Realisation of abelian varieties as automorphism groups
aa r X i v : . [ m a t h . AG ] F e b Realisation of Abelian varieties as automorphism groups
Mathieu Florence Abstract.
Let A be an Abelian variety over a field F . We show that A is isomorphic tothe automorphism group scheme of a smooth projective F -variety if, and only if,Aut gp ( A ) is finite. This result was proved by Lombardo and Maffei [7] in the case F = C , and recently by Blanc and Brion [1] in the case of an algebraically closed F . Contents
1. Introduction. 21.1. Sketch of our construction. 22. Notation. 22.1. Geometry over F . 22.2. Frobenius. 32.3. Abelian varieties. 32.4. Barycentric operations. 33. Statement of the theorem. 44. Auxiliary results. 44.1. Blowups. 44.2. (Semi-)abelian varieties. 55. Proof of the implication 1) ⇒ X . 75.2. Proof that Aut ( X ) ≃ A . 96. Acknowledgments. 11Bibliography 11 Partially supported by the French National Agency (Project GeoLie ANR-15-CE40-0012). Introduction.
Let F be a field, with algebraic closure F . Let X be a projective variety over F . The automorphism group functor Aut ( X ) is represented by a group scheme,locally of finite type over F ([8], Theorem 3.7). Conversely, given a group scheme G , of finite type over F , it is natural to ask whether G can be realised as theautomorphism group of such an X . When G = A is an abelian variety, thisquestion was recently considered in [7]. When F = C , Lombardo and Maffei provethat A is the automorphism group of a projective smooth complex variety, if andonly if Aut gp ( A ) is finite. They use analytic methods. Their result was extendedto F algebraically closed of any characteristic in [1], using algebro-geometrictechniques: blowups, Lie algebra computations and modding out actions of finitegroup schemes. Making a different use of these tools, we provide a generalisationof this result, to the case of all ground fields F .1.1. Sketch of our construction.
Let
A/F be an abelian variety over a field F , such that G := Aut gp ( A ) is finite.We first introduce an integer n ≥
1, invertible in F , such that G acts faithfully onthe n -torsion subgroup A [ n ]( F ).Then, we pick an abelian variety B /F , enjoying the following properties.1) The abelian varieties A and B are ‘orthogonal’, in the sense thatHom gp ( A, B ) = 0 , where homomorphisms are taken over F .2) There exists an injection (of algebraic F -groups) ι : A [ n ] ֒ → B . We denote by B /F the abelian variety fitting into the ‘diagonal’ extension0 −→ A [ n ] t ( t,ι ( t )) −→ A × B π −→ B −→ . Using point 1) above, we prove that automorphisms of (the variety) B are diag-onal: they come from automorphisms of A × B , respecting the embedded A [ n ].Next, we build an appropriate smooth closed F -subvariety Z ⊂ B , stable bytranslations by A ≃ π ( A ) ⊂ B .We define a smooth F -variety X as the blowup X := Bl Z B . The natural arrow A −→ Aut ( B ) , given by translations, lifts to an arrow τ : A −→ Aut ( X ) . We show that τ is an isomorphism of algebraic groups over F .2. Notation.
