Abelian varieties as automorphism groups of smooth projective varieties in arbitrary characteristics
aa r X i v : . [ m a t h . AG ] F e b ABELIAN VARIETIES AS AUTOMORPHISM GROUPS OFSMOOTH PROJECTIVE VARIETIES IN ARBITRARYCHARACTERISTICS
J´ER´EMY BLANC AND MICHEL BRION
Abstract.
Let A be an abelian variety over an algebraically closed field.We show that A is the automorphism group scheme of some smooth pro-jective variety if and only if A has only finitely many automorphisms asan algebraic group. This generalizes a result of Lombardo and Maffei forcomplex abelian varieties. Contents
1. Introduction 12. Preliminaries and notation 23. Proof of Theorem A(i) 34. Proof of Theorem A(ii): first steps 45. Proof of Theorem A(ii): the construction of Y P ) r Y Introduction
Let X be a projective algebraic variety over an algebraically closed field.The automorphism group functor of X is represented by a group scheme Aut X ,locally of finite type (see [Gro61, p. 268] or [MO67, Thm. 3.7]). Thus, theautomorphism group Aut( X ) is the group of k -rational points of a smoothgroup scheme that we will still denote by Aut( X ) for simplicity. One mayask which smooth group schemes are obtained in this way, possibly imposingsome additional conditions on X such as smoothness or normality. It is knownthat every finite group G is the automorphism group scheme of some smooth Mathematics Subject Classification.
Key words and phrases.
Abelian varieties, automorphism group schemes, Albanesemorphism.The first author is supported by the Swiss National Science Foundation Grant “Birationaltransformations of threefolds” 200020 178807. projective curve X (see e.g. the main result of [MR92]). The case of a complexabelian variety A was treated recently by Lombardo and Maffei in [LM20]; theyshowed that A = Aut( X ) for some complex projective manifold X if and onlyif A has only finitely many automorphisms as an algebraic group. In this note,we generalize their result as follows: Theorem A.
Let A be an abelian variety over an algebraically closed field.Denote by Aut gp ( A ) the group of automorphisms of A as an algebraic group.(i) If A = Aut( X ) for some projective variety X , then Aut gp ( A ) is finite.(ii) If Aut gp ( A ) is finite, then there exists a smooth projective variety X such that A = Aut X .Like in [LM20], the proof of the first assertion is easy, and the second oneis obtained by constructing X as a quotient ( A × Y ) /G , where G ⊂ A is afinite subgroup, Y is a smooth projective variety such that G = Aut Y , andthe quotient is taken for the diagonal action of G on A × Y . In [LM20], G isa cyclic group of prime order ℓ , and Y a surface of degree ℓ in P equippedwith a free action of G . As the construction of Y does not extend readily toprime characteristics, we take for G the n -torsion subgroup scheme A [ n ] foran appropriate integer n , and for Y an appropriate rational variety.A different construction of a variety X satisfying the second assertion hasbeen obtained independently by Mathieu Florence, see [Flo21]; it works overan arbitrary field.Let us briefly describe the structure of this note. Section 2 is a short intro-duction to basic notation and reminders on abelian varieties. In Section 3, wetake an abelian variety A with Aut gp ( A ) infinite, assume that A = Aut( X ) forsome projective variety X , and derive a contradiction. In Section 4, we takean abelian variety A with Aut gp ( A ) finite and prove that for each prime num-ber ℓ different from the characteristic of the ground field, for each m ≥ Y with Aut Y ≃ A [ ℓ m ],one has Aut X = A where X is the smooth projective variety ( A × Y ) /A [ ℓ m ]. Then, Section 5 isdevoted to an explicit construction of Y .2. Preliminaries and notation
We begin by fixing some notation and conventions which will be usedthroughout this note. The ground field k is algebraically closed, of char-acteristic p ≥
0. A variety X is a separated integral scheme of finite typeover k . By a point of X , we mean a k -rational point.We use [Mum08] as a general reference for abelian varieties. We denote by A such a variety of dimension g ≥
1, with group law + and neutral element 0.
BELIAN VARIETIES AS AUTOMORPHISM GROUPS 3
Then Aut( A ) = A ⋊ Aut gp ( A ) , where A acts on itself by translations. Moreover, Aut gp ( A ) = Aut( A,
0) (thegroup of automorphisms fixing the neutral element), see [Mum08, §
4, Cor. 1].For any positive integer n , we denote by A [ n ] the n -torsion subgroup schemeof A , i.e., the schematic kernel of the multiplication map n A : A −→ A, a na.
Clearly, A [ n ] is stable by Aut gp ( A ). Also, recall from [Mum08, S 6, Prop.]that A [ n ] is finite; moreover, A [ n ] is the constant group scheme ( Z /n ) g if n is prime to p .We denote by q : A −→ A/A [ n ] , a ¯ a the quotient morphism. Then n A factors as q followed by an isomorphism A/A [ n ] ≃ −→ A . 3. Proof of Theorem A (i)In this section, we choose an abelian variety A such that Aut gp ( A ) is infinite,and proceed to the proof of Theorem A(i). We will need: Lemma 3.1.
