Geometrical description of smooth projective symmetric varieties with Picard number one
aa r X i v : . [ m a t h . AG ] D ec GEOMETRICAL DESCRIPTION OF SMOOTHPROJECTIVE SYMMETRIC VARIETIES WITHPICARD NUMBER ONE
Alessandro Ruzzi
Abstract
In [Ru2] we have classified the smooth projective symmetric G -varietieswith Picard number one (and G semisimple). In this work we give a ge-ometrical description of such varieties. In particular, we determine theirgroup of automorphisms. When this group, Aut ( X ), acts non-transitivelyon X , we describe a G -equivariant embedding of the variety X in a ho-mogeneous variety (with respect to a larger group). keywords : Symmetric varieties, Fano varieties.Mathematics Subject Classification 2000: 14M17, 14J45, 14L30 A Gorenstein normal algebraic variety X over C is called a Fano varietyif the anticanonical divisor is ample. The Fano surfaces are classically calledDel Pezzo surfaces. The importance of Fano varieties in the theory of higherdimensional varieties is similar to the importance of Del Pezzo surfaces in thetheory of surfaces. Moreover Mori’s program predicts that every uniruled varietyis birational to a fiberspace whose general fiber is a Fano variety (with terminalsingularities).Often it is useful to subdivide the Fano varieties in two kinds: the Fanovarieties with Picard number equal to one and the Fano varieties whose Picardnumber is strictly greater of one. For example, there are many results whichgive an explicit bound to some numerical invariants of a Fano variety (dependingon the Picard number and on the dimension of the variety). Often there is anexplicit expression for the Fano varieties of Picard number equal to one andanother expression for the remaining Fano varieties.We are mainly interested in the smooth projective spherical varieties withPicard number one. The smooth toric (resp. homogeneous) projective varietieswith Picard number one are just projective spaces (resp. G/P with G sim-ple and P maximal). B. Pasquier has recently classified the smooth projectivehorospherical varieties with Picard number one (see [P]). In a previous work wehave classified the smooth projective symmetric G -varieties with Picard num-ber one and G semisimple (see [Ru2]). One can easily show that they are allFano, because the canonical bundle cannot be ample. We have also obtained apartial classification of the smooth Fano complete symmetric varieties with Pi-card number strictly greater of one (see [Ru1]). Our classification of the smooth1rojective symmetric varieties with Picard number one is a combinatorial one,so we are naturally interested to give a geometrical description of such varieties.In particular, we have proved that, given a symmmetric space G/H , there is atmost a smooth completion X of G/H with Picard number one and X must beprojective (see [Ru2], Theorem 3.1). We will prove that the automorphism groupof a such variety X can act non-transitively on X only if the rank of X is 2. Itwould be interesting to find a reason for such exceptionality of the rank 2 case.Unfortunately, our prove does not explain completely this fact, because there isa part of the proof that it is a case-to-case analysis. The homogeneousness ofthe rank one varieties was proved first by Ahiezer in [A1].More precisely we have proved that: Theorem 1
Let X be a smooth projective completion of a symmetric space G/H with Picard number one (where G is semisimple and simply connected).Then Aut ( X ) does not act transitively on X if and only if: i) the restrictedroot system has type either A or G and ii) the subgroup H is the subgroup ofinvariants G θ . There are six varieties which are not homogenous; their connected automor-phism group
Aut ( X ) is isomorphic to G up to isogeny. These varieties canbe realized as intersection of hyperplane sections of a homogeneous projectivevariety with Picard number one. Moreover, they are someway related to theexceptional groups; in particular there are two varieties related to G and fourvarieties related to the magic square of Freudenthal. The non-homogeneousvarieties with restricted root system of type A are obtainable as hyperplanesections of the Legendrian varieties in the third row of the Freudenthal magicsquare. Moreover each one is contained in the the others with bigger dimension.The completion of SL was already studied by J. Buczy´nski (see [Bu]).More precisely we will prove the following theorems: Theorem 2
Let
G/G θ be a symmetric space whose restricted root system hastype G . We have the following possibilities for the smooth completion of G/H with Picard number one:1. the smooth equivariant completion with Picard number one of the sym-metric variety G / ( SL × SL ) of type G . In this case, Aut ( X ) = G .The involution θ can be extended to an involution of SO , whose invari-ant subgroup is S ( O × SO ) . The unique equivariant smooth comple-tion of SO /N SO ( S ( O × SO )) with Picard number one is isomorphicto G (7) ⊂ P and X is the intersection of G (7) with a 27-dimensionalsubspace of P . If we interprete C as the subspace of imaginary elementsof the complexified octonions O , then X parametrizes the subspaces W of C such that W ⊕ C is a subalgebra of O isomorphic to the complexifiedquaternions.2. the smooth equivariant completion with Picard number one of the symmet-ric ( G × G ) -variety G . In this case, Aut ( X ) is generated by G × G and . The involution θ can be extended to an involution of SO × SO , withinvariant subgroup equal to the diagonal. The unique equivariant smoothcompletion of SO with Picard number one is isomorphic to IG (14) ⊂ P and X is the intersection of IG (14) with a 49-dimensional subspace of P . Theorem 3
Let
G/G θ be a symmetric space whose restricted root system hastype A . We have the following possibilities for the smooth completion of G/H with Picard number one:1. the smooth equivariant completion with Picard number one of the symmet-ric variety SL /SO of type AI ; it is an hyperplane section of LG (6) .2. the smooth equivariant completion with Picard number one of the symmet-ric variety SL ; it is an hyperplane section of G (6) .3. the smooth equivariant completion with Picard number one of the symmet-ric variety SL /Sp of type AII ; it is an hyperplane section of S .4. the smooth equivariant completion with Picard number one of the sym-metric variety E /F of type EII ; it is an hyperplane section of E /P ≡ G ( O ) .Moreover, SL /SO ⊂ SL ⊂ SL /Sp ⊂ E /F and also their smoothcompletions with Picard number one are contained nested in each other: SL /SO (cid:31) (cid:127) / / (cid:127) _ (cid:15) (cid:15) SL (cid:31) (cid:127) / / (cid:127) _ (cid:15) (cid:15) SL /Sp (cid:31) (cid:127) / / (cid:127) _ (cid:15) (cid:15) E /F (cid:127) _ (cid:15) (cid:15) LG (6) (cid:31) (cid:127) / / G (6) (cid:31) (cid:127) / / S (cid:31) (cid:127) / / G ( O ) If G/H is different from SL , then the automorphism group of X is generatedby Aut ( X ) and θ . If, instead, G/H = SL then Aut ( X ) has index four in Aut ( X ) . We give also an explicit description of the smooth projective symmetric vari-eties with Picard number one over which
Aut ( X ) acts transitively: in particular,we will describe their connected automorphism group and the immersion of G/H in Aut ( X ) /Stab Aut ( X ) ( H/H ).In the first section we introduce the notation and recall some general factsabout symmetric varieties. In the second one we cite a theorem of D. Ahiezerabout the homogeneousness of the rank one varieties. In the third one we studythe varieties of rank two which does not belong to an infinite family; in particularwe consider all the varieties which are not homogeneous. In the forth sectionwe study the remaining varieties; in particular we will study all the varieties ofrank at least three. 3
Introduction and notations
In this section we introduce the necessary notations. The reader interested tothe embedding theory of spherical varieties can see [Br3] or [T2]. In [V1] isexplained such theory in the particular case of the symmetric varieties.
Let G be a connected semisimple algebraic group over C and let θ be an involu-tion of G . Let H be a closed subgroup of G such that G θ ⊂ H ⊂ N G ( G θ ). Wesay that G/H is a symmetric homogeneous variety. An equivariant embeddingof
G/H is the data of a G -variety X together with an equivariant open immer-sion G/H ֒ → X . A normal G -variety is called a spherical variety if it contains adense orbit under the action of an arbitrarily chosen Borel subgroup of G . Onecan show that an equivariant embedding of G/H is a spherical variety if andonly if it is normal (see [dCP1], Proposition 1.3). In this case we say that itis a symmetric variety. We say that a subtorus of G is split if θ ( t ) = t − forall its elements t . We say that a split torus of G of maximal dimension is amaximal split torus and that a maximal torus containing a maximal split torusis maximally split. One can prove that any maximally split torus is θ stable (see[T2], Lemma 26.5). We fix arbitrarily a maximal split torus T and a maximallysplit torus T containing T . Let R G be the root system of G with respect to T and let R G be the subroot system composed by the roots fixed by θ . We set R G = R G \ R G . We can choose a Borel subgroup B containing T such that, if α is a positive root in R G , then θ ( α ) is negative (see [dCP1], Lemma 1.2). Onecan prove that BH is dense in G (see [dCP1], Proposition 1.3). Now, we want to define the colored fan associated to a symmetric variety. Let D ( G/H ) be the set of B -stable prime divisors of G/H ; its elements are calledcolors. Since
BH/H is an affine open set, the colors are the irreducible com-ponents of (
G/H ) \ ( BH/H ). We say that a spherical variety is simple if itcontains one closed orbit. Let X be a simple symmetric variety with closedorbit Y . Let D ( X ) be the subset of D ( G/H ) consisting of the colors whoseclosure in X contains Y . We say that D ( X ) is the set of colors of X . To eachprime divisor D of X , we can associate the normalized discrete valuation v D of C ( G/H ) whose ring is the local ring O X,D . One can prove that D is G -stableif and only if v D is G -invariant, i.e. v D ( s · f ) = v D ( f ) for each s ∈ G and f ∈ C ( G/H ). Let N be the set of all G -invariant valuations of C ( G/H ) takingvalue in Z and let N ( X ) be the set of the valuations associated to the G -stableprime divisors of X . Observe that each irreducible component of X \ ( G/H )has codimension one, because
G/H is affine. Let S := T / T ∩ H ≃ T · x ,4here x = H/H denotes the base point of
G/H . One can show that the group C ( G/H ) ( B ) / C ∗ is isomorphic to the character group χ ( S ) of S (see [V1], § G/H as the rankof χ ( S ). We can identify the dual group Hom Z ( C ( G/H ) ( B ) / C ∗ , Z ) with thegroup χ ∗ ( S ) of one-parameter subgroups of S ; so we can identify χ ∗ ( S ) ⊗ R with Hom Z ( χ ( S ) , R ). The restriction map to C ( G/H ) ( B ) / C ∗ is injective over N (see [Br3], § N with a subset of χ ∗ ( S ) ⊗ R .We say that N is the valuation semigroup of G/H . Indeed, N is the semigroupconstituted by the vectors in the intersection of the lattice χ ∗ ( S ) with an ap-propriate rational polyhedral convex cone C N , called the valuation cone. Foreach color D , we define ρ ( D ) as the restriction of v D to χ ( S ). In general, themap ρ : D ( G/H ) → χ ∗ ( S ) ⊗ R is not injective.Let C ( X ) be the cone in χ ∗ ( S ) ⊗ R generated by N ( X ) and ρ ( D ( X )). Wedenote by cone ( v , ..., v r ) the cone generated by the vectors v , ..., v r . Given acone C in χ ∗ ( S ) ⊗ R and a subset D of D ( G/H ), we say that (
C, D ) is a coloredcone if: 1) C is generated by ρ ( D ) and by a finite number of vectors in N ; 2) therelative interior of C intersects C N . The map X → ( C ( X ) , D ( X )) is a bijectionfrom the set of simple symmetric varieties to the set of colored cones (see [Br3], § e X (not necessarily simple), let { Y i } i ∈ I be theset of G -orbits. Observe that e X contains a finite number of G -orbits, thus e X i := { x ∈ e X | G · x ⊃ Y i } is open in e X and is a simple symmetric varietywhose closed orbit is Y i . We define D ( e X ) as the set S i ∈ I D ( e X i ). The family { ( C ( e X i ) , D ( e X i )) } i ∈ I is called the colored fan of e X and determines completely e X (see [Br3], § e X is complete if and only if C N ⊂ S i ∈ I C ( e X i ) (see [Br3], § To describe the sets N and ρ ( D ( G/H )), we need to associate a root systemto
G/H . The subgroup χ ( S ) of χ ( T ) has finite index, so we can identify χ ( T ) ⊗ R with χ ( S ) ⊗ R . Because T is θ -stable, θ induces an involution of χ ( T ) ⊗ R which we call again θ . The inclusion T ⊂ T induces an isomorphismof χ ( T ) ⊗ R with the ( − χ ( T ) ⊗ R under the action of θ (see[T2], § W G the Weyl group of G (with respect to T ) and let( · , · ) be the Killing form over span R ( R G ). We denote with the same symbolthe restriction of ( · , · ) to χ ( T ) ⊗ R , thus we can identify χ ( T ) ⊗ R with itsdual χ ∗ ( T ) ⊗ R . The set R G,θ := { β − θ ( β ) | β ∈ R G }\ { } is a root systemin χ ( S ) ⊗ R (see [V1], § G, θ ); we call the non zero β − θ ( β ) the restricted roots. Usually we denote by β (respectively by α ) a root of R G (respectively of R G,θ ); often we denote by ̟ (respectively by ω ) a weight of R G (respectively of R G,θ ). In particular, wedenote by ̟ , ..., ̟ n the fundamental weight of R G (we have chosen the basisof R G associated to B ). Notice however that the weights of R G,θ are weights5f R G . The involution ι := − ̟ · θ of χ ( T ) preserves the set of simple roots;moreover ι coincides with − θ modulo the lattice generated by R G (see [T2], p.169). Here ̟ is the longest element of the Weyl group of R G . We denote by α , ..., α s the elements of the basis { β − θ ( β ) | β ∈ R G simple } \ { } of R G,θ . Let b i be equal to if 2 α i belongs to R G,θ and equal to one otherwise; for each i wedefine α ∨ i as the coroot b i ( α i ,α i ) α i . The set { α ∨ , ..., α ∨ s } is a basis of the dual rootsystem R ∨ G,θ , namely the root system composed by the coroots of the restrictedroots. We call the elements of R ∨ G,θ the restricted coroots. Let ω , ..., ω s bethe fundamental weights of R G,θ with respect to { α , ..., α s } and let ω ∨ , ..., ω ∨ s be the fundamental weights of R ∨ G,θ with respect to { α ∨ , ..., α ∨ s } . Let C + bethe positive closed Weyl chamber of χ ( S ) ⊗ R ; we call − C + the negative Weylchamber. Given a dominant weight λ of G , we denote by V ( λ ) the irreduciblerepresentation of highest weight λ .We want to give a description of the weight lattice of R G,θ . We say that adominant weight ̟ ∈ χ ( T ) is a spherical weight if V ( ̟ ) contains a non-zerovector fixed by G θ . In this case, V G θ is one-dimensional and θ ( ̟ ) = − ̟ . Thus,we can think ̟ as a vector in χ ( S ) ⊗ R . One can show that the lattice generatedby the spherical weights coincides with the weight lattice of R G,θ . See [CM],Theorem 2.3 or [T2], Proposition 26.4 for an explicit description of the sphericalweights. Such description implies that the set of dominant weights of R G,θ isthe set of spherical weights and that C + is the intersection of χ ( S ) ⊗ R with thepositive closed Weyl chamber of the root system R G . N and D ( G/H )The set N is equal to − C + ∩ χ ∗ ( S ); in particular, it consists of the lattice vectorsof the rational polyhedral convex cone C N = − C + . The set ρ ( D ( G/H )) is equalto { α ∨ , ..., α ∨ s } (see [V1], § i , thefibre ρ − ( α ∨ i ) contains at most 2 colors (see [V1], § α i is exceptional if there are two distinct simpleroots β i and β i in R G such that: 1) β i − θ ( β i ) = β i − θ ( β i ) = α i ; 2) either θ ( β i ) = − β i or θ ( β i ) = − β i and ( β i , β i ) = 0. In this case we say thatalso α ∨ i , θ and all the equivariant embeddings of G/H are exceptional. If
G/H is exceptional, then ρ is not injective. We say that G/H contains a Hermitianfactor if the center of [
G, G ] θ has positive dimension. If G/H does not containa Hermitian factor, then ρ is injective (see [V1], § ρ isinjective, we denote by D α ∨ the unique color contained in ρ − ( α ∨ ). Let X be a symmetric variety and let Y be a closed orbit of X . One can showthat there is a unique B -stable affine open set X B that intersects Y and isminimal for this property. Moreover, the complement X \ X B is the union ofthe B -stable prime divisors not containing Y (see [Br3], § P of X B is a parabolic subgroup containing B . Suppose X smooth6r non-exceptional and let L be the standard Levi subgroup of P . Then L is θ -stable, θ ( P ) = P - and the P -variety X B is the product R u P × Z of theunipotent radical of P and of an affine L -symmetric variety Z containing x (see [Br3], § Y is projective, then Z contains a L -fixed point whose stabilizer in G is P - . Given a root β , let U β be the unipotent one-dimensional subgroup of G corresponding to β . Given µ ∈ χ ∗ ( T ) ⊗ Q ≡ χ ( T ) ⊗ Q , we denote by P ( µ ) the parabolic subgroup of G generated by T and by the subgroups U β corresponding to the roots β such that( β, µ ) ≥
0. Given a parabolic subgroup P = P ( µ ), sometimes we denote by P - the opposite standard parabolic subgroup, namely P ( − µ ). In [Ru2] we have proved the following combinatorial classification of the smoothcomplete symmetric varieties with Picard number one.
Unless otherwise speci-fied, we always suppose G simply connected. Notice that this assumption is notrestrictive (see, for example, [V1], § Theorem 1.1
Let G be a semisimple, simply connected group and let G/H bea homogeneous symmetric variety. Suppose that there is a smooth, completeembedding X of G/H with Picard number one. Then: • Fixed
G/H , there is, up to equivariant isomorphism, at most one embed-ding with the previous properties. Moreover, it is projective and containsat most two closed orbits. • The number of colors of
G/H is equal to the rank l of G/H ; in particularthere are no exceptional roots. • We have the following classification depending on the type of the restrictedroot system R G,θ :(i) If R G,θ has type A × A , then χ ( S ) has basis { ω , ω + ω } ; in partic-ular H has index two in N G ( G θ ) . Moreover, X has two closed orbits;the maximal colored cones of the colored fan of X are ( cone ( α ∨ , − ω ∨ − ω ∨ ) , { D α ∨ } ) and ( cone ( α ∨ , − ω ∨ − ω ∨ ) , { D α ∨ } ) .(ii) If l = 1 , then G/H can be isomorphic neither to SL n +1 /S ( L × L n ) , nor to SL /SO . With such hypothesis, G/H has a unique nontrivial embedding which is simple, projective, smooth and with Picardnumber one.(iii) If R G,θ has type A l with l > , we have the following possibilities:(a) H = N G ( G θ ) and X is simple. In this case X is associated eitherto the colored cone ( cone ( α ∨ , ..., α ∨ l − , − ω ∨ ) , { D α ∨ , ..., D α ∨ l − } ) orto the one ( cone ( α ∨ , ..., α ∨ l , − ω ∨ l ) , { D α ∨ , ..., D α ∨ l } ) ; b) H = G θ and l = 2 . In this case X has two closed orbits. Themaximal colored cones of the colored fan of X are ( cone ( α ∨ , − ω ∨ − ω ∨ ) , { D α ∨ } ) and ( cone ( α ∨ , − ω ∨ − ω ∨ ) , { D α ∨ } ) .(iv) If R G,θ has type B , then X is simple and we have the followingpossibilities:(a) H = N G ( G θ ) and X is associated to ( cone ( α ∨ , − ω ∨ ) , { D α ∨ } ) ;(b) H = G θ and X is associated to ( cone ( α ∨ , − ω ∨ ) , { D α ∨ } ) . More-over G/H cannot be Hermitian.(v) If R G,θ has type B l with l > , then H = N G ( G θ ) , X is simple andis associated to ( cone ( α ∨ , ..., α ∨ l − , − ω ∨ ) , { D α ∨ , ..., D α ∨ l − } ) .(vi) If R G,θ has type C l , then H = G θ , X is simple and corresponds to ( cone ( α ∨ , ..., α ∨ l − , − ω ∨ ) , { D α ∨ , ..., D α ∨ l − } ) . Moreover, G/H cannotbe Hermitian.(vii) If R G,θ has type BC l with l > , then H = N G ( G θ ) = G θ , X is simpleand corresponds to ( cone ( α ∨ , ..., α ∨ l − , − ω ∨ ) , { D α ∨ , ..., D α ∨ l − } ) .(viii) If R G,θ has type D l with l > , then χ ∗ ( S ) is freely generated by ω ∨ , ..., ω ∨ l − , ω ∨ l − + ω ∨ l , ω ∨ l ; in particular H has index two in N G ( G θ ) . X has two closed orbits; the maximal colored cones of the colored fanof X are ( cone ( α ∨ , ..., α ∨ l − , − ω ∨ ) , { D α ∨ , ..., D α ∨ l − } ) and ( cone ( α ∨ , ...,α ∨ l − , α ∨ l , − ω ∨ ) , { D α ∨ , ..., D α ∨ l − , D α ∨ l } ) .(ix) If R G,θ has type D , then H has index two in N G ( G θ ) and X has twoclosed orbits. If χ ∗ ( S ) = Z ω ∨ i ⊕ Z ω ∨ ⊕ Z ( ω ∨ j + ω ∨ k ) ⊕ Z ω ∨ k , then themaximal colored cones of the colored fan of X are ( cone ( α ∨ i , α ∨ , α ∨ j , − ω ∨ i ) , { D α ∨ i , D α ∨ , D α ∨ j } ) and ( cone ( α ∨ i , α ∨ , α ∨ k , − ω ∨ i ) , { D α ∨ i , D α ∨ ,D α ∨ k } ) .(x) If R G,θ has type G then H = N G ( G θ ) = G θ , X is simple and isassociated to ( cone ( α ∨ , − ω ∨ ) , { D α ∨ } ) .(xi) If the type of R G,θ is different from the previous ones, then there isno a variety X with the requested properties. We will need a description of some exceptional groups via complex compositionalgebras and Jordan algebras. The interested reader can see [LM] and [Ad] fora detailed exposition of the facts which we recall here.Let A be a complex composition algebra, i.e. A = A R ⊗ R C where A R isa real division algebra (namely A R = R , C , H or O ). If a ∈ A , let ¯ a be itsconjugate; we denote by Im A the subspace of pure imaginary element, i.e. theelements a such that ¯ a = − a . Let J ( A ) be the space of A -Hermitian matricesof order three, with coefficients in A : 8 ( A ) = r ¯ x ¯ x x r ¯ x x x r , r i ∈ C , x i ∈ A . J ( A ) has the structure of a Jordan algebra with the multiplication A ◦ B := ( AB + BA ), where AB is the usual matrix multiplication. There is a welldefined cubic form on J ( A ), which we call the determinant. Given P ∈ J ( A ),its comatrix is defined by com P = P − ( trace P ) P + 12 (( trace P ) − trace P ) I and characterized by the identity com ( P ) P = det ( P ) I . In particular, P isinvertible if and only if det ( P ) is different from 0.Let SL ( A ) ⊂ GL C ( J ( A )) be subgroup preserving the determinant; J ( A )is an irreducible SL ( A ) representation. We let SO ( A ) denote the group ofcomplex linear transformations preserving the Jordan multiplication; it is alsothe subgroup of SL ( A ) preserving the quadratic form Q ( A ) = trace ( A ).We call Z ( A ) := C ⊕ J ( A ) ⊕ J ( A ) ∗ ⊕ C ∗ the space of Zorn matrices. Thespace sp ( A ) := C ∗ ⊕ J ( A ) ∗ ⊕ ( Lie ( SL ( A )) + C ) ⊕ J ( A ) ⊕ C has a structureof Lie algebra and Z has a natural structure of (simple) sp ( A )-module. Thereis a unique closed connected subgroup of GL A ( Z ( A )) with Lie algebra sp ( A );we denote it by Sp ( A ). Moreover, there is a Sp ( A )-invariant symplectic formon Z ( A ).The closed Sp ( A )-orbit in P ( Z ( A )) is the image of the Sp ( A )-equivariantrational map: φ : P ( C ⊕ J ( A )) P ( Z ( A ))( x, P ) → ( x , x P, x com ( P ) , det ( P )) . Furthermore, if C is interpreted as a space of diagonal matrix (in J ( A )) and( I, P ) is interpreted as a matrix of three row vectors in A , then the previousmap is the usual Plucker map. The condition P ∈ J ( A ) can be interpretedas the fact that the three vectors defined by the matrix ( I, P ) are orthogonalwith respect to the Hermitian symplectic two-form w ( x, y ) = t xJ ¯ y , where J =( I - I ). Therefore, it is natural to see the closed Sp ( A )-orbit in P ( Z ( A )) as theGrassmannian LG ( A , A ) of symplectic 3-planes in A .Explicitly, we have the following possibilities: 1) if A R = R then SL ( A ) is SL , SO ( A ) is SO and Sp ( A ) is Sp ; 2) if A R = C then SL ( A ) is SL × SL , SO ( A ) is SL and Sp ( A ) is SL ; 3) if A R = H then SL ( A ) is SL , SO ( A )is Sp and Sp ( A ) is Spin ; 4) if A R = O then SL ( A ) is E , SO ( A ) is F and Sp ( A ) is E . We conclude this section with a description of the projective homogeneous vari-eties with Picard number one for the classic groups. Let V be a n -dimensional9ector space, we will denote by G m ( V ) the Grassmannian of the m -dimensionalsubspace of V . Let q be a non-degenerate symmetric bilinear form on V and let SO ( V, q ) be the corresponding special orthogonal group. We will say that a sub-space of V is isotropic if the restriction of q to it is zero. Fix an integer m suchthat 2 m ≤ r and let IG m ( V ) ⊂ G m ( V ) be the algebraic subvariety whose pointsare identified with isotropic m -dimensional subspaces of V . The group SO ( V, q )acts on IG m ( V ) in the natural manner and the action is transitive if 2 m < n .Instead, if 2 m = n , IG m ( V ) consists of two isomorphic SO ( V, q )-orbits (eachof them being a connected component of IG m ( V )); we denote a such orbit by S m ( V ) (or by S m if V = C m and q is the standard symmetric bilinear form).The variety IG m ( V ) is called the isotropic Grassmannian of m -dimensionalisotropic subspace , while S m ( V ) is called the spinorial variety of order m . In ananalogous manner, let q ′ be a non-degenerate skew-symmetric bilinear form on V and fix an integer m such that 2 m ≤ n . We denote by LG m ( V ) the algebraicsubvariety of G m ( V ) whose points are identified with isotropic m -dimensionalsubspaces of V ; LG m ( V ) is called the Lagrangian Grassmannian . In this section we describe the varieties with rank one. For any homogenoussymmetric variety
G/H with rank one there is a unique equivariant completion X . It is smooth, projective and has Picard number at most two. Moreover, X has exactly two orbits: an open orbit and a closed orbit of codimension one.Thus X is a homogeneous variety by the following theorem due to D. Ahiezer. Theorem 2.1 (see [A1], Theorem 4)
Let G be a semisimple group and let H be a closed reductive subgroup. Let X be an equivariant (normal) completionof G/H such that X \ ( G/H ) is a G -orbit (of codimension one). Then X isa homogeneous space for a larger group. If G/H is a symmetric variety and G acts effectively on G/H , we have the following possibilities:1. G is SL × SL , H is SL and X is { ( x, t ) | det ( x ) = t } ⊂ P ( S C ⊕ C ) ;2. G is P SL × P SL , H is P SL and X is P ( S C ) ;3. G is SL n , H is GL n − , θ has type AIV (or AI if n = 2 ) and X is P ( C n ) × P (( C n ) ∗ ) ;4. G is P SL , H is P SO , θ has type AI and X is P ( sl ) ;5. G is Sp n , H is Sp × Sp n − , θ has type CII and X is G (2 n ) ;6. G is SO n , H is SO n − , θ has type BII or DII and X is { ( x, t ) | q ( x, x ) = t } ⊂ P ( C n ⊕ C ) , where q is the standard symmetric bilinear form;7. G is SO n , H is S ( O × O n − ) , θ has type BII or DII , and X is P ( C n ) ;8. G is F , H is Spin , θ has type F II and X is E /P ω ≡ P O ; Varieties of rank two
In the following of this work we always suppose that
G/H has rank strictlygreater than one.
In this section we describe the varieties of rank two which donot belong to an infinite family. Explicitly, we consider completion of the fol-lowing homogenous varieties: 1) the symmetric variety SL /N SL ( SL × SL )of type AIII ; 2) the symmetric ( SL × SL )-variety SL ; 3) the symmet-ric variety SL /SO of type AI ; 4) the symmetric variety SL /Sp of type AII ; 5) the symmetric variety E /F of type EIV (with E simply connected);6) the symmetric variety E /N E ( F ) of type EIV ; 7) the symmetric variety Sp / ( Sp × Sp ) of type CII ; 8) the symmetric variety G ; 9) the symmetricvariety G / ( Sl × Sl ) of type G ; 10) the symmetric varieties whose restrictedroot system has type A × A . In the last case G/N G ( G θ ) is isomorphic to SO n /N SO n ( SO n − ) × SO m /N SO m ( SO m − ); n, m are strictly greater than twoand H has index two in N G ( G θ ).In this section we do not consider the completions of the symmetric varieties Spin and SO /N SO ( GL ) because: 1) Spin is isomorphic to Sp and wewill study it together with the completion of Sp n ; 2) SO / N SO ( GL ) isisomorphic to SO /N SO ( SO × SO ) and we will study it together with thecompletion of SO n /N SO n ( SO × SO n − ). We begin with some general considerations; in particular we do not supposeyet that X has rank two. We prove that a smooth projective symmetricvariety X with Picard number one is a homogeneous variety if and only if H ( G/P - , N G/P - , X ) = 0 for an appropriate closed orbit G/P - of X . Further-more, we explain how to study the automorphism group of X (if such variety isnot homogenous). Lemma 3.1
Let X be a smooth projective symmetric variety with Picard num-ber one and let Y be a proper G -stable closed subvariety of X . If Y is smooth,then it is a G -orbit.Proof. Recall that the minimal B -stable affine open set X B which intersects Y is a product P u × Z , where Z is an affine L -symmetric variety. One can easilyshow that Z is an irreducible L -representation (see [Ru2], pages 9 and 17 or[Ru2], Theorem 2.2). Observe that Y is smooth if and only if Y ∩ Z is smooth.But Y ∩ Z is a cone because the center of L acts non-trivially on it. Thus Y ∩ Z is smooth if and only if it is a subrepresentation of Z . Since Z is irreducible, Y ∩ Z is smooth if and only if it is a point. In such case Y is a closed orbit. (cid:3) We use the previous lemma only to prove following corollary.
Corollary 3.1
Let X be a smooth projective symmetric variety with Picardnumber one. If Aut ( X ) does not act transitively over X , then Aut ( X ) stabi-lizes a closed G -orbit. roof. Let Y be a minimal closed G -stable subvariety stabilized by Aut ( X ).Suppose by contradiction that Y is not a G -orbit, then it is singular by theLemma 3.1. Thus the singular locus of Y is not empty and is stabilized by Aut ( X ). In particular, its irreducible components are Aut ( X )-stable, closedsubvarieties of X , properly contained in Y ; a contradiction. (cid:3) A closed orbit
G/P - of X is stabilized by Aut ( X ) if and only if H ( G/P - ,N G/P - ,X ) = 0, where N G/P - ,X is the normal bundle to G/P - (see [A2], § . X B is P u × Z ; moreover, the intersection of G/P - with X B is P u × { } . Thus we can identify Z with the fibre of the normal bundle N G/P - , X over P - /P - . We know by the Borel-Weil theorem (see [A2], § H ( G/P - ,N G/P - , X ) = 0 if and only if the highest weight of Z is not dominant.Now, we want to explain how to calculate the highest weight ω of Z when therank of G/H is two. The center of L has dimension one and R [ L,L ] ,θ has rankone; moreover, if α is the simple restricted root of R [ L,L ] ,θ , then ( ω, α ∨ ) = 1(see [Ru2], Theorem 2.2). Let ̟ ∨ be a generator of χ ∗ ( Z ( L ) / ( Z ( L ) ∩ H ));suppose also that there exists x ′ := lim t → ̟ ∨ ( t ) · x in the closure of T · x in Z . Then ( ω, ̟ ∨ ) = 1 because ( Z ( L ) / ( Z ( L ) ∩ H )) ∪ x ′ is contained in Z (and Z has a unique isotopic component as Z ( L ) -representation). To determine thesign of ̟ ∨ , we will use the fact that ( λ, ̟ ∨ ) ≥ λ ∈ χ ( S ) ∩ σ ∨ , where σ is the cone associated to the closure of T · x in Z and σ ∨ is its dual cone. In[Ru2] is proved that σ ∨ is equal to W L,θ · C ( X ) ∨ , where W L,θ is the Weyl groupof the restricted root system of L (see the proof of Lemma 2.8 in [Ru2]).In the following of this section, let X be a smooth projective symmetricvariety with Picard number one, over which Aut ( X ) acts non-transitively. Wewill prove in § X has rank 2; for the moment let us assume it. Suppose alsothat Aut ( X ) stabilizes all the closed G -orbits in X . We define the canonicalcompletion of G/H as the simple symmetric variety associated to the cone( − C + , ∅ ). Let e X be the decoloration of X , namely the minimal toroidal varietywith a proper map π : e X → X which extents the identity over G/H : it is thecanonical variety if X is simple and the blow-up of the canonical variety in theclosed orbit otherwise. Moreover, e X is the blow-up of X along the closed orbits(see Theorem 3.3 in [Br2]); in particular, e X is smooth and the G -stable divisorof X is the image of a G -stable divisor of e X . Let G/ e P - be a closed orbit of e X .We claim that, if Aut ( G/ e P - ) = G/Z ( G ), then Lie ( Aut ( X )) is isomorphicto Lie ( G ). The group Aut ( X ) is isomorphic to Aut ( e X ). Indeed Aut ( X ) iscontained in Aut ( e X ) because the closed orbits are stable under the action of Aut ( X ). Moreover, by a result of A. Blanchard, Aut ( e X ) acts on X in such away that the projection is equivariant (see [A2], § . e X is complete and P ic ( e X ) is discrete, the group Aut ( e X ) is linear algebraic and its Lie algebra isthe space of global vector fields, namely H ( X, T e X ).We want to prove that Aut ( e X ) stabilizes all the G -orbits in e X . This factimplies that Aut ( e X ) is reductive by a result of M. Brion (see [Br4], Theorem4.4.1). First, suppose X simple; we will prove in § H is N G ( G θ ). Thus12 X is a wonderful G -variety. Moreover, Aut ( e X ) is semisimple, e X is a wonderful Aut ( X )-variety and the set of colors is D ( G/H ) (see [Br4], Theorem 2.4.2). In[Br4] it is also determined the automorphism group of the wonderful completionof a simple adjoint group G (see [Br4], Example 2.4.5); in particular, it is provedthat Aut ( G ) is G × G if G = P SL (2). Coming back to our problem, thewonderful G -variety e X has two stable prime divisors: the exceptional divisor E and the strict transform e D of the G -stable divisor D of X . The divisor e D corresponds to a simple restricted root (see [dCP1], Lemma 2.2), which is nota dominant weight because the restricted root system is irreducible of rank 2(see Theorem 1.1). Thus e D is fixed by Aut ( e X ) (see [Br4], Theorem 2.4.1), so Aut ( e X ) fixes all the G -orbits in e X .We can proceed similarly in the case where X is non-simple. Indeed, using[Br1], Proposition 3.3, [V1], § H ( e X, O ( F )) = C s F for each G -stable divisor F of e X (here s F isa global section with divisor F ). Since ξ · s F is a scalar multiple of s F for any ξ ∈ H ( e X, T e X ) = Lie ( Aut ◦ ( e X )), F is stabilized by Aut ◦ ( e X ) as in the proof ofTheorem 2.4.1 in [Br4] (see also Proposition 4.1.1 in [BB]).Notice that an element of Z ( G ) acts trivially on G/H (and on e X ) if andonly if it belongs to H , thus Aut ◦ ( e X ) contains G/ ( Z ( G ) ∩ H ). Assume that Aut ( G/ e P - ) is G/Z ( G ), then the restriction ψ : Aut ◦ ( e X ) → Aut ◦ ( G/ e P - ) = G/Z ( G ) is an isogeny, because Aut ( e X ) stabilizes all the G -orbits. Indeed, ker ψ centralizes G/ ( Z ( G ) ∩ H ) (because Aut ( e X ) is reductive) and stabilizes G/H , so it is contained in the finite group
Aut G ( G/H ) ∼ = N G ( H ) /H . Therefore Aut ◦ ( X ) is G/ ( Z ( G ) ∩ H ). Remark 3.1.1
To summarize, given X over which Aut ( X ) acts non-transitively, we have to prove: • Aut ( X ) stabilizes all the closed G -orbits in X ; • X has rank 2; • Aut ( G/ e P - ) = G/Z ( G ) for a closed orbit G/ e P - of e X . If the previous conditions are verified, then
Aut ( X ) is G/ ( Z ( G ) ∩ H ) . In the following of this section we suppose that the previous conditionsare verified and we try to study the full automorphism group
Aut ( X ). Thisgroup permutes the Aut ( X )-orbits; in particular it stabilizes the open orbitand the codimension one orbit. Furthermore, Aut ( X ) stabilizes G/H , thus itis contained in
Aut ( G/H ). We will prove the converse. Notice that
Aut ( X )is contained in Aut ( e X ), because e X is the blow-up of X along a Aut ( X )-stable(eventually non-connected) subvariety. Remark 3.1.2
The automorphism group of e X is isomorphic to the auto-morphism group of G/H . Indeed,
Aut ( e X ) stabilizes G/H , so it is contained in
Aut ( G/H ). Moreover, we can extend every automorphism of
G/H to an auto-morphism of e X , because e X is normal and toroidal (these facts imply that each13non-open) G -orbit O is contained in the closure of an orbit of dimension equalto dim O + 1). To study Aut ( X ), we have to determine the automorphisms of e X which descend to X .Now, we study Aut ( e X ) (and Aut ( G/H )). For any ϕ in Aut ( e X ) we define˜ ϕ ∈ Aut alg ( Aut ◦ ( e X )) ≡ Aut alg ( G ) by˜ ϕ ( g ) · x = ϕ ( g · ϕ − ( x )) , g ∈ Aut ( X ) , x ∈ e X. The kernel of ϕ → ˜ ϕ is the group of equivariant automorphism of e X (and G/H ), thus it is isomorphic to N G ( H ) /H . We will prove that N G ( H ) /H issimple (or trivial); moreover Z ( G/Z ( G ) ∩ H ) is trivial only if N G ( H ) /H is it.Thus ker ( ϕ → ˜ ϕ ) is Z ( G/Z ( G ) ∩ H ) = Z ( G ) / ( Z ( G ) ∩ H ).Let T ′′ be a maximal torus of H , then its centralizer T ′ := C G ( T ′′ ) is amaximal torus of G (see [T2], Lemma 26.2); moreover T ′′ contains a regularone-parameter subgroup λ of G . Thus B ′ := P ( λ ) is a Borel subgroup of G and B ′′ := ( B ′ ∩ H ) is a Borel subgroup of H . The group Aut alg ( G ) is generatedby G := G/Z ( G ) and E = { ψ ∈ Aut alg ( G ) : ψ ( T ′ ) = T ′ and ψ ( B ′ ) = B ′ } ;the intersection of E with G is T ′ := T ′ /Z ( G ), so Aut alg ( G ) is the semidirectproduct of G and E ′ := { ψ ∈ E : ψ ( t ) = t ∀ t ∈ T ′ } . Observe that every ψ ∈ E induces an automorphism of the Dynkin diagram (with respect to T ′ and B ′ ), moreover such automorphism is trivial if and only if ψ belongs to G .Furthermore, Aut ( e X ) is generated by G/ ( Z ( G ) ∩ H ) and by the stabilizer of x in Aut ( e X ). Notice that, given ϕ ∈ Aut x ( e X ), then e ϕ belongs to E , up tocomposing ϕ with an element of H . Remark 3.1.3
Suppose that
Aut G ( e X ) is Z ( G/ ( Z ( G ) ∩ H )); in particularit is contained in Aut ( e X ). Then Aut ( G/H ) is generated by
Aut ( G/H ) = G/ ( Z ( G ) ∩ H ) and by the subgroup K = { ϕ ∈ Aut ( G/H ) : ϕ ( x ) = x and e ϕ ∈ E } . Moreover the map ϕ → ˜ ϕ restricted to K is injective, because Z ( Aut ( G/H ))does not fix x . Observe that any automorphism of G stabilizing H induces anautomorphism of G/H , so K is isomorphic to K ′ := { ψ ∈ E : ψ ( H ) = H } .Besides θ belongs to K (see [T2], Lemma 26.2).Now, we explain how to prove that every automorphism of e X descends to anautomorphism of X . Let ϕ in K , then ϕ ( g e P ′ ) = e ϕ ( g ) e P ′ for every g e P ′ ∈ G/ e P ′ .First, suppose e X (and X ) simple; in particular, ϕ stabilizes the closed orbit of e X . Let x be a point (in the closed orbit of e X ) fixed by B ′ and let e P ′ be thestabilizer of x in G . Then ϕ fixes x , because e ϕ ∈ E . Let G/P ′ be the imageof G/ e P ′ ⊂ e X in X ; we have to prove that e ϕ ( P ′ ) is P ′ . Thus, it is sufficient that ϕ induces the trivial automorphism of the Dynkin diagram (with respect to T ′ and B ′ ).Finally, suppose that X contains two closed orbits. In this case the restrictedroot system has type A , because of Theorem 1.1 and Proposition 3.4. Further-more, the closed orbits are G/P ( ω ) and G/P ( ω ); in particular, they are not G -isomorphic. Let O and O be the inverse images in e X respectively of G/P and G/P ; one can easily show that they are isomorphic. We want to prove that14ny automorphism in K exchange O with O . Let e D be the strict trasform of D and let E , E be the exceptional divisors of e X ; we can suppose that O i is theintersection of e D with E i . Thus it is sufficient to prove that ϕ exchange E with E . For the moment, assume this fact and let x i be a point in O i stabilized by B ′ , let P ′ i be the stabilizer of π ( x i ) and let P ′ be the stabilizer of x (and x ).We have ϕ ( x ) = x , because e ϕ belongs to E . Now, it is sufficient to prove that e ϕ is associated to a non-trivial automorphism of the Dynkin diagram. Indeed,in this case e ϕ ( P ′ ) is a parabolic subgroup of G containing B ′ , distinct by P ′ and with the same dimension of P ′ ; thus it is P ′ . Remark 3.1.4
Summarizing, to prove that
Aut ( X ) coincides with Aut ( G/H )(and with
Aut ( e X )), it is sufficient to show that: • N ( H ) /H is simple or trivial; • if N ( H ) /H is non-trivial, then also Z ( G ) / ( Z ( G ) ∩ H ) is non-trivial; • if X is simple (and non-homogeneous), then R G,θ has type G . Moreover,given ϕ ∈ K then e ϕ induces a trivial automorphism of the Dynkin diagram(associated to T ′ and B ′ ); • if X is non-simple and ϕ ∈ K , then ϕ exchange E with E . Moreover, e ϕ induces a non-trivial automorphism of the Dynkin diagram (associated to T ′ and B ′ ).To study Aut ( X ), we have to determine the group K ⊂ Aut ( G/H ). We willprove that θ is always contained Aut ( X ). If, moreover, G/H is not isomorphicto SL , then K ′ ∩ E ′ is E ′ ; thus the map ϕ → e ϕ is surjective (if G/H = SL ).In following subsections we will study Aut ( X ) by a case-to-case analysis. We begin studying the symmetric varieties of rank 2 which are homogeneous.
Proposition 3.1
The smooth completion of SL /N SL ( S ( L × L )) with Picardnumber one is isomorphic to G (6) .Proof. The symmetric varieties SL /N SL ( S ( L × L )) has type AIII .The group SL acts on the six-dimensional space V ( C ), thus it acts on G ( V C ). The stabilizer of the space generated by e ∧ e and e ∧ e is N SL ( S ( L × L )), thus SL /N SL ( S ( L × L )) is contained in G ( V C ).Moreover SL /N SL ( S ( L × L )) has the same dimension of G ( V C ), so theGrasmannian is the unique smooth completion of SL /N SL ( S ( L × L )) withPicard number one. (cid:3) Proposition 3.2
The smooth completion of E /N E ( F ) with Picard numberone is isomorphic to P ( J ( O )) . roof. The symmetric varieties E /N E ( F ) has type EIV and J ( O ) isthe 27-dimensional irreducible representation of E corresponding to the firstfundamental weight. The subgroup F of E is isomorphic to the group of au-tomorphism of J ( O ); in particular F fixes the identity matrix. Thus P ( J ( O ))contains the 26-dimensional variety G/H . (cid:3) Proposition 3.3
The smooth completion of Sp /N Sp ( Sp × Sp ) with Picardnumber one is isomorphic to E /P ≡ P ( O ) .Proof. The symmetric variety Sp /N Sp ( Sp × Sp ) has type CII . Let( G ′ , σ ) be an involution of type EI , where G ′ is the simply connected simplegroup of type E and G ′ σ has type C . Choose a maximally σ -split torus and aBorel subgroup of G ′ as in § σ acts as − id over R G ′ and the parabolicsubgroup P := P ( ̟ ) of G ′ is opposed to σ ( P ). Furthermore, P ∩ σ ( P ) is a Levisubgroup of P containing P σ ; the derived subgroup of P ∩ σ ( P ) has type D while ( P σ ) has type B × B . Hence G ′ σ /P σ is isomorphic to Sp /N Sp ( Sp × Sp ), up to quotient by a finite group. Notice that G ′ /P is a smooth completionof G ′ σ /P σ with Picard number one. The variety G ′ σ /P σ cannot be isomorphicto Sp / ( Sp × Sp ) because the unique smooth completion of Sp / ( Sp × Sp )with Picard number one is isomorphic to G (8) (see Proposition 4.5 in § (cid:3) Now, we study the symmetric varieties whose restricted root system is re-ducible. If the restricted root system of
G/H has type A , then G/N G ( G θ )is isomorphic to SO n /N SO n ( SO n − ) for an appropriate n ≥
3. If n = 3, thetype of ( G, θ ) is AI ; if n = 4 then G/N G ( G θ ) is isomorphic to P SL ; if n = 6,the type of ( G, θ ) is
AII ; if n = 2 k with k ≥
4, the type of (
G, θ ) is
DII ; if n = 2 k + 1 with k ≥
2, the type of (
G, θ ) is
BII . Proposition 3.4 If R G,θ has type A × A , then G/N G ( G θ ) is isomorphic to SO n /N SO n ( SO n − ) × SO m /N SO m ( SO m − ) and the smooth projective embed-ding of G/H with Picard number one is isomorphic to IG ( n + m ) . First we want to describe H : write ( G, θ ) = ( G , θ ) × ( G , θ ) and let g i ∈ N G i ( G θi ) be a representant of the non-trivial element of N G i ( G θi ) /G θi for each i . Then H is generated by G θ and ( g , g ). Proof.
The group SO n × SO m is contained in SO n + m . Let { e , ..., e n + m } be an orthonormal basis of C n + m such that SO n ⊂ GL ( span C { e , ..., e n } ) and SO m ⊂ GL ( span C { e n +1 , ..., e n + m } ). The connected component of the stabilizerof e + ie n +1 in SO n × SO m is SO n − × SO m − (see also [dCP1], Lemma1.7). Furthermore, SO ( C n + m ) · [ e + ie n +1 ] ⊂ P ( C n + m ) is IG ( n + m ) because e + ie n +1 is an anisotropic vector. (Notice that if R G,θ has type A × A , thenthere is a unique subgroup G θ ⊂ H ⊂ N G ( G θ ) such that G/H has a smoothcompletion with Picard number one). (cid:3) .3 Restricted root system of type G Now, we study the symmetric varieties whose restricted root system has type G .First of all, we determine their connected automorphism group; in particular,we prove that such varieties are non-homogeneous. Lemma 3.2
Suppose that R G,θ has type G . Then the connected automorphismgroup Aut ( X ) is isomorphic to G .Proof of Lemma 3.2. First, we prove that
Aut ◦ ( X ) does not act transitivelyover X . This simple variety is associated to the colored cone ( σ ( α ∨ , − ω ∨ ) , { D α ∨ } ) and its closed orbit is isomorphic to G/P ( − ω ). Let ω be the highestweight of Z (with respect to L ). The simple restricted root of R [ L,L ] ,θ is α ,so ( ω, α ∨ ) = 1 (see § Z ( L ) / ( Z ( L ) ∩ H ) is the one-parametersubgroup of T /T ∩ H corresponding to − ω ∨ ; indeed ( − ω , ω ∨ ) