } is a discrete subgroup in the commutative connected complex Lie group ˜ G ( H, ker( H )) . (iii) The commutative connected complex Lie group ˜ G ( H, ker( H )) / ˜ L is isomor-phic to the quotient (ker( H ) × C ∗ ) / ¯ L where ¯ L := { ( l, α ( l )) | l ∈ L } is adiscrete subgroup in ker( H ) × C ∗ .Proof. We have already seen that ˜ L meets mult ( C ∗ ) only at the identity and ˜ L ⊂ ˜ G ( H, L ). Since L ⊥ E contains L , we have˜ G ( H, L ) = κ − ( L ) ⊂ κ − (cid:0) L ⊥ E (cid:1) = ˜ G (cid:0) H, L ⊥ (cid:1) and therefore ˜ L ⊂ ˜ G (cid:0) H, L ⊥ (cid:1) . Recall that E ( L ⊥ E , L ) ⊂ Z . So, if ˜ l ∈ ˜ L, φ ∈ ˜ G ( H, L ⊥ E )then E (cid:16) κ (˜ l ) , κ ( φ ) (cid:17) ∈ Z and therefore ˜ l and φ commute (see the very end of Subsect.2.3). This implies that ˜ L is a central subgroup of ˜ G (cid:0) H, L ⊥ E (cid:1) . In order to check thediscreteness of ˜ L , recall that κ ( ˜ L ) = L is a discrete subgroup in L ⊥ E . Hence, if ˜ L is not discrete, the intersection ˜ L T ker( κ ) is infinite. However, ker( κ ) = mult ( C ∗ )and we know that ˜ L meets mult ( C ∗ ) only at a single element. The obtainedcontradiction proves that ˜ L is discrete. This proves (i). Since L = L T ker( H ),(ii) follows from (i) combined with (28) applied to Π = ker( H ). Now (iii) followsfrom (ii) combined with Example 2.6. (cid:3) Remark 4.6.
The same arguments prove that ˜ G ( H, L ) is a central subgroup of˜ G (cid:0) H, L ⊥ E (cid:1) . In fact, the natural “multiplication map” mult ( C ∗ ) × ˜ L ( H, α ) → ˜ G ( H, L )is a group isomorphism.
Applying short exact sequence (19) to Π = L ⊥ E , we get a short exact sequenceof complex Lie groups(49) 1 → C ∗ mult −→ ˜ G (cid:0) H, L ⊥ E (cid:1) κ → L ⊥ E → mult ( C ∗ ) is a central subgroup of ˜ G (cid:0) H, L ⊥ E (cid:1) . Each u ∈ L ⊥ E liftsto B H,u ∈ ˜ G (cid:0) H, L ⊥ E (cid:1) and the commutator pairing L ⊥ E × L ⊥ E → mult ( C ∗ ) OMPLEX TORI 15 attached to (49) coincides with u , u mult (cid:16) e π i E ( u ,u ) (cid:17) ∀ u , u ∈ L ⊥ E . Recall that ˜ L ⊂ ˜ G (cid:0) H, L ⊥ E (cid:1) ⊂ ˜ G ( H, V ) ⊂ Aut ( V L ) , is a central discrete subgroup ˜ L of ˜ G (cid:0) H, L ⊥ E (cid:1) that acts discretely on V L and L ( H, α ) = V L / ˜ L = V L / ˜ L ( H, α ) . This gives us the natural embedding of the complex Lie quotient group G ( H, α ) := ˜ G (cid:0) H, L ⊥ E (cid:1) / ˜ L = ˜ G (cid:0) H, L ⊥ E (cid:1) / ˜ L ( H, α )into the group Aut( L ( H, α )) of holomorphic automorphisms of the total space of L ( H, α ). Further, we will identify G ( H, α ) = ˜ G (cid:0) H, L ⊥ E (cid:1) / ˜ L with its (isomorphic)image in Aut( L ( H, α )). It follows from Lemma 3.5(iii) that the action map G ( H, α ) × L ( H, α ) → L ( H, α )is holomorphic.
It follows from (49) and (45) that G ( H, α ) = ˜ G (cid:0) H, L ⊥ E (cid:1) / ˜ L is included in ashort exact sequence of complex Lie groups(50) 1 → C ∗ → G ( H, α ) ¯ κ → L ⊥ E /L (= K ( L ( H, α ) ) → . Here the image of C ∗ is a central subgroup in G ( H, α ), each λ ∈ C ∗ → G ( H, α ) ⊂ Aut( L ( H, α ))acts on the total space of L ( H, α ) as the multiplication by λ at every fiber of L ( H, α ) → X , i.e, λ is mapped to mult L ( H,α ) ( λ ); the surjective complex Lie grouphomomorphism ¯ κ : G ( H, α ) = ˜ G (cid:0) H, L ⊥ E (cid:1) / ˜ L → L ⊥ E /L kills the image of C ∗ and sends a coset B H,u ˜ L to u + L for each u ∈ L ⊥ .Clearly, the commutator pairing attached to central extension (50) is(51) e H,α : L ⊥ /L × L ⊥ /L → C ∗ , u + L, u + L e π i E ( u ,u ) ∀ u + L, u + L ∈ L ⊥ /L. Remark 4.9. (i) Clearly, the restriction of e H,α to L ⊥ ,E /L × L ⊥ ,E /L coin-cides with the nondegenerate pairing (47).(ii) Clearly, ker( H ) /L lies in the kernel of e H,α . Combining this with (i), weobtain that ker( H ) /L coincides with the kernel of e H,α , since L ⊥ E /L = (ker( H ) /L ) ⊕ ( L ⊥ ,E /L ) . Theorem 4.10.
The identity component G ( H, α ) of G ( H, α ) coincides with thepreimage κ − (ker( H ) /L ) of ker( H ) /L ⊂ L ⊥ E /L ⊂ V /L = X and is canonically isomorphic as a complex Lie group to the quotient ˜ G ( H, ker( H )) / ˜ L .In particular, G ( H, α ) is a central subgroup of G ( H, α ) that is isomorphic as acomplex Lie group to (ker( H ) × C ∗ ) / ¯ L . Proof.