Geometry over F . Let F be a field, with algebraic closure F , and separableclosure F s ⊂ F . We denote by F [ ǫ ], ǫ = 0, the F -algebra of dual numbers. Weuse it for differential calculus.By ‘variety over F ’, we mean ‘separated F -scheme of finite type’. An algebraic F -group (or simply F -group) is an F -group scheme of finite type. It is often assumed to be reduced, hence smooth over F .Let X be a variety over F . For a field extension E/F , we denote by X E := X × F E the E -variety obtained from X by extending scalars. We put X := X × F F .If X is smooth over F , we denote by T X −→ X the tangent bundle of X . A globalsection of the tangent bundle is called a vector field on X .We denote by Aut( X ) the (abstract) group of automorphisms of the F -variety X ,and by Aut( X ) the group of automorphisms of the F -variety X .If X/F is a projective variety, we denote by
Aut ( X ) the F -group scheme ofautomorphisms of X ; it is locally of finite type over F . By [2], Lemma 3.1, thereis a canonical isomorphism H ( X, T X ) ∼ −→ Lie(
Aut ( X )) . If an abstract group G acts on a variety X , and if Z ⊂ X is a closed subvariety, wedenote by Stab G ( Z ) ⊂ G , or simply by Stab( Z ) ⊂ G when no confusion arises, thesubgroup of transformations leaving Z (globally) invariant. Let G/F be a groupscheme, which is locally of finite type. In the situation where G acts on X , we usethe notation Stab G ( Z ) ⊂ G for the closed F -subgroup scheme defined by Stab G ( Z )( A ) = { g ∈ G ( A ) , g ( Z A ) = Z A } , for all commutative F -algebras A . That it is representable follows from [3], II 1.3.6.2.2. Frobenius. If F has characteristic p >
0, we put X (1) := X × Frob F -extension of scalars taken with respect to Frob : F x x p −→ F . Recall the Frobeniushomomorphism Frob X : X −→ X (1) ; it is a morphism of F -varieties, functorial in X . If X/F is an algebraic group, it is a group homomorphism.2.3.
Abelian varieties. If A and B are Abelian varieties over F , we denote byHom gp ( A, B ) the group of homomorphisms of algebraic F -groups, from A to B .We denote by Hom gp ( A, B ) the group of homomorphisms of algebraic F -groups,from A to B . These are finite free Z -modules. We adopt the similar notation forendomorphisms (End gp ) and automorphisms (Aut gp ). For an integer n ≥
1, wedenote by A [ n ] the n -torsion of A , seen as a finite group scheme over F .2.4. Barycentric operations.
Let A be an abelian variety over F . Then A comes naturally equipped with barycentric operations with integer coefficients.More precisely, for a positive integer n , denote by Z n ⊂ Z n the subset consistingof relative integers α = ( α , . . . , α n ), with α + . . . + α n = 1. For α ∈ Z n , there isa barycentric operation B α : A n −→ A, ( x , . . . , x n ) α x + . . . + α n x n . The B α ’s, for various α ′ s , satisfy the usual associativity relations. Note that thisconstruction applies more generally to torsors under commutative algebraic F -groups.An F -subvariety X ⊂ A which is stable under all barycentric operations will becalled barycentric . Note that X is barycentric if and only if it is a translate ofan algebraic F -subgroup −→ X ⊂ A . Checking this fact is left as an exercise forthe reader. Of course, X ( F ) might be empty. If X is geometrically reduced and geometrically connected, so is −→ X - hence −→ X is an abelian subvariety of A .Let A and B be two abelian varieties over F . Recall the essential factHom F − var ( A, B ) = B ( F ) × Hom gp ( A, B ) . In particular, morphisms (of varieties) between abelian varieties commute to thebarycentric operations B α .If X ⊂ A is a geometrically reduced closed F -subvariety, the smallest geometricallyreduced barycentric F -subvariety containing X is called the barycentric envelopeof X . We denote it by E ( X ).Assume now that X is geometrically reduced and geometrically connected. Pick n ≥ α ∈ Z n . Consider B α ( X n ) ⊂ A as a geometrically reduced andgeometrically connected closed subvariety of A . Then, if n and α are chosen so that B α ( X n ) is of maximal dimension, we have B α ( X n ) = E ( X ). Thus, E ( X ), beinggeometrically connected and geometrically reduced, is a translate of an abeliansubvariety of A . 3. Statement of the theorem.Theorem . Let A be an Abelian variety, over a field F . The following areequivalent:1) The group G := Aut gp ( A ) is finite.2) There exists a smooth projective F -variety X , such that A is isomorphic to Aut ( X ) (as algebraic groups over F ). Note that 2) ⇒
1) can be checked over F , which follows from [1], Theorem A.Our task in this paper is to prove the converse implication.4. Auxiliary results.
Blowups.
This section contains two elementary lemmas on automorphismsof blowups, which we provide with short proofs. A good recent reference on thistopic, also containing more advanced material, is section 2 of [9].