For any positive integer n , the kernel of the restriction map ρ n : Aut gp ( A ) −→ Aut gp ( A [ n ]) is infinite.Proof. Note that ρ n extends to a ring homomorphism σ n : End gp ( A ) −→ End gp ( A [ n ])with an obvious notation. Moreover, the image of σ n is a finitely generatedabelian group (as a quotient of End gp ( A )) and is killed by n ; thus, this imageis finite. So the image of ρ n is finite as well. (cid:3) We assume, for contradiction, the existence of a projective variety X suchthat A = Aut( X ); in particular, X is equipped with a faithful action of A .By [Bri10, Lem. 2.2], there exist a finite subgroup scheme G of A and an A -equivariant morphism f : X → A/G , where A acts on A/G via the quotientmap. Denote by n the order of G ; then G is a subgroup scheme of A [ n ]. Bycomposing f with the natural map A/G → A/A [ n ], we may thus assume that G = A [ n ].We now adapt the proof of [LM20, Thm. 2.2]. Let Y be the schematic fiberof f at ¯0. Then Y is a closed subscheme of X , stable by the action of A [ n ]. J´ER´EMY BLANC AND MICHEL BRION
Form the cartesian square X ′ f ′ / / r (cid:15) (cid:15) A q (cid:15) (cid:15) X f / / A/A [ n ] . Then X ′ is a projective scheme equipped with an action of A ; moreover, f ′ is an A -equivariant morphism and its fiber at 0 may be identified to Y . Itfollows that the morphism A × Y −→ X ′ , ( a, y ) a · y is an isomorphism with inverse X ′ −→ A × Y, x ′ ( f ′ ( x ′ ) , − f ′ ( x ′ ) · x ′ ) . So we may identify X ′ with A × Y ; then r is invariant under the action of A [ n ]via g · ( a, y ) = ( a − g, g · y ). Since q is an A [ n ]-torsor, so is r . In particular, X = ( A × Y ) /A [ n ] and the stabilizer in A of any y ∈ Y is a subgroup schemeof A [ n ].By Lemma 3.1, we may choose a nontrivial v ∈ Aut gp ( A ) which restrictsto the identity on A [ n ]. Then v × id is an automorphism of A × Y thatcommutes with the action of A [ n ]. Since r is an A [ n ]-torsor and hence acategorical quotient, it follows that v × id ∈ Aut( A × Y ) factors through aunique u ∈ Aut( X ), which satisfies u ( a · y ) = v ( a ) · y for all a ∈ A and y ∈ Y .As Aut( X ) = A , we have u ∈ A . For any a, b ∈ A and y ∈ Y , we have b · ( a · y ) = ( a + b ) · y . Choosing b = u , v ( a ) − a − u fixes every point of Y for any a ∈ A . Taking a = 0, it follows that u fixes Y pointwise, and hence u ∈ A [ n ]. So v ( a ) − a ∈ A [ n ] for any a ∈ A , i.e., v − id factors through ahomomorphism A → A [ n ].Since A is smooth and connected, it follows that v − id = 0, a contradiction.4. Proof of Theorem A (ii) : first steps
We assume from now on that Aut gp ( A ) is finite. Recall that q : A → A/A [ n ]is the quotient morphism (see Section 2). Lemma 4.1. (i)
The map q ∗ : Aut gp ( A ) → Aut gp ( A/A [ n ]) is an isomorphism for anyinteger n ≥ . (ii) Let ℓ = p be a prime number. Then ρ ℓ m : Aut gp ( A ) → Aut gp ( A [ ℓ m ]) isinjective for m ≫ .Proof. (i) Since Aut gp ( A/A [ n ]) ≃ Aut gp ( A ) is finite, it suffices to show that q ∗ is injective. Let u ∈ Aut gp ( A ) such that q ∗ ( u ) = id. Then u ( a ) − a ∈ A [ n ] forany a ∈ A , that is, u − id factors through a homomorphism A → A [ n ]. As in BELIAN VARIETIES AS AUTOMORPHISM GROUPS 5 the very end of the proof of Theorem A(i) the smoothness and connectednessof A yield u = id.(ii) Let T ℓ ( A ) = lim ← A [ ℓ m ]; then T ℓ ( A ) is a Z ℓ -module and the naturalmap Aut gp ( A ) → Aut Z ℓ ( T ℓ ( A )) is injective (see [Mum08, §
19, Thm. 3]). Thus, T m ≥ Ker( ρ ℓ m ) = { id } . Since the Ker( ρ ℓ m ) form a decreasing sequence, we getKer( ρ ℓ m ) = { id } for m ≫ (cid:3) Next, consider a smooth projective variety Y equipped with an action ofthe finite group G = A [ n ], for some integer n prime to p . Then G acts freelyon A × Y via g · ( a, y ) = ( a − g, g · y ). The quotient X = ( A × Y ) /G existsand is a smooth projective variety (see [Mum08, §
7, Thm.]). The A -actionon A × Y via translation on itself yields an action on X . The projectionpr A : A × Y → A yields a morphism f : X −→ A/G which is A -equivariant, where A acts on A/G via the quotient map q . More-over, f is smooth and its schematic fiber at ¯0 is G -equivariantly isomorphicto Y . Lemma 4.2.
Assume that Y is rational. (i) The map f is the Albanese morphism of X . (ii) The neutral component
Aut ( Y ) is a linear algebraic group.Proof. (i) Let B be an abelian variety, and u : X → B a morphism. Composing u with the quotient morphism A × Y → X yields a G -invariant morphism v : A × Y → B . As Y is rational, v factors through a morphism A → B , whichmust be G -invariant. So u factors through a morphism A/G → B .(ii) By a theorem of Nishi and Matsumura (see [Bri10] for a modern proof),there exist a closed affine subgroup scheme H ⊂ Aut ( Y ) such that the homo-geneous space Aut ( Y ) /H is an abelian variety, and an Aut ( Y )-equivariantmorphism u : Y → Aut ( Y ) /H . As Y is rational, this forces H = Aut ( Y ). (cid:3) As a consequence of Lemma 4.2, if Y is rational then f induces a homo-morphism f ∗ : Aut( X ) −→ Aut(
A/G ) , and hence an exact sequence1 −→ Aut
A/G ( X ) −→ Aut( X ) f ∗ −→ A/G ⋊ Aut gp ( A/G ) , where Aut A/G ( X ) denotes the group of relative automorphisms. The A -actionon X yields a homomorphism G → Aut
A/G ( X ). Moreover, the image of f ∗ contains the group A/G of translations, and hence equals
A/G ⋊ Γ, where Γdenotes the subgroup of Aut gp ( A/G ) consisting of automorphisms which liftto X . J´ER´EMY BLANC AND MICHEL BRION
Lemma 4.3.
Let G = A [ ℓ m ] , where ℓ, m satisfy the assumptions of Lemma . .Let Y be a smooth projective rational G -variety such that Aut( Y ) = G . (i) The map G → Aut
A/G ( X ) is an isomorphism. (ii) The group Γ is trivial.Proof. (i) Let u ∈ Aut
A/G ( X ). Then u restricts to an automorphism of Y (the fiber of f at 0), and hence to a unique g ∈ G . Replacing u with g − u ,we may assume that u fixes Y pointwise. For any a ∈ A and y ∈ Y , we have f ( u (( a, y )) = f (( a, y ) = ¯ a , where ( a, y ) denotes the image of ( a, y ) in X . As f is A -equivariant, it follows that ( − a ) · u (( a, y )) ∈ Y . This defines a morphism F : A × Y −→ Y, ( a, y ) ( − a ) · u (( a, y ))such that F (0 , y ) = u ( y ) = y for all y ∈ Y . As A is connected, this definesin turn a morphism (of varieties) A → Aut ( Y ), which must be constant byLemma 4.2(ii). So u (( a, y )) = a · y = ( a, y ) identically, i.e., u = id.(ii) Let γ ∈ Γ; then there exists u ∈ Aut( X ) such that f ∗ ( u ) = γ . Since γ (¯0) = ¯0, we see that u stabilizes Y ; thus, u | Y = g for a unique g ∈ G . Also,there exists v ∈ Aut gp ( A ) such that q ∗ ( v ) = γ (Lemma 4.1(i)). Thus, we have f ( u ( a, y )) = γf ((( a, y )) = v ( a ), i.e., ( − v ( a )) · u (( a, y )) ∈ Y for all a ∈ A and y ∈ Y . Arguing as in the proof of (i), it follows that u (( a, y )) = v ( a ) · g ( y )identically. In particular, g ( a · y ) = v ( a ) · g ( y ) for all a ∈ G and y ∈ Y .Since G is commutative, we obtain v ( a ) = a for all a ∈ G . Thus, v = id byLemma 4.1(ii). So γ = id as well. (cid:3) Proposition 4.4.
Under the assumptions of Lemma . , the A -action on X yields an isomorphism A → Aut( X ) . If in addition G = Aut Y , then A → Aut X is an isomorphism as well.Proof. We have a commutative diagram of exact sequences0 / / G / / (cid:15) (cid:15) A / / (cid:15) (cid:15) A/G / / (cid:15) (cid:15) / / Aut
A/G ( X ) / / Aut( X ) f ∗ / / Aut(
A/G ) . By Lemma 4.3, the left vertical map is an isomorphism and the image of f ∗ isthe group A/G of translations. This yields the first assertion.To show the second assertion, it suffices to show that the induced homomor-phism of Lie algebras Lie( A ) → Lie(Aut X ) is an isomorphism when G = Aut Y .Recall that Lie(Aut X ) is the space of global sections of the tangent bundle T X BELIAN VARIETIES AS AUTOMORPHISM GROUPS 7 (see e.g. [MO67, Lem. 3.4]). Moreover, as f is smooth, we have an exactsequence 0 −→ T f −→ T X df −→ f ∗ ( T A/G ) −→ , where T f denotes the relative tangent bundle. Since T A/G is the trivial bundlewith fiber Lie(
A/G ), this yields an exact sequence0 −→ H ( X, T f ) −→ H ( X, T X ) −→ Lie(
A/G )such that the composition Lie( A ) → H ( X, T X ) → Lie(
A/G ) is Lie( q ). So itsuffices in turn to show that H ( X, T f ) = 0.We have a cartesian diagram A × Y pr A / / (cid:15) (cid:15) A (cid:15) (cid:15) X f / / A/G, where the vertical arrows are G -torsors. This yields an isomorphism H ( X, T f ) ≃ H ( A × Y, T pr A ) G and hence H ( X, T f ) ≃ H ( A × Y, pr ∗ Y ( T Y )) G ≃ ( O ( A ) ⊗ H ( Y, T Y )) G ≃ H ( Y, T Y ) G . As G = Aut Y , we have H ( Y, T Y ) = 0; this completes the proof. (cid:3) Proof of Theorem A (ii) : the construction of Y In this section, we fix integers n, r ≥
2, where p does not divide n , andconstruct a smooth projective rational variety Y of dimension r such thatAut Y = ( Z /n ) r .We define G = { ( µ , . . . , µ r ) ∈ k r | µ ni = 1 for each i ∈ { , . . . , r }} ≃ ( Z /n ) r and let G act on ( P ) r by G × ( P ) r → ( P ) r (( µ , . . . , µ r ) , ) ([ u : µ v ] , . . . , [ u r : µ r v r ])For each i ∈ { , . . . , r } , we denote by ℓ i ⊂ ( P ) r the closed curve isomorphicto P given by the image of P → ( P ) r ([ u : v ]) ([0 : 1] , . . . , [0 : 1] , [ u : v ] , [0 : 1] , . . . , [0 : 1])where the [ u : v ] is at the place i . The curves ℓ , . . . , ℓ r ⊂ ( P ) r generate thecone of curves of ( P ) r .For each i ∈ { , . . . , r } , the curve ℓ i is stable by G and the action of G on ℓ i corresponds to a cyclic action of order n on P , given by [ u : v ] [ µu : v ], J´ER´EMY BLANC AND MICHEL BRION where µ ∈ k , µ n = 1. All orbits are of size n , except the two fixed points[0 : 1] and [1 : 0].We choose s = ( s , . . . , s r ) to be a sequence of positive integers, all distinct,such that s i · n ≥ i if r = 2, and consider a finite subset∆ ⊂ ℓ ∪ · · · ∪ ℓ r ⊂ ( P ) r , stable by G , given by a union of orbits of size n . For each i ∈ { , . . . , r } ,we define ∆ i ⊂ ℓ i to be a union of exactly s i ≥ n , andchoose then ∆ = S ri =1 ∆ i . We moreover choose the points such that thegroup H = { h ∈ Aut( P ) | h (∆ i ) = ∆ i , h ([0 : 1]) = [0 : 1] } only consists of { [ u : v ] [ µu : v ] | u n = 1 } . As the unique point of intersections of the ℓ i isfixed by G , each point of ∆ lies on exactly one of the curves ℓ i . This gives∆ = ⊎ ri =1 ∆ i Let π : Y → ( P ) r be the blow-up of ∆. As ∆ is G -invariant, the action of G lifts to an action on Y . We want to prove that the resulting homomorphism G → Aut Y is an isomorphism.5.1. Intersection on ( P ) r . For i = 1 , . . . , r , we denote by H i ⊂ ( P ) r thehypersurface given by H i = { ([ u : v ] , . . . , [ u r : v r ]) ∈ ( P ) r | u i = 0 } . Then H , . . . , H r generate the cone of effective divisors on ( P ) r , and we have H i · ℓ i = 1 , H i · ℓ j = 0for all i, j ∈ { , . . . , r } with i = j . Moreover, the canonical divisor class of( P ) r satisfies K ( P ) r = − H − H − · · · − H r , so K ( P ) r · ℓ i = − i ∈ { , . . . , r } .We also observe that ℓ i ⊂ H j for all i, j ∈ { , . . . , r } with i = j and that ℓ i H i .5.2. Intersection on Y . For i = 1 , . . . , r , denote by ˜ ℓ i , ˜ H i ⊂ Y the stricttransforms of ℓ i and H i .For each p ∈ ∆, we denote by E p = π − ( p ) the exceptional divisor, isomor-phic to P r − , and choose a line e p ⊂ E p .A basis of the Picard group of Y is given by the union of ˜ H , . . . , ˜ H r and ofall exceptional divisors E p , with p ∈ ∆. A basis of the vector space of curves(up to numerical equivalence) is given by ˜ ℓ , . . . , ˜ ℓ r and by all e p with p ∈ ∆.We have e p · E p = − , e p · E q = 0for all p, q ∈ ∆, p = q . Lemma 5.1.