It follows from Theorem 4.5 that G ( H, α ) is the image of ˜ G ( H, ker( H )) in˜ G (cid:0) H, L ⊥ E (cid:1) / ˜ L = G ( H, α ) and this image coincides with˜ G ( H, ker( H )) / (cid:16) ˜ L \ ˜ G ( H, ker( H )) (cid:17) = ˜ G ( H, ker( H )) / ˜ L ⊂ ˜ G (cid:0) H, L ⊥ E (cid:1) / ˜ L = G ( H, α ) . Since ker( H ) /L is the identity component of L ⊥ E /L , (50) implies that G ( H, α ) ⊂ κ − (ker( H ) /L ). The connectedness of C ∗ implies that its image in G ( H, α ) (see(50)) lies in G ( H, α ) . The exactness of (50) implies that in order to prove thedesired equality, it suffices to check that for each u + L ∈ ker( H ) /L (with u ∈ ker( H )), there is u ∈ G ( H, α ) with ¯ κ ( u ) = u + L . Thanks to (19) and (49), u := B H,u ˜ L ∈ ˜ G ( H, ker( H )) / ˜ L = G ( H, α ) does the trick. Now the last assertion of Theorem follows from Theorem 4.5(iii). (cid:3) Theorem 4.11.
Let ˜ B be a subgroup of G ( H, α ) and B := ¯ κ ( ˜ B ) ⊂ L ⊥ /L be its image, which is a subgroup in L ⊥ /L . (0) If N is a subgroup of B then ˜ N := ¯ κ − ( N ) \ ˜ B = { u ∈ ˜ B ⊂ G ( H, α ) | ¯ κ ( u ) ∈ B } is a normal subgroup in ˜ B . In addition, if ˜ B is finite then [ B : N ] and [ ˜ B : ˜ N ] coincide. (i) ˜ B is commutative if and only if B is isotropic with respect to e H,α . (ii) Suppose that H = 0 and B ⊂ L ⊥ ,E /L ⊂ L ⊥ /L. Then ˜ B is commutative if and only if B is isotropic with respect to e E . Ifthis is the case then the index (cid:2)(cid:0) L ⊥ ,E /L (cid:1) : B (cid:3) is divisible by r (cid:16) L ⊥ ,E /L (cid:17) . iii) Suppose that H = 0 and n is a positive integer such that E ( L, L ) ⊂ n Z . Let ˜ A be a commutative subgroup of G ( H, α ) and let A := ¯ κ ( ˜ A ) ⊂ L ⊥ /L be its image, which is a subgroup in L ⊥ /L . If A ⊂ L ⊥ ,E /L then the index (cid:2)(cid:0) L ⊥ ,E /L (cid:1) : B (cid:3) is divisible by n . iv) Suppose that H = 0 and (52) pr : L ⊥ /L = (ker( H ) /L ) ⊕ ( L ⊥ ,E /L ) → ( L ⊥ ,E /L ) be the projection map. Let us put B := pr ( B ) = pr ¯ κ ( ˜ B ) ⊂ L ⊥ ,E /L . Then ˜ B is commutative if and only if B is isotropic with respect to e E . Ifthis is the case then the index [( L ⊥ ,E /L ) : B ] is divisible by q L ⊥ ,E /L ) . OMPLEX TORI 17
Proof. (0). Since B ⊂ L ⊥ ,E /L is commutative, its every subgroup, including N , isnormal in B . Let us consider the surjective homomorphism(53) ¯ κ : ˜ B → B and denote its kernel by ˜ B , which is a normal subgroup in ˜ B . The surjectivity of(53) implies that the preimage ˜ N ⊂ ˜ B of N is also normal in ˜ B ; in addition, ˜ N contains ˜ B , which is normal in ˜ N . The surjection (53) induces group isomorphisms˜ B/ ˜ B ∼ = B, ˜ N / ˜ B ∼ = N. If ˜ B is finite then all the other groups involved are also finite and B ) = B ) · B ) , N ) = B ) · N ) , which implies that[ ˜ B : ˜ N ] = B ) N ) = B ) · B ) B ) · N ) = B ) N ) = [ B : N ] . It follows that [ ˜ B : ˜ N ] = [ B : N ]. This completes the proof of (0).(i) follows from the description (51) of e H,α as the commutator pairing attachedto the central extension (50).The first assertion of (ii) follows from (i) and Remark 4.9. The second assertionof (ii) follows from the first one and the nondegeneracy of e E .(iii) follows from (ii) combined with Lemma 4.4.(iv) By (i), ˜ B is commutative if and only if B is isotropic with respect e H,α . Let x , x ∈ B ⊂ L ⊥ /L = ker( H ) /L ⊕ L ⊥ ,E /L ⊂ X. We have x = h + l , x = h + l ; h j ∈ ker( H ) /L , l j ∈ L ⊥ ,E /L . By Remark 4.9, each h j lies in the kernel of e H,α . This implies that e H,α ( x , x ) = e H,α ( l , l ) = e E ( l , l ) . This implies that B is isotropic with respect to e H,α if and only if B is isotropicwith respect to e E . The remaining assertion about the index follows from the nondegeneracy of e E . (cid:3) We call G ( H, α ) the theta group of L ( H, α ). Recall (Remark 4.2) that(54) K ( L ( H, α )) = L ⊥ /L ⊂ V /L = X. Clearly, G ( H, α ) ⊂ S ( L ( H, α )) . More precisely, all elements of mult L ( H,α ) ( C ∗ ) are liftings of the identity automor-phism T e of X (where e is the zero of group law on X ) while B H,u ˜ L is a lifting of T x where x = u + L ∈ L ⊥ E /L = K ( L ( H, α )) ⊂ V /L = X. It follows that ¯ κ : G ( H, α ) → L ⊥ E /L = K ( L ( H, α )) ⊂ X coincides with the restriction of ρ = ρ L ( H,α ) : S ( L ( H, α )) → X (defined in Subsection 1.3) to G ( H, α ) ⊂ S ( L ( H, α )).
Theorem 4.13.
The identity component G ( H, α ) = ¯ κ − (ker( H ) /L ) of the com-plex Lie group G ( H, α ) is the center of G ( H, α ) , which is included in the short exactsequence of complex Lie groups → C ∗ → G ( H, α ) κ → ker( H ) /L → . In particular, G ( H, α ) is commutative if H = 0 .Proof of Theorem 4.13. It follows from the results of Subsection 4.8 that u ∈ G ( H, α )lies in the center of G ( H, α ) if and only if x := ¯ κ ( u ) ∈ L ⊥ E /L satisfies x = v + L with v ∈ L ⊥ E , e π i E ( v,w ) = 1 ∀ w ∈ L ⊥ E , i.e.,(55) E ( v, w ) = Im( H ( v, w )) ∈ Z ∀ w ∈ L ⊥ E . Clearly, each v ∈ ker( H ) satisfies (55) and therefore the center of G ( H, α ) contains¯ κ − (cid:16) ker( H ) / (cid:16) ker( H ) \ L (cid:17)(cid:17) = ¯ κ − (ker( H ) /L ) = G ( H, α ) . In particular, if H = 0 thenker( H ) = V, L = L, L ⊥ E = V, ker( H ) /L = L ⊥ E /L ;hence G ( H, α ) coincides with its central subgroup G ( H, α ) and therefore is com-mutative.Now suppose that (in the notation of Subsection 3.6) v ker( H ) ⊕ (cid:0) ⊕ g − g i =1 U i (cid:1) . This implies that H = 0 and there exist u ∈ ker( H ) and u i ∈ d i U i (for all i with1 ≤ i ≤ g − g ) such that not all u i ∈ U i and u = u + g − g X i =1 u i . Suppose that u j U j for a certain j ∈ { , . . . , g − g } . Then u j = a j e j + b j f j where a j , b j ∈ d j Z and, at least, one of a j , b j is not an integer . Recall that1 d j e j , d j f j ∈ L ⊥ E . However, E (cid:18) u, d j e j (cid:19) = b j , E (cid:18) u, d j f j (cid:19) = a j and therefore (55) does not hold. This implies that if u is a central element of G ( H, α ) then v ∈ ker( H ) ⊕ (cid:0) ⊕ g − g i =1 U i (cid:1) , i.e.,¯ κ ( u ) ∈ ker( H ) /L , which means that u ∈ G ( H, α ) . This completes the proof. (cid:3) Theorem 4.14. If L = L ( H, α ) then G ( H, α ) = S ( L ( H, α )) . In particular, (56) ρ L ( H,α ) = ¯ κ, K ( L ( H, α )) = ¯ κ ( G ( H, α )) = ρ L ( H,α ) ( S ( L ( H, α ))) . OMPLEX TORI 19
Remark 4.15.
Recall (Remark 4.2) that ker( H ) /L is the identity component of K ( L ( H, α )). Now results of Subsection 4.12 combined with Theorems 4.14 implythat(57) G ( H, α ) = ¯ κ − (ker( H ) /L ) = ρ − (cid:0) K ( L ( H, α )) (cid:1) = S ( L ( H, α )) . of Theorem 4.14. We write p : L → X for the structure morphism from the totalspace of the holomorphic line bundle to its base.Let u ∈ S ( L ) ⊂ Aut( L ). By definition of S ( L ), there is x ∈ X such that u : L → L is a lifting of T x : X → X . In particular, the restriction of u to the fibersof L induces the linear isomorphisms u z : L z ∼ = L z + x between the fibers of L at z and x + z for all z ∈ X .It follows from [12, Ch. 1, Sect. 2, proposition 2.14] (applied to f = T x ) thatthere exist an induced holomorphic line bundle T ∗ x L = { ( l , z ) ∈ L × X | p ( l ) = z + x } over X with the structure morphism T ∗ x L → X, ( l , z ) z, and a holomorphic map of total spaces of holomorphic line bundles over X ( T x ) ∗ : T ∗ x L → L , ( l , z ) l that lifts T x and induces C -linear isomorphisms between the corresponding fibers( T ∗ x L ) z and L z + x for all z ∈ X . Clearly, ( T x ) ∗ is a biholomorphic isomorphism:indeed, its inverse is defined by l ( l , z ) = ( l , p ( l ) − x ) . It follows that the composition( T x ) ∗ ◦ u − : L → L → T ∗ x L is an isomorphism of holomorphic line bundles L and T ∗ x L over X . Therefore holo-morphic line bundles L ( H, α ) = L and T ∗ x L ( H, α ) = T ∗ x L over X are isomorphic.Hence x ∈ K ( L ( H, α )) = L ⊥ E /L ⊂ V /L = X. Pick u ∈ L ⊥ E with u + L = x . Then u B − H,u is a holomorphic automorphism of L ( H, α ) that leaves invariant every fiber L ( H, α ) z and acts on it as a C -linearautomorphism. By compactness and connectedness of X , there is a nonzero scalar λ ∈ C ∗ such that u B − H,u acts as multiplication by λ on every fiber. It followsthat u B − H,u lies in G ( H, α ). Since B H,u lies in G ( H, α ) as well, we conclude that u ∈ G ( H, α ). (cid:3) Remark 4.16.
It follows from Theorem 4.14 combined with (50) and the resultsof Subsection 4.12 that S ( L ( H, α )) = G ( H, α ) is included in a short exact sequenceof complex Lie groups(58) 1 → C ∗ → S ( L ( H, α ))(= G ( H, α )) ρ =¯ κ −→ K ( L ( H, α )) → . Proofs of Theorem 1.5 and 1.7
Definition 5.1.
Let L be a holomorphic line bundle on X . Let us choose anisomorphism of holomorphic line bundles ψ : L ∼ = L ( H, α ) for suitable A.H. date(
H, α ) where (
H, α ) is uniquely determined by the isomorphism class of L . ByRemark 1.4 combined with Theorem 4.14, there is a certain group isomorphism ψ S : S ( L ) ∼ = S ( L ( H, α ) = G ( H, α ) that does not depend on a choice of ψ . Thenthere is the canonical structure of a complex Lie group on S ( L ) such that the groupisomorphism ψ S : S ( L ) ∼ = G ( H, α ) is an isomorphism of complex Lie groups.
Corollary 5.2.
The action map S ( L ) × L → L and the group homomorphism ρ L : S ( L ) → X are holomorphic.Proof. We may assume that L = L ( H, α ) and therefore S ( L ) = G ( H, α ). Then ourassertion follows from the results of Subsection 4.7 and Theorem 4.14. (cid:3)
Proof of Theorem 1.5. (0) and (i) are contained in Corollary 5.2 and Theorem 4.14.(iii) follows from the very Definition 5.1. In order to check (ii), let us assume that L = L ( H, α ) and therefore S ( L ) = G ( H, α ) , ρ L = ¯ κ, S ( L ) = G ( H, α ) , K ( L ) = ker( H ) /L . Then all the assertions of (ii) follow from Theorems 4.10 and Theorem 4.13. (cid:3)
Proof of Theorem 1.7.
Denote by V the rank 2 vector bundle V = L ⊕ X over X .By definition, Y L is the projectivization of V .First, let us define an embedding S ( L ) ֒ → Aut ( V ) = Aut ( L ⊕ X ) . In order to do that, recall that each u ∈ S ( L ) ⊂ Aut ( L ) is a lifting of of T x : X → X where x = ρ L ( u ) ∈ X . This allows us to define the action of u on X = X × C as¯ κ ( u ) : X × C → X × C , ( z, λ ) ( z + ρ L ( u ) , λ ) ∀ z ∈ X, λ ∈ C . By Corollary 5.2, ρ L is a homomorphism of complex Lie groups, hence is holomor-phic and therefore the corresponding action map S ( L ) × X → X , u , ( z, λ ) ( z + ρ L ( u ) , λ )is holomorphic. This gives us a (non-injective) group homomorphism¯ κ : S ( L ) → Aut ( X ) , whose image meets “scalar automorphisms” mult X ( C ∗ ) only at the identity auto-morphism of X . Taking the “direct sum” of the embedding S ( L ) ⊂ Aut ( L ) with¯ κ , we get a group embedding ¯ κ : S ( L ) ֒ → Aut ( L ⊕ X ) = Aut ( V ) , whose image also meets precisely one element of mult V ( C ∗ ), namely the identityautomorphism of V . Clearly, the corresponding action map S ( L ) ×V → V , u , ( l z ; ( z, λ )) ( u ( l z ); ( z + ρ L ( u , λ )) ∀ z ∈ X, l z ∈ L z , λ ∈ C , u ∈ S ( L )is holomorphic as well, since the action map S ( L ) × L → L is holomorphic, thanksto Corollary 5.2. It is also clear that ¯ κ ( u ) is a lifting of T x with x = ρ L ( u ). Itfollows that the group homomorphismΥ L : S ( L ) → Aut( P ( V )) = Aut( Y L ) OMPLEX TORI 21 induced by ¯ κ is an embedding, the corresponding action map S ( L ) × Y L → Y L isholomorphic and Υ L ( u ) is a a lifting of T x with x = ρ L ( u ). (cid:3) Jordan properties of theta groups
We keep the notation and assumption of Section 3.
Theorem 6.1.
Suppose that H = 0 . Then G ( H, α ) is Jordan and its Jordanconstant is q L ⊥ ,E /L ) = g − g Y i =1 d i ( E ) . Corollary 6.2.
Let H = 0 and n be a positive integer such that E ( L, L ) ⊂ n Z . Then the Jordan constant J G ( H,α ) of G ( H, α ) is divisible by n . In particular, J G ( H,α ) ≥ n .Proof of Corollary 6.2. By Lemma 4.4, L ⊥ ,E /L ) is divisible by n . Now thedesired result follows readily from Theorem 6.1. (cid:3) We will need the following Lemma that will be proven at the end of this section.
Lemma 6.3.
Let ∆ be a finite subgroup in K ( L ( H, α )) . Then there exists a finitesubgroup ˜∆ in G ( H, α ) such that ¯ κ ( ˜∆) = ∆ .Proof of Theorem 6.1. Let ˜ B be a finite subgroup in G ( H, α ). Let us consider itsimages B = ¯ κ ( ˜ B ) ⊂ K ( L ( H, α )) = (ker( H ) /L ) ⊕ ( L ⊥ ,E /L ) , B = pr ( B ) = pr ¯ κ ( ˜ B ) ⊂ L ⊥ ,E /L . Let A be a maximal isotropic subgroup in B with respect to e E . The nondegen-erate pairing e E gives rise to an embedding B /A ֒ → Hom( A , C ∗ ) , b + A
7→ { a e E ( a, b ) ∀ a ∈ A } . Since the orders of finite commutative groups A and Hom( A , C ∗ ) coincide, B /A )divides A ) and therefore B /A ) divides B ), which in turn, divides L ⊥ ,E /L ). It follows that the index[ B : A ] = B /A ) does not exceed (actually divides) q L ⊥ ,E /L ). Let us consider the subgroup˜ A := (pr ¯ κ ) − ( A ) \ ˜ B. Since A is isotropic, it follows from Theorem 4.11(iv) that ˜ A is a commutativesubgroup. Since A is obviously normal in (commutative) B , the preimage ˜ A of A with respect to surjective ˜ B pr ¯ κ −→ B is a normal subgroup of ˜ B , whose index does not exceed (actually equals) [ B : A ],which, in turn, does not exceed q L ⊥ ,E /L ). It follows that G ( H, α ) is Jordanand its Jordan constant does not exceed q L ⊥ ,E /L ). We need to prove that theJordan constant is, at least, q L ⊥ ,E /L ). In order to do that, let us consider the finite subgroup∆ := L ⊥ ,E /L = { } ⊕ (cid:0) L ⊥ ,E /L (cid:1) ⊂ (ker( H ) /L ) ⊕ ( L ⊥ ,E /L ) = K ( L ( H, α )) . By Lemma 6.3, there is a finite subgroup ˜∆ ⊂ G ( H, α ) such that¯ κ ( ˜∆) = ∆ . Let A ′ ⊂ ˜∆ be a commutative normal subgroup of ˜∆. By Theorem 4.11(ii), thesubgroup A = ¯ κ ( A ′ ) ⊂ ∆ = L ⊥ ,E /L is an isotropic subgroup with respect to e E and the index [ (cid:0) L ⊥ ,E /L (cid:1) : A ] is divisibleby q L ⊥ ,E /L ). Let us define˜ A := ¯ κ − ( A ) \ ˜∆ ⊂ ˜∆ . By Theorem 4.110, ˜ A is a normal subgroup of ˜∆ and[ ˜∆ : ˜ A ] = [ L ⊥ ,E /L : A ] . This implies that [ ˜∆ : ˜ A ] is divisible by q L ⊥ ,E /L ).Clearly, ˜ A contains A ′ . This implies that [ ˜∆ : A ′ ] is divisible by [ ˜∆ : ˜ A ] andtherefore is divisible by q L ⊥ ,E /L ). However, if U is a maximal isotropic sub-group in L ⊥ ,E /L then U ) = q L ⊥ ,E /L ) = [ L ⊥ ,E /L : U ] . Let us put ˜ U := ¯ κ − ( U ) \ ˜∆ ⊂ ˜∆ . By Theorem 4.11(0,ii), ˜ U is a commutative normal subgroup in L ⊥ ,E /L of in-dex q L ⊥ ,E /L ). It follows that the Jordan constant of G ( H, α ) is, at least, q L ⊥ ,E /L ). This completes the proof. (cid:3) Proof of Lemma 6.3.
In what follows we identify C ∗ with its image in G ( H, α ) andview it as a certain central subgroup of G ( H, α ). Let d be the exponent of ∆. Letus consider the finite multiplicative group µ d of all d th roots of unity and the finitemultiplicative group µ d of all d th roots of unity in C . We have µ d ⊂ µ d ⊂ C ∗ ⊂ G ( H, α ) . For each x ∈ ∆ choose its lifting ˜ x ∈ G ( H, α ) with the same order as x andsuch that the lifting ] ( − x ) of − x coincides with ˜ x − . (This is possible, since C ∗ isa central divisible subgroup in G ( H, α ).) Let us consider the finite set˜∆ = { γ ˜ x | γ ∈ µ d , x ∈ ∆ } ⊂ G ( H, α ) . Clearly, ¯ κ ( γ ˜ x ) = x and therefore ¯ κ ( ˜∆) = ∆. It remains to check that ˜∆ is asubgroup in G ( H, α ). Let x , x ∈ ∆ and x = x + x ∈ ∆. We need to compare˜ x ˜ x and ˜ x in G ( H, α ). Notice that there is γ ∈ C ∗ such that˜ x = γ ˜ x ˜ x . OMPLEX TORI 23
In addition, ˜ x d = ˜ x d = ˜ x d = 1 ∈ C ∗ ⊂ G ( H, α ) . On the other hand, we have γ := ˜ x ˜ x ˜ x − ˜ x − ∈ µ d ∈ C ∗ ⊂ G ( H, α ) , since the orders of both ˜ x and ˜ x divide d . It follows that the images of ˜ x and˜ x in the quotient G ( H, α ) /µ d do commute and therefore the image of ˜ x ˜ x in G ( H, α ) /µ d has order dividing d . This means that( ˜ x ˜ x ) d ∈ µ d and therefore ( ˜ x ˜ x ) d = 1 . It follows that 1 = ˜ x d = ( γ ˜ x ˜ x ) d = γ d ( ˜ x ˜ x ) d = γ d · . This implies that γ d = 1 and therefore˜ x ˜ x = γ − ˜ x ∈ ˜∆ . It follows that ˜∆ is a subgroup. (See also [4, Sect. 4, p. 132, ex. 3].) (cid:3) Proof of Theorem 1.8
We may and will assume that L = L ( H, α ). We keep the notation and assump-tions of Section 6.By Theorem 4.14, S ( L ) = G ( H, α ). By Theorem 4.13, G ( H, α ) := ¯ κ − (ker( H ) /L )is the center of G ( H, α ).Suppose that H = 0. Then K ( L ( H, α )) = X [5, Cor. 1.9 on p. 7]; in particular,it is connected, i.e., the number of its connected components is 1. On the otherhand, by Theorem 4.13, G ( H, α ) is commutative and therefore its Jordan constantis 1. This gives us the desired result when H = 0.Suppose that H = 0. By Theorem 6.1, the Jordan constant of G ( H, α ) is r (cid:16) L ⊥ ,E /L (cid:17) . By Remark 4.2, L ⊥ ,E /L is isomorphic to the group K ( L ( H, α )) /K ( L ( H, α )) of connected component of K ( L ( H, α )). This impliesthat the Jordan constant of G ( H, α ) is p K ( L ( H, α )) /K ( L ( H, α )) ). This com-pletes the proof. 8. CP -bundles over complex tori We start with the following elementary but useful observations that allow usto handle the groups of bimeromorphic automorphisms of CP -bundles, using aninformation about the groups of biholomorphic automorphisms. Remarks 8.1.
Let L and N be holomorphic line bundles over X . Assume that L admits a nonzero holomorphic section say, s . Let n be a positive integer.(0) Clearly, L n also admits a nonzero holomorphic section s ⊗ n . (i) The holomorphic C -linear map of rank 2 holomorphic vector bundles on X N ⊕ X → ( N ⊗ L ) ⊕ X , ( t x ; x, λ ) ( t x ⊗ s ( x ); x, λ ) ∀ x ∈ X, t x ∈ N x , λ ∈ C induces a bimeromorphic isomorphism of the corresponding CP -bundles P ( N ⊕ X ) = Y N and P (( N ⊗ L ) ⊕ X ) = Y N ⊗L over X . Therefore thegroups Bim( Y N ) and Bim( Y N ⊗L ) are isomorphic.Taking into account that L n also admits a nonzero holomorphic section,we obtain that the groups Bim ( Y N ) and Bim( Y N ⊗L n ) are isomorphic forall positive integers n .(ii) It follows from (i) applied to N = X combined with Example 1.2 that forall positive integers n the CP -bundles X × CP and Y L n are bimeromorphic,and therefore the groups Bim( X × CP ) and Bim( Y L n ) are isomorphic.(iii) It follows from (i) and (ii) that for all positive integers n the group Bim ( Y N )contains a subgroup isomorphic to S ( N ⊗ L n ) and the group Bim( X × CP )contains a subgroup isomorphic to S ( L n ). We will use this observation(together with Theorem 1.5) in the proof of Theorems 1.9 and 1.10. Proof of Theorem 1.9.