Lemma . Let
Y ֒ → D be a closed immersion of smooth F -varieties, such thatall connected components of Y have codimension ≥ in D .Denote by β : X := Bl Y ( D ) −→ D the blowup of Y inside D .The F -variety X is smooth.Let f be an automorphism of the F -variety D . Then, f lifts via β to an automor-phism of X , if and only if f ( Y ) = Y . Proof. If f ( Y ) = Y , then f lifts to an automorphism of X by the universalproperty of the blowup.Conversely, assume that f lifts to an automorphism φ of X , so that we have acommutative square X φ / / (cid:15) (cid:15) X (cid:15) (cid:15) D f / / D. To check that f ( Y ) = Y , can assume that F = F . It then suffices to prove that Y ⊂ D and f ( Y ) ⊂ D have the same set of F -rational points. This is clear, since the fiber of β over a point x ∈ D ( F ) is either a point if s / ∈ Y ( F ), or a projectivespace of dimension ≥ x ∈ Y ( F ). (cid:3) Lemma 4.1 has an infinitesimal analogue, as follows.
Lemma . Let
Y ֒ → D be a closed immersion of smooth F -varieties, such thatall connected components of Y have codimension ≥ in D .Denote by β : X := Bl Y ( D ) −→ D the blowup of Y inside D .Let s : D −→ T D be a vector field on D . Then, s lifts to a vector field on X , ifand only if s | Y takes values in T Y . Proof.
Denote by i : E ֒ → X the exceptional divisor. The restriction β | X − E : E − X −→ D − Y is an isomorphism.We thus have a natural injective F -linear arrow ρ : H ( X, T X ) −→ H ( D − Y, T D ) = H ( D, T D ) ,σ σ | X − E . Note that the equality H ( D − Y, T D ) = H ( D, T D ) follows from the fact that Y ⊂ D has codimension ≥
2. On E , we have a natural extension of vector bundles0 −→ T E −→ i ∗ ( T X ) −→ N E/X −→ , where N E/X ≃ O E ( −
1) is the normal bundle of E in X . Since Y has codimension ≥ D , we have H ( E, O E ( − β over geometric points of Y , which are projective spaces of dimension ≥
1. Hence, σ | E takes values in T E . Consequently, ρ ( σ ) | Y takes values in T Y .Conversely, let s : D −→ T D be a vector field on D . Then s corresponds toan automorphism ψ of the F [ ǫ ]-scheme D × F F [ ǫ ], reducing to the identity at ǫ = 0. Assume that s | Y takes values in T Y . Then, ψ restricts to an automorphismof the closed subscheme Z × F F [ ǫ ] ⊂ D × F F [ ǫ ]. By the universal property(and compatibility with base change) of the blowup, ψ lifts, via β × F F [ ǫ ], to anautomorphism of X × F F [ ǫ ]. Equivalenty, s lifts, via β , to a vector field on X . (cid:3) (Semi-)abelian varieties. The next Lemma is borrowed from [2], Lemma5.3. We provide here a different proof. In practice, we will apply it to abelianvarieties, in which case it is due to Chow.
Lemma . Assume that F has characteristic p > .Let A, B be semi-abelian varieties over F . Then, all elements of Hom gp ( A, B ) aredefined over the separable closure F s ⊂ F . Proof.
We have to show the following. Let
E/F be a purely inseparable algebraicextension. Let g : A E −→ B E be a homomorphism of algebraic groups over E .Then g is defined over F . Without loss of generality, we can assume that E/F isfinite. By induction, we reduce to the case where E = F ( p √ a ) /F is a primitivepurely inseparable extension of height one. Note that Frob : E −→ E takes valuesin F . Hence, g (1) : A (1) E −→ B (1) E is defined over F . The Frobenius homomor-phism A −→ A (1) presents A (1) as a quotient of A , by a finite (characteristic)sub-group scheme µ A ⊂ A [ p ]; similarly for B . Modding out futher, we get thatthe E -morphism A/A [ p ] −→ B/B [ p ], induced by g , is defined over F . Via the identification A/A [ p ] ∼ −→ A given by ‘multiplication by p ’, this isomorphism isactually g itself. The Lemma is proved. (cid:3) Lemma . Denote by S ( F s , the set of isogeny classes of (absolutely) simpleabelian surfaces over F s . Then S ( F s , is infinite. Proof.