For all i, j ∈ { , . . . , r } with i = j , the following hold: BELIAN VARIETIES AS AUTOMORPHISM GROUPS 9 (i) ˜ H i = π ∗ ( H i ) − P p ∈ ∆ ∩ H i E p = π ∗ ( H i ) − P s = i P p ∈ ∆ s E p . (ii) ˜ ℓ i · E p = 1 if p ∈ ∆ i and ˜ ℓ i · E p = 0 if p ∈ ∆ \ ∆ i . (iii) ˜ H i · ˜ ℓ i = 1 . (iv) ˜ H i · ˜ ℓ j = −| ∆ j | = − ns j .Proof. (i) follows from the fact that H i is a smooth hypersurface of ( P ) r andthat ∆ ∩ H i = S s = i ∆ s .(ii): follows from the fact that ℓ i is a smooth curve, passing through allpoints of ∆ i and not through any point of ∆ \ ∆ i .(iii): With (i) and (ii), we get ˜ H i · ˜ ℓ i = H i · ℓ i = 1.(iv): With (i) and (ii), we get ˜ H i · ˜ ℓ j = H i · ℓ j − | ∆ j | = −| ∆ j | = − ns j . (cid:3) Lemma 5.2.
For all i ∈ { , . . . , r } and each p ∈ ∆ \ ∆ i , we take the irreduciblecurve γ p,i ⊂ ( P ) r passing through p and being numerically equivalent to ℓ i . (i) Let j ∈ { , . . . , r } be such that p ∈ ∆ j . The j -th coordinate of γ p,i is theone of p , its i -th coordinate is free, and all others are [0 : 1] . (ii) The strict transform ˜ γ p,i of γ p,i on Y is numerically equivalent to ˜ ℓ i + P q ∈ ∆ i e q − e p and satisfies ˜ γ p,i · E p = 1 and ˜ γ p,i · E q = 0 for all q ∈ ∆ \ { p } .Proof. (i): We write p = ( p , . . . , p r ) ∈ ( P ) r . Since γ p,i ⊂ ( P ) r is a curveequivalent to ℓ i and passing through p , it has to be γ p,i = { ( p , . . . , p i − , t, p i +1 , . . . , p r ) ∈ ( P ) r | t ∈ P } ≃ P . Moreover, for each s ∈ { , . . . , r } \ { j } , we have p s = [0 : 1], as p ∈ ∆ j ⊂ ℓ j .This completes the proof of (i).(ii): We want to prove that ˜ γ p,i ≡ ˜ ℓ i + P q ∈ ∆ i e q − e p . For each divisor D on( P ) r , we have ˜ γ p,i · π ∗ ( D ) = π (˜ γ p,i ) · D = γ p,i · D (˜ ℓ i − e p ) · π ∗ ( D ) = π (˜ ℓ i ) · D = ℓ i · D = γ p,i · D We moreover have (with Lemma 5.1(ii))˜ γ p,i · E p = 1 = E p · (˜ ℓ i + X q ∈ ∆ i e q − e p ) , ˜ γ p,i · E q = 0 = E q · (˜ ℓ i + X q ∈ ∆ i e q − e p ) , for all q ∈ ∆ \ { p } . (cid:3) Lemma 5.3.
Let γ ⊂ Y be an irreducible curve. Then, one of the followingholds: (i) We have γ ≡ de p for some d ≥ and some p ∈ ∆ ( where ≡ denotesnumerical equivalence ) ; (ii) There are non-negative integers a , . . . , a r and { µ p } p ∈ ∆ such that γ ≡ r X i =1 a i ˜ ℓ i + X p ∈ ∆ µ p e p and such that a + · · · + a r ≥ . (iii) There are j ∈ { , . . . , r } , q ∈ ∆ j and integers a , . . . , a r ≥ such that γ ≡ a j e q + X i = j a i ˜ γ q,i and such that P i = j a i ≥ .Proof. Suppose first that γ is contained in some E p , where p ∈ ∆. In thiscase, γ is a curve of degree d ≥ E p ≃ P r − (if r = 2,then γ = e p = E p and d = 1), and thus γ ≡ de p . This gives Case (i).We may now assume that γ is not contained in E p for any p ∈ ∆. Hence, γ is the strict transform of the irreducible curve π ( γ ) ⊂ ( P ) r , numericallyequivalent to P ri =1 a i ℓ i , with a , . . . , a r ≥ P ri =1 a i ≥
1. For each p ∈ ∆,we write ǫ p = E p · γ ≥ ♠ ) γ ≡ r X i =1 a i ˜ ℓ i + r X i =1 X p ∈ ∆ i ( a i − ǫ p ) e p . Intersecting both sides of ( ♠ ) with the divisor π ∗ ( D ), for any divisor D on( P ) r , gives π ( γ ) · D = P a i ℓ i · D . Moreover, for each p ∈ ∆, there is j ∈{ , . . . , r } such that p ∈ ∆ j . Intersecting E p with both sides of ( ♠ ), we obtain E p · γ = ǫ p Lemma 5.1(ii) = E p · ( r P i =1 a i ˜ ℓ i + r P i =1 P p ∈ ∆ i ( a i − ǫ p ) e p ). This completes theproof of ( ♠ ).For each p ∈ ∆, we denote by i ∈ { , . . . , r } the integer such that p ∈ ∆ i and by H p ⊂ ( P ) r the hypersurface consisting of points q ∈ ( P ) r having thesame i -th coordinate as p . Hence p i ∈ H p , H p ∩ ∆ = { p } and H p ∼ H i . Thestrict transform of H p , that we write ˜ H p , satisfies ˜ H p ∼ π ∗ ( H i ) − E p . Thisgives( ♥ ) ˜ H p · γ = a i − E p · γ = a i − ǫ p . Suppose first that ˜ H p · γ ≥ p ∈ ∆. This means (with ( ♥ )), that a i − ǫ p ≥ i ∈ { , . . . , r } and each p ∈ ∆ i . Hence all coefficients in( ♠ ) are non-negative, so we obtain (ii).Suppose now that ˜ H q · γ < q ∈ ∆. This implies that γ ⊂ ˜ H q .As H q ∩ ∆ = { q } , we obtain E p ∩ ˜ H q = ∅ for each p ∈ ∆ \ { q } , which yields BELIAN VARIETIES AS AUTOMORPHISM GROUPS 11 ǫ p = E p · γ = 0. Writing j ∈ { , . . . , r } the element such that q ∈ ∆ j , the j -thcomponent of π ( γ ) ⊂ ( P ) r is constant, so a j = π ∗ ( H j ) · γ = H j · π ( γ ) = 0.We now prove that( ♦ ) γ ≡ ( − ǫ q + X i = j a i ) e q + X i = j a i ˜ γ q,i Intersecting both sides of ( ♦ ) with the divisor π ∗ ( D ), for any divisor D on( P ) r , gives π ( γ ) · D = P a i ℓ i · D . Intersecting E q with both sides gives ǫ q = ǫ q ,since E q · ˜ γ q,i = 1 for each i = j (Lemma 5.2(ii)). Intersecting with E p for p ∈ ∆ \ { q } gives ǫ p = 0. This completes the proof of ( ♦ ).As the j -th component of π ( γ ) ⊂ ( P ) r is constant, there is an integer i ∈ { , . . . , r } \ { j } such that the i -th component of π ( γ ) is not constant. Thisimplies that π ( γ ) H i , so ˜ γ ˜ H i . We obtain0 ≤ ˜ H i · γ Lemma 5.1(i) = ( π ∗ ( H i ) − X s = i X p ∈ ∆ s E p ) · γ = a i − ǫ q . Hence, the coefficents of ( ♦ ) are non-negative, giving (iii). (cid:3) Proposition 5.4.
Let γ ⊂ Y be an irreducible curve. Then, the following areequivalent: (i) For all effective -cycles γ , γ on Y such that γ ≡ γ + γ , we have γ = 0 or γ = 0 . (ii) γ is numerically equivalent to ˜ ℓ i for some i ∈ { , . . . , r } , to ˜ γ p,i for some i ∈ { , . . . , r } , p ∈ ∆ \ ∆ i , or to e p for some p ∈ ∆ . (iii) γ is either equal to ˜ ℓ i for some i ∈ { , . . . , r } , or equal to ˜ γ p,i for some i ∈ { , . . . , r } , p ∈ ∆ \ ∆ i , or is a line in E p , for some p ∈ ∆ .Proof. (i) ⇒ (ii): By Lemma 5.3, γ ≡ γ + · · · + γ s where s ≥ γ , . . . , γ s belong to { ˜ ℓ i | i ∈ { , . . . , r }} ∪ { e p | p ∈ ∆ } ∪ { ˜ γ p,i | i ∈ { , . . . , r } , p ∈ ∆ \ ∆ i } . As (i) is satisfied, we have s = 1, which implies(ii).(ii) ⇒ (iii): Suppose first that γ ≡ e p for some p ∈ ∆. For an ample divisor D on ( P ) r , we have 0 = e p · π ∗ ( D ) = π ∗ ( γ ) · D , which implies that γ iscontracted by π . Hence, γ is a curve of degree d ≥ E q , q ∈ ∆, andis thus equivalent to de q . As − E p · e p = E p · γ , we have q = p and d = 1.Suppose now that γ ≡ ˜ ℓ i for some i ∈ { , . . . , r } . For each j ∈ { , . . . , r } with j = i , we have ˜ H i · γ = ˜ H i · ˜ ℓ j Lemma 5.1(iv) = − ns j <
0. Hence, π ( γ ) ⊂ T j = i H j = ℓ i . As π ( γ ) · H i = π ∗ ( H i ) · γ = π ∗ ( H i ) · ˜ ℓ i = 1, we have π ( γ ) = ℓ i and ˜ γ = ˜ ℓ i .In the remaining case, γ ≡ ˜ γ p,i for some i ∈ { , . . . , r } and some p ∈ ∆ \ ∆ i . Hence, π ( γ ) is numerically equivalent to π (˜ γ p,i ), which is equivalentto ℓ i (Lemma 5.2(ii)). Hence, all coordinates of π ( γ ) except the i -th one are constant. As γ · E p = ˜ γ p,i · E p = 1 (again by Lemma 5.2(ii)), the point p belongs to both π ( γ ) and γ p,i , which yields π ( γ ) = γ p,i and thus γ = ˜ γ p,i .(iii) ⇒ (i): We take effective 1-cycles γ , γ on Y such that γ ≡ γ + γ andprove that one of the two is zero, using (iii).For each i ∈ { , . . . , r } , we write a i = π ∗ ( H i ) · γ , b i = π ∗ ( H i ) · γ and c i = π ∗ ( H i ) · γ and obtain a i = b i + c i . As H i is nef, π ∗ ( H i ) is nef, so a i , b i , c i ≥
0. Moreover, γ satisfying (iii), we have P ri =1 a i = 1, which impliesthat, up to exchanging γ and γ , we may assume that P ri =1 a i = P ri =1 b i and P ri =1 c i = 0. In particular, γ is a sum of irreducible curves contained in theexceptional divisor E p , p ∈ ∆.Suppose first that γ = e q for some q ∈ ∆. This gives P ri =1 a i = P ri =1 b i = P ri =1 c i = 0, which implies that both γ and γ are sums of irreducible curvescontained in the exceptional divisor E p , p ∈ ∆. For each p ′ ∈ ∆ and eachirreducible curve c ⊂ E p ′ of degree d ≥ P p ∈ ∆ E p · c = d . As P p ∈ ∆ E p · γ = 1, this gives γ = 0 or γ = 0.We may now take s ∈ { , . . . , r } and either γ = ˜ ℓ s or γ = ˜ γ p,s for some p ∈ ∆ \ ∆ s . This gives b s = 1, c s = 0 and b i = c i = 0 for all i ∈ { , . . . , r }\{ s } .Lemma 5.3 implies that γ is equivalent to a sum of curves contained in { ˜ ℓ i | i ∈ { , . . . , r }} ∪ { e p | p ∈ ∆ } ∪ { ˜ γ p,i | i ∈ { , . . . , r } , p ∈ ∆ \ ∆ i } . As b s = 1and b i = 0 for all i ∈ { , . . . , r } \ { s } , we have γ ≡ α + β , where α is eitherequal to ˜ ℓ s or ˜ γ p,s for some p ∈ ∆ \ ∆ s and where β is a non-negative sum of e p , p ∈ ∆. For each p ∈ ∆, we obtain E p · γ = E p · α + E p · β + E p · γ ≤ E p · α. We now use the fact that we know the intersection of α and γ with E p (whichis given either by Lemma 5.1(ii) or by Lemma 5.2(ii), depending if the curveis equal to ˜ ℓ s or ˜ γ p,s ).If γ = ˜ γ p,s for some p ∈ ∆ \ ∆ s , then 1 = E p · γ ≤ E p · α , which implies that α = ˜ γ p,s . If γ = ˜ γ s , then 1 = E q · γ ≤ E q · α for each q ∈ ∆ s , which impliesthat α = ˜ γ s . In both cases, we get α = γ , which implies that E p · γ = 0 foreach p ∈ ∆, and thus that γ = 0, as desired. (cid:3) Theorem 5.5.
The map G → Aut Y is an isomorphism.Proof. We first show that G ∼ → Aut( Y ). Let α ∈ Aut( Y ). For each irreduciblecurve γ ⊂ Y that satisfies Proposition 5.4(i), the curve α ( γ ) also satisfiesProposition 5.4(i). Hence, the union F ⊂ Y of all curves satisfying thisassertion is also stable by Aut( Y ).By Proposition 5.4, we have F = ( [ p ∈ ∆ E p ) ∪ ( r [ i =1 ˜ ℓ i ) ∪ ( r [ i =1 ( [ p ∈ ∆ \ ∆ i ˜ γ p,i )) . BELIAN VARIETIES AS AUTOMORPHISM GROUPS 13
We observe that the above union is the decomposition of F into irreduciblecomponents. Hence, α permutes the irreducible components. We now makethe following observations:(i) For each i ∈ { , . . . , r } , ˜ ℓ i intersects exactly n · s i other irreduciblecomponents of F , namely the E p with p ∈ ∆.(ii) For each p ∈ ∆ i , the divisor E p intersects exactly r other irreduciblecomponents of F namely the curve ˜ ℓ i and the curves ˜ γ p,j with j ∈ { , . . . , r } \{ i } .(iii) For each i ∈ { , . . . , r } and p ∈ ∆ \ ∆ i , the curve ˜ γ p,i intersects exactly n · s i +1 other irreducible components of F . Writing j ∈ { , . . . , r } the elementsuch that p ∈ ∆ j , the curve intersects E p and all curves ˜ γ q,j for each q ∈ ∆ i .If r ≥
3, the exceptional divisors E p are the irreducible components ofmaximal dimension of F , so g permutes them. If r = 2, then g also permutesthe E p , as these are the only irreducible components of F that intersect exactly2 other irreducible components of F (we assumed n · s i ≥ i in thecase r = 2). In any case, g permutes the exceptional divisors E p and is thusthe lift of an automorphism ˆ g of ( P ) r : we observe that the birational self-mapˆ g = πgπ − of ( P ) r restricts to an automorphism on the complement of ∆,and as ∆ has codimension ≥
2, ˆ g is an automorphism. We then use again thethree observations above to see that g (˜ ℓ i ) = ˜ ℓ i for each i ∈ { , . . . , r } , as the s i are all distinct. Hence, ˆ g ( ℓ i ) = ˆ g ( ℓ i ) for each i . This implies that ˆ g is of theform ( P ) r → ( P ) r (( µ , . . . , µ r ) , ) ([ u : µ v + κ u ] , . . . , [ u r : µ r v r + κ r u r ])for some µ , . . . , µ r ∈ k ∗ and κ , . . . , κ r ∈ k .For each i ∈ { , . . . , r } , the restriction of ˆ g to ℓ i corresponds to the auto-morphism [ u : v ] [ u i : µ i v + κ i u i ]. As it has to stabilize the set ∆ i , we have κ i = 0 and µ i ∈ k ∗ is of order n . This yields the isomorphism G ≃ Aut( Y ).To complete the proof, it suffices to show that Aut Y is constant, or equiva-lently that its Lie algebra is trivial. (We refer to [Mart20, § Y ) = H ( Y, T Y ), where T Y denotes the tangent sheaf. In other terms, Lie(Aut Y )consists of the global vector fields on Y . Denoting by E = ⊎ p ∈ ∆ E p the excep-tional divisor, we have an exact sequence of sheaves on Y −→ T Y,E −→ T Y −→ M p ∈ ∆ N E p /Y −→ , where T Y,E is the sheaf of vector fields that are tangent to E , and N E p /Y denotes the normal sheaf. Moreover, for any p ∈ ∆, we have E p ≃ P r − and this identifies N E p /Y with O P r − ( − H ( E p , N E p /Y ) = 0. As a consequence, H ( Y, T Y,E ) ∼ → H ( Y, T Y ). Viewing vector fields as derivationsof the structure sheaf O Y , this yieldsDer( O Y , O Y ( − E )) ∼ → Der( O Y ) , where the left-hand side denotes the Lie algebra of derivations which stabilizethe ideal sheaf of E .The blow-up π : Y → ( P ) r contracts E to ∆ and satisfies π ∗ ( O Y ) = O ( P ) r ;also, π ∗ ( O Y ( − E )) = I ∆ (the ideal sheaf of ∆). So π induces a homomor-phism of Lie algebras π ∗ : Der( O Y ) → Der( O ( P ) r ), which is injective as π is birational. Moreover, π ∗ sends Der( O Y , O Y ( − E )) into Der( O ( P ) r , I ∆ ), theLie algebra of vector fields on ( P ) r which vanish at each p ∈ ∆. So it sufficesto show that each such vector field is zero.We haveDer( O ( P ) r ) = H (( P ) r , T ( P ) r ) = r M i =1 H ( P , T P ) = Lie(Aut P ) r . Moreover, Lie(Aut P ) = M ( k ) / k id, the quotient of the Lie algebra of 2 × ξ = ( ξ , . . . , ξ r ) ∈ Der( O ( P ) r ), with represen-tative ( A , . . . , A r ) ∈ M ( k ) r . Then ξ vanishes at p = ([ x : y ] , . . . , [ x r : y r ])if and only if ( x i , y i ) is an eigenvector of A i for each i ∈ { , . . . , r } . Thus, if ξ ∈ Der( O ( P ) r , I ∆ ), then (0 ,
1) is an eigenvector of each A i , i.e., A i is lowertriangular. In addition, each point of ∆ i yields an eigenvector of A i . So each A i is scalar, and ξ = 0 as desired. (cid:3) References [Bri10] Michel Brion,
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