Since X has positive algebraic dimension, it follows fromthe results of [3, Ch. 2, Sect. 6] that there exists a surjective holomorphic ho-momorphism ψ : X → A to a positive-dimensional complex abelian variety A .There exists a very ample holomorphic line bundle M on A such that the groupH ( A, M ) of global sections of M has C -dimension at least 2. Let us considerthe induced holomorphic line bundle ψ ∗ M on X . Since ψ is surjective, the groupH ( X, ψ ∗ M ) of global sections of ψ ∗ M also has C -dimension at least 2, becauseH ( A, M ) embeds into H ( X, φ ∗ M ). There exists an A.-H. data ( H, α ) on X such ψ ∗ M is isomorphic to L ( H, α ). This implies that L ( H, α ) has at least two linearlyindependent nonzero holomorphic sections. Now if H = 0 then L ( H, α ) = L (0 , α )and one of the following two conditions holds:(i) α = α , i.e., L ( H, α ) = L (0 , α ) is isomorphic to X andH ( X, L ( H, α )) = H ( X, L (0 , α )) = H ( X, X ) = C . (ii) α = α . It follows from [5, Th. 2.1] thatH ( X, L ( H, α )) = H ( X, L (0 , α )) = { } . Since the C -dimension of H ( X, L ( H, α )) is at least 2, neither (i) nor (ii) holds.This implies that H = 0.Let n be a positive integer. Then nH = 0 and the holomorphic line bundle L ( nH, α n ) ∼ = L ( H, α ) ⊗ n over X also admits a nonzero holomorphic section. Notice that E n := Im( nH ) = nE where E = Im( H ) . In particular, E n ( L, L ) ⊂ n Z . It follows from Corollary 6.2 applied to L ( nH, α n )that the Jordan constant of G ( nH, α n ) is at least n . By Theorem 1.7, there is agroup embedding G ( nH, α n ) ֒ → Aut ( P ( L ( nH, α n ) ⊕ X )) . By Remark 8.1, Bim( X × CP ) and Bim ( P ( L ( nH, α n ) ⊕ X )) are isomorphic. Thisimplies that for all positive integers n the group Bim( X × CP ) contains a subgroup,whose Jordan constant is at least n . It follows that Bim( X × CP ) is not Jordan. (cid:3)
OMPLEX TORI 25 Pencils of Hermitians forms
In order to prove Theorem 1.10, we need to construct families of Hermitianforms and corresponding alternating forms. We keep the notation and assumptionsof Section 3.Let H : V × V → C be an Hermitian form. Let us consider its imaginary part E : V × V → R , ( u, v ) Im( H ( u, v )) , which is an alternating R -bilinear form on V . Let us assume that E ( L, L ) ⊂ Z . Definition 9.1.
We say that H is dominated by H ifker( H ) ⊂ ker( H ) . For every positive integer n let us consider the Hermitian form H n := H + n H : V × V → C , whose imaginary part E n := Im( H n ) = E + n E : V × V → R is an alternating R -bilinear form on V . Clearly, for all nE n ( L, L ) ⊂ Z . If H is dominated by H then every H n is also dominated by H . Theorem 9.3.
Suppose that H = 0 (i.e., g > g ) and that H is dominated by H .Then for all but finitely many n (59) ker( H ) = ker( H ) and the restriction (60) E n (cid:12)(cid:12) L : L × L → Z of E n to L is a nondegenerate alternating bilinear form.Proof. Let ˜ E be the square matrix of E (cid:12)(cid:12) L of order 2 g − g with respect to thebasis { ¯ l , . . . , ¯ l g − g } of L . (Recall that ˜ E is the matrix of E (cid:12)(cid:12) L with respect tothe same basis.) Then for all n the matrix ˜ E + n ˜ E coincides with the matrix ˜ E n of E n (cid:12)(cid:12) L . with respect to { ¯ l , . . . , ¯ l g − g } . Recall that the matrix ˜ E is nondegenerateand consider the polynomial(61) f H, H ( T ) := det( ˜ E ) · det (cid:16) ˜ E − ˜ E + T (cid:17) ∈ Q [ T ] . Clearly, f H, H ( T ) is a degree 2 g − g polynomial with (positive) leading coefficientdet( ˜ E ). We havedet( ˜ E n ) = det( ˜ E + n ˜ E ) = det (cid:16) ˜ E (cid:16) ˜ E − ˜ E + n (cid:17)(cid:17) =det( ˜ E ) det (cid:16) ˜ E − ˜ E + n (cid:17) = f H, H ( n ) . In other words,(62) det( ˜ E n ) = f H, H ( n ) . Since f H, H ( T ) is not a constant, det( ˜ E n ) = 0 for all but finitely many n . Let us assume that det( ˜ E n ) = 0, which is true for all but finitely many positiveintegers n . Then E n (cid:12)(cid:12) L is nondegenerate . It follows that the restriction of E n to L ⊗ R is nondegenerate as well. On the other hand, the restriction of E n to ker( H )is identically 0. This implies that ker( E n ) = ker( H ) and thereforeker( H n ) = ker( E n ) = ker( H ) . (cid:3) Definition 9.4.
Suppose that H = 0 and n is a positive integer such that ker( H ) =ker( H ) and E n (cid:12)(cid:12) L is a nondegenerate (By Theorem 9.3, these properties hold forall but finitely many positive integers n .) Let us define L ⊥ ,E n as L ⊥ ,E n = { ˜ x ∈ L ⊗ R | E n (˜ x, l ) ∈ Z ∀ l ∈ L } . Remark 9.5.
Applying arguments of Subsection 4.1 to nondegenerate E n (cid:12)(cid:12) L (instead of E (cid:12)(cid:12) L ), we obtain that L ⊥ ,E n is a free Z -module of rank 2 g − g thatlies in L ⊗ Q and contains L as a subgroup of finite index, i.e., the quotient L ⊥ ,E n /L is a finite commutative group.It follows from (62) and the arguments of Subsection 4.1 applied to E n (insteadof E ) that(63) L ⊥ ,E n /L ) = det( ˜ E n ) = f H, H ( n ) . Since f H, H ( T ) is a polynomial of positive degree, if n tends to infinity then L ⊥ ,E n /L )also tends to infinity, i.e., q L ⊥ ,E n /L ) tends to infinity as well. Theorem 9.6.
Let H = 0 be a semi-positive Hermitian form, H a Hermitianform that is dominated by H , ( H , α ) an A.-H. data, L ( H , α ) the correspondingholomorphic line bundle on X and Y L ( H ,α ) the corresponding CP -bundle on X .Then the group Bim( Y L ( H ,α ) ) is not Jordan.Proof. Replacing H by 2 H , we may and will assume that its imaginary part E satisfies E ( L, L ) ⊂ Z . Then ( H, α ) is an A.-H. data. Since H is semi-positive, itfollows from [5, Th. 2.1 on p. 9] that the holomorphic line bundle L ( H , α ) admitsa nonzero holomorphic section. Since for all positive integers nH n = H + nH, α = α · α n , we obtain that L ( H n , α ) ∼ = L ( H , α ) ⊗ L ( H, α ) n . It follows from Remark 8.1 that the groups Bim( Y L ( H ,α ) ) and Bim( Y L ( H n ,α ) ) areisomorphic. On the other hand, by Theorem 1.7, Aut( Y L ( H n ,α ) ) contains a sub-group isomorphic to G ( H n , α ). In light of Theorems 9.3 and Theorem 6.1 (ap-plied to ( H n , α )), for all sufficiently large n the Jordan constant of G ( H n , α )) is q L ⊥ ,E n /L ). It follows from Remark 9.5 that the Jordan constant of G ( H n , α )tends to infinity when n tends to infinity. Since each G ( H n , α ) ֒ → Aut( Y L ( H n ,α ) ) ⊂ Bim( Y L ( H n ,α ) )is isomorphic to a certain subgroup of Bim( Y L ( H ,α ) ), we conclude that the Jordanconstant of Bim( Y L ( H ,α ) ) is ∞ , i.e., Bim( Y L ( H ,α ) ) is not Jordan. (cid:3)
OMPLEX TORI 27
Complex tori and abelian varieties
A complex abelian variety A of positive dimension is a complex torus W/ Γwhere W is a C -vector space of finite positive dimension and Γ ⊂ W is a discreteadditive group of maximal rank 2dim C ( W ). In addition, there exists a polarization ,i.e., a positive-definite Hermitian form H A : W × W → C such that Im( H A ( γ , γ )) ∈ Z ∀ γ , γ ∈ Γ . Proof of Theorem 1.10.
Every surjective holomorphic homomorphism ψ : X → A is induced by a certain surjective C -linear map ¯ ψ : V → W such that ψ ( L ) ⊂ Γ inthe sense that ψ ( v + L ) = ¯ ψ ( v ) + Γ ∈ W/ Γ = A ∀ v + L ∈ V /L = X. Every holomorphic line bundle M on A is isomorphic to L ( H A , β ) for a certainA.-H. data ( H A , β ) where the Hermitian form H A : W × W → C satisfies E A ( γ , γ ) := Im( H A ( γ , γ )) ∈ Z ∀ γ , γ ∈ Γand the map β : Γ → U(1) satisfies β ( γ + γ ) = ( − E A ( γ ,γ ) β ( γ ) β ( γ ) ∀ γ , γ ∈ Γ . In addition, it follows from [2, Lemma 2.3.4 on p. 33] that the induced holomorphicline bundle ψ ∗ M on X is isomorphic to L ( H , α ) where H : V × V → C , H ( v , v ) = H A ( ¯ ψ ( v ) , ¯ ψ ( v )); α : L → U(1) , α ( l ) = β ( ¯ ψ ( l )) . Clearly, ker( ¯ ψ ) ⊂ ker( H ) ⊂ V. Let us choose a polarization H A on A and consider the Hermitian form H : V × V → C , H ( v , v ) = H A ( ¯ ψ ( v ) , ¯ ψ ( v )) . Clearly, H = 0, it is semi-positive and for all l , l ∈ L Im( H ( l , l )) = Im( H A ( ¯ ψ ( l ) , ¯ ψ ( l )) ∈ Z , because ¯ ψ ( l ) , ¯ ψ ( l ) ∈ Γ. On the other hand, since H A is positive and thereforenondegenerate, ker( H ) = ker( ¯ ψ ). This implies that ker( H ) ⊂ ker( H ), i.e., H is dominated by H . It follows from Theorem 9.6 that the group Bim( Y L ( H ,α ) ) is not Jordan for every holomorphic line bundle L ( H , α ) where α : L → U(1) is any mapsuch that ( H , α ) is an A.-H. data. On the other hand, every holomorphic linebundle on X that is isomorphic to L ( H , α ) ⊗ F with F ∈ Pic ( X ) is isomorphicto L ( H , α ) for suitable α . In order to finish the proof, one has only to recall that L ( H , α ) is isomorphic to ψ ∗ M . (cid:3) Proof of Theorem 1.12.
A nonzero complex subtorus X ⊂ X and the quotient A = X/X admit the following description. There exists a nonzero C -vector subspace U ⊂ V such that L U = L T U is a lattice of rank 2dim C ( W ) in U , the quotient L/L U is a lattice of rank 2dim C ( V /U ) in the nonzero C -vector space W := V /U and X = U/L U ⊂ V /L = X, A = (
V /U ) / ( L/L U ) = W/ Γwhere Γ :=
L/L U ⊂ V /U = W. We may assume that F = L ( H , α ) for a certain A.-H. date ( H , α ) on X where H is an Hermitian form H : V × V → C , whose imaginary part E := Im( H ) : V × V → R is integer valued on L × L . The restriction of F to X lies in Pic ( X ). It followsfrom [2, Lemma 2.3.4 on p. 33] that H ( U, U ) = { } . This implies that H induces the biadditive form S : U × W (= V /U ) → C , S ( u, v + U ) = H ( u, v )such that S ( λu, w ) = λS ( u, w ) , S ( u, λw ) = ¯ λ · S ( u, w )for all u ∈ U, w ∈ W, λ ∈ C . In addition,(64) Im( S ( l, γ )) ∈ Z ∀ l ∈ L U ⊂ U, γ ∈ Γ ⊂ W. Clearly,(65) S = 0 if and only if U ⊂ ker( H ) . Let us consider the dim C ( W )-dimensional C -vector space Hom anti-lin ( W, C ) of C -antilinear maps h : W → C , h ( w + w ) = h ( w ) + h ( w ) , h ( λw ) = ¯ λ · h ( w ) ∀ w , w , w ∈ W, λ ∈ C and the latticeΓ anti-lin := { h ∈ Hom anti-lin ( W, C ) | Im( h ( γ )) ∈ Z ∀ γ ∈ Z } ⊂ Hom anti-lin ( W, C )of rank 2dim C ( W ). The form S defines the C -linear homomorphism of vector spaces a S : U → Hom anti-lin ( W, C ) , u
7→ { w S ( u, w ) } . Clearly,(66) S = 0 if and only if a S = 0 . In light of (64), a U ( L U ) ⊂ Γ anti-lin . This implies that a S induces a holomorphichomomorphism of complex tori b S : U/L U → Hom anti-lin ( W, C ) / Γ anti-lin , u + L U a S ( u ) + Γ anti-lin . Recall that
U/L U = X and W/ Γ is our complex abelian variety A . It is provenin [7, Sect. 3] that Hom anti-lin ( W, C ) / Γ anti-lin is the dual abelian variety ˆ A of A .We are given that Hom( X , A ) = { } . Since every abelian variety and its dual areisogenous, Hom( X , ˆ A ) = { } as well. It follows that b S = 0. This means that the C -vector subspace a S ( U ) lies in the lattice Γ anti-lin and therefore a S = 0. By (66), OMPLEX TORI 29 S = 0. Now it follows from (65) that U ⊂ ker( H ). This implies that there is anHermitian form H A : W × W → C on W = V /U such that(67) H A ( v + U, v + U ) = H ( v , v ) ∀ v , v ∈ V ; v + U, v + U ∈ V /U = W. We haveIm ( H A ( l + L U , l + L U )) = Im( H ( l , l )) ∈ Z ∀ l , l ∈ L ; l + L U , l + L U ∈ L/L U = Γ . By [5, Sect. 1.4, Lemma 1.6], there exists a map β : Γ → U(1) such that ( H A , β ) isan A.-H. data on A . Let L ( H A , β ) be the corresponding holomorphic line bundleon A . The inverse image ψ ∗ L ( H A , β ) on X is a holomorphic line bundle on X thatis isomorphic to some L ( H ′ , α ′ ). It follows from [2, Lemma 2.3.4 on p. 33] that theHermitian form H ′ on V and the map α ′ : L → U(1) are as follows:(68) H ′ ( v , v ) = H A ( v + U, v + U ) ∀ v , v ∈ U.α ′ ( l ) = β ( l + L U ) ∀ l ∈ L. It follows from (68) and (67) that H ′ = H . This means that ψ ∗ L ( H A , β ) is iso-morphic to L ( H , α ). Since F = L ( H, α ), it is isomorphic to ψ ∗ L ( H A , β ) ⊗ F where F = L (0 , α/α ) ∈ Pic ( X ). Now the desired result follows from Theorem1.12. (cid:3)
11. Pic and theta groups In this section we revisit theta groups that correspond to the case H = 0. Themain idea is to identify the theta group of a line bundle from Pic and the totalspace of the bundle with zero section removed. (See [13, 15] where the case ofabelian varieties was discussed.)Recall that a holomorphic line bundle L over X lies in Pic ( X ) if the correspond-ing Hermitian form H is zero, i.e., L ∼ = L (0 , α ). If this is the case then α : L → U(1) ⊂ C ∗ is a group homomorphism and L (0 , α ) is the quotient of the direct product V × C modulo the following action of L .( v, c ) ( v + l, α ( l ) c ) ∀ l ∈ L ; v ∈ V, c ∈ C . On the other hand, the C ∗ -bundle L (0 , α ) ∗ over X obtained from L (0 , α ) by remov-ing zero section may be viewed as the quotient ( V × C ∗ ) / ˜ L of the commutativecomplex Lie group V × C ∗ by its discrete subgroup ˜ L := { ( l, α ( l )) | l ∈ L } ⊂ L × C ∗ . In particular, L (0 , α ) ∗ carries the natural structure of a commutative complex Liegroup. It is included in the short exact sequence of commutative complex Lie groups1 → C ∗ → L (0 , α ) ∗ → ( V /L =) X → . Notice that the natural faithful action of V × C ∗ on V × C descends to the faithful ac-tion of L (0 , α ) ∗ on L (0 , α ), so one may view L (0 , α ) ∗ as a subgroup of Aut( L (0 , α )). Remark 11.1.
Clearly, L (0 , α ) ∗ ⊂ S ( L (0 , α )) ⊂ Aut( L (0 , α )) and for each c ∈ C ∗ (0 , c ) ˜ L ∈ L (0 , α ) ∗ ⊂ S ( L (0 , α )) ⊂ Aut( L (0 , α ))acts as multiplication by c in all fibers of L (0 , α ) → X . Theorem 11.2. L ( α, ∗ = S ( L (0 , α )) . In particular, S ( L (0 , α )) is commutative. Proof.
Let u ∈ S ( L (0 , α )). Then there is y ∈ X such that u is a lifting of T y . Choose˜ y ∈ L (0 , α ) ∗ that lifts T y as well. For example, take v ∈ V such that y = V + L and put ˜ y = ( v,
1) ˜ L ∈ ( V × C ∗ ) / ˜ L. Then u ˜ y − is an automorphism of L (0 , α ) that sends every fiber of L (0 , α ) → X into itself and acts on each such fiber as a C -linear automorphism. This meansthat there is a holomorphic function f on X that does not vanish and such that u ˜ y − acts on the fiber L (0 , α ) z as the multiplication by f ( z ) ∈ C ∗ for all z ∈ X .The compactness and connectedness of X implies that there is c ∈ C ∗ such that f ( z ) = c for all z ∈ X . It follows from Remark 11.1(ii) that u ˜ y − ∈ L (0 , α ) ∗ . Since˜ y ∈ L (0 , α ) ∗ , we have u = (cid:0) u ˜ y − (cid:1) ˜ y ∈ L (0 , α ) ∗ . This completes the proof. (cid:3) References [1] T. Bandman, Yu.G. Zarhin,
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