Since F s is separably closed, ‘simple=absolutely simple’ for abelianvarieties over F s (use Lemma 4.3). Without loss of generality, we assume that F s is the algebraic closure of its prime subfield. Over Q , we can then use theexistence of abelian surfaces with a prescribed CM type. Over F p , we can useHonda-Tate theory. For concrete constructions, and more general results, we referto [9], Theorem 1 (where F s = Q ), and [6], Theorem 13 (where F s = F p ). (cid:3) Lemma . Let B be an abelian variety over F , whose simple factors (over F ) areof dimensions ≥ . (Equivalently: all F -homomorphisms from an elliptic curve to B are constant.)Then, there exists a smooth F -subvariety Y ⊂ B , which is a disjoint union ofsmooth F -curves, and of a separable closed point, such that Stab ( Y ) = { Id } ⊂ Aut ( B ) . Proof.
Assume first that B is F -simple, in the sense that it has no non-trivialproper abelian F -subvariety. Let C ⊂ B be any geometrically irreducible smooth F -curve, which exists by Bertini’s theorem (in the delicate case where F is finite,this follows from [10] and [4], Theorems 1.1). Then, C has genus g ≥
2. Indeed,using our assumption on B , every F -morphism from an elliptic curve to B isconstant. Hence, the group Aut( C ) is finite. We even have a stronger statement:the F -group scheme Aut ( C ) is finite ´etale over F . Indeed, Lie( Aut ( C )) is thespace of vector fields on C , which vanishes since g ≥ E ( C ) = B . The barycentric envelope E ( C ) is a translate of anabelian subvariety B ′ ⊂ B . Since B is F -simple, we get B ′ = B , hence E ( C ) = B .Now, let g ∈ Aut ( B )( F [ ǫ ]) = B ( F [ ǫ ]) × Aut gp ( B ) be such that g | C × F F [ ǫ ] = Id | C × F F [ ǫ ] . Because g commutes to barycentric operations, g acts as the identity on the closedsubscheme E ( C ) × F F [ ǫ ] ⊂ B × F F [ ǫ ]. Since E ( C ) = B , it follows that g = Id.Thus, we get a natural embedding of F -group schemes H := Stab
Aut ( B ) ( C ) ֒ → Aut ( C ) . In particular, H is finite ´etale over F . Let E/F be a finite separable field extension,such that H ( E ) = H ( F ). Denote byΦ := [ h ∈ H ( E ) ,h = e B h ⊂ B be the (strict) closed subscheme, consisting of points fixed by at least one non-trivial element h ∈ H ( E ). It is defined over F by Galois descent. There exists afinite separable field extension L/E , and a point b = 0 ∈ B ( L ), which does not liein Φ( L ), nor in C ( L ). We then have a separable zero-cycle [ b ] in the F -variety B ,of degree [ L : F ]. Define Y ⊂ B as the disjoint union of [ b ] and C . We claim that Y has the required property. Indeed, let f ∈ Aut ( B )( F [ ǫ ]) be an automorphismstabilizing Y - or more accurately, Y × F F [ ǫ ] ⊂ B × F F [ ǫ ]. Then, f permutes the two connected components of the scheme Y × F F [ ǫ ]. For dimension reasons, itpreserves C × F F [ ǫ ] on the one hand, and [ b ] × F F [ ǫ ] on the other hand. Fromthe first fact, we know that f belongs to H ( F ); in particular, it is defined over E ,hence over L . From the latter fact, we get f ( b ) = b , hence f = Id. The Lemma isproved in this case.Assume now that B = B × . . . B n , with the B i ’s are F -simple abelian varieties.We can then adapt the preceding proof, as follows. For each i , let C i ⊂ B i , L i /E i /F and b i ∈ B ( L i ) be as in the first part of the proof. We can fulfill theextra requirements that no C i passes through 0, and that no C i is F -isomorphicto a C j , when i = j . We can also assume that L i = L and E i = E are independentof i . Set b := ( b , . . . , b n ) ∈ B ( L ) . Define Y to be the disjoint union of [ b ], and of the n curves C i ≃ { } × . . . × { } × C i × { } × . . . × { } ֒ → B. Verifying that Y enjoys the required property is left to the reader.In general, write B = ( Q r B j ) /µ , where S , . . . , S r are F -simple abelian varieties,and where µ is a finite F -subgroup, intersecting trivially each coordinate axis. Wecan choose Y ֒ → B × . . . B n as in the previous part of the proof, and such that the composite Y ֒ → B × . . . B n can −→ ( r Y B j ) /µ = B is a closed immersion, identifying Y to a smooth closed subvariety of B .An automorphism of B stabilizing Y ⊂ B then lifts, via the quotient can , to anautomorphism of B × . . . B n stabilizing Y ⊂ B × . . . B n . We conclude as before.Details are left to the reader. (cid:3) Proof of the implication ⇒ . Let
A/F be an abelian variety, such that G := Aut( A ) is finite. We give aconstruction of a smooth projective F -variety X , such that A = Aut ( X ), inseveral steps.5.1. Construction of X . Denote by g the dimension of A .Let n ≥ F , such that the action of G on A [ n ]( F s ) ≃ ( Z /n ) g is faithful. Such an n exists: use that G is finite, andthat torsion points of order prime to char( F ) in A ( F ) are Zariski-dense in A .Let B s be a simple abelian surface over F s , such that Hom gp ( A, B ) =
Hom gp ( B, A ) = 0. Since A has a finite number of simple components (upto isogeny), which are all defined over F s by Lemma 4.3, the existence of B s follows from Lemma 4.4.Let E/F , be the finite Galois extension, with group Γ, which is minimal w.r.t.the following properties.1) The extension
E/F splits the F -group of multiplicative type A [ n ].In other words, A [ n ]( E ) ≃ ( Z /n ) g .2) The abelian variety B s is defined over E : there exists an abelian E -variety B E , such that B E × E F s ≃ B s .3) Same as 1), for B E : we have B [ n ]( E ) ≃ ( Z /n ) .Using 1), we view A [ n ]( E ) as a ( Z /n )[Γ]-module.Introduce the Weil restriction of scalars B := R E/F ( B gE ) . Geometrically, we have B ≃ B mgs , where m is the cardinality of Γ.We have B [ n ] = R E/F (( Z /n ) g ) , so that E/F splits B [ n ], and B [ n ]( F ) is a free ( Z /n )[Γ]-module of rank 4 g . Lemma . There exists an embedding of ( Z /n Z )[Γ] -modules A [ n ]( E ) ֒ → B [ n ]( E ); that is to say, an embedding of finite ´etale F -group schemes ι : A [ n ] ֒ → B [ n ] . Proof.
Recall the perfect duality ( . ) ∨ := Hom( ., Z /n ), in the category of ( Z /n )[Γ]-modules. Pick a generating set t , . . . , t g of the Z /n -module A [ n ]( E ) ∨ - which isfree of rank 2 g . Introduce the surjection of ( Z /n )[Γ]-modules( Z /n )[Γ] g −→ A [ n ]( E ) ∨ ,e i t i , where e i denotes the i -th element of the canonical basis. Dualizing it yields aninjection of ( Z /n )[Γ]-modules ι : A [ n ]( E ) −→ ( Z /n )[Γ] g ≃ B [ n ] , concluding the construction. (cid:3) Form the exact sequence of algebraic F -groups0 −→ A [ n ] a ( a,ι ( a )) −→ A × B π −→ B −→ . Its cokernel B is an abelian variety over F .We have F -embeddings A a ( a, ֒ → B and B b (0 ,b ) ֒ → B . Introduce the quotient q : B −→ B := B /A ≃ B /ι ( A [ n ]) . Let Y ⊂ B be a smooth F -subvariety, enjoying the properties of Lemma 4.5,where we take B to be our B , and set Y := Y .Put Y := q − ( Y ) . The restriction q | Y : Y −→ Y is an A -torsor.We now define X := Bl Y ( B )to be the blowup of Y in B .5.2. Proof that
Aut ( X ) ≃ A . Translating by elements of A inside B yields a natural arrow A −→ Aut ( B ) . Since Y ⊂ B is stable by these translations, we get an induced arrow of F -groupschemes τ : A −→ Aut ( X ) . It is clear that τ is an embedding. I claim that it is an isomorphism.Let us first check that it induces a bijection A ( F ) ∼ −→ Aut( X ) = Aut ( X )( F ) . Pick φ ∈ Aut( X ). It induces a birational isomorphism f of the F -variety B ,which is a regular isomorphism since B is an abelian variety. Thus, we get acommutative diagram X φ / / (cid:15) (cid:15) X (cid:15) (cid:15) B f / / B , where the vertical arrows are the structure morphism of the blowup.Using Lemma 4.1, we get f ( Y ) = Y . We know that f ( x ) = g ( x ) + t , where g ∈ Aut gp ( B ), and t ∈ B ( F ). We have to show that g = Id and t ∈ A ( F ). To do so, we can assume without loss of generality that t ∈ B ( F ).We then have to prove g = Id and t ∈ A ( F ) ∩ B ( F ) = ι ( A [ n ])( F ) . Geometrically, B ≃ B mgs . Since Hom gp ( A, B s ) = Hom gp ( B s , A ) = 0, we get Hom gp ( A, B ) = Hom gp ( B , A ) = 0 . Therefore g leaves A ⊂ B and B ⊂ B stable.We infer that g lifts, via π , to a diagonal group automorphism δ = ( h, g )of A × B , which automatically leaves the diagonally embedded A [ n ] stable.Consider the automorphism of B given by f ( b ) := g ( b ) + t , and the diagonal automorphism of A × B given by∆( a, b ) := ( h ( a ) , f ( b )) . Since δ leaves A × ι ( A [ n ]) ⊂ A × B stable, there exists f ∈ Aut( B ) such thatthe diagram A × B / / π (cid:15) (cid:15) A × B π (cid:15) (cid:15) B f / / q (cid:15) (cid:15) B q (cid:15) (cid:15) B f / / B commutes.Because f ( Y ) = Y , we get f ( Y ) = Y . By Lemma 4.5, we conclude that f = Id. Hence, we have t ∈ ι ( A [ n ])( F ) and g = Id. Since δ preserves thediagonally embedded A [ n ], we get that h , restricted to A [ n ] ⊂ A , is the identity.Since G acts faithfully on A [ n ], we conclude that h = Id. Hence, g = Id as well,and our job is done.We have proved that τ induces a bijection on F -points. If F has characteristiczero, this is enough to conclude that τ is an isomorphism of algebraic F -groups.In general, it remains to check that the F -linear map on tangent spaces d e ( τ ) : Lie( A ) −→ Lie(
Aut ( X ))is bijective. Recall that Lie( Aut ( X ) is the space of vector fields on X ; that is,global section of the tangent bundle T X −→ X . Let s : X −→ T X be such a section. Restricting s to the complement of the exceptional divisor, weget a global section σ ′ of the tangent bundle of B − Y . Since B is an abelianvariety, its tangent bundle is trivial, so that σ ′ is given by an arrow of F -varieties σ ′ : B − Y −→ A (Lie( B )) , with target an affine space of dimension dim( B ). Since Z has codimension ≥ B , σ ′ extends to a morphism σ : B −→ A (Lie( B )) , which is constant because B /F is proper. Write σ = t , with t ∈ Lie( B ). Toconclude, we have to show t ∈ Lie( A ).For y ∈ B ( F ), denote by α y : B −→ B the F -morphism given by x x + y. Recall that the linear isomorphisms d y α − y : T y ( B ) ∼ −→ Lie( B ) ⊗ F F are used to trivialize the tangent bundle of B .Since σ lifts to a section of the tangent bundle of the blowup Bl Y ( B ), Lemma4.2 implies, when y ∈ Y ( F ), that t belongs to d y α − y ( T y ( Y )) ⊂ Lie( B ) ⊗ F F .
Taking a y lying above (via q ) an isolated separable point of Y , we conclude that t ∈ Lie( A ), as desired. Acknowledgments.
We thank Michel Brion for his careful reading, and for helpful suggestions con-cerning blowups. We thank Frans Oort for helping us understand why Lemma 4.4is true, and Daniel Bertrand for pointing out an existing reference for its proof.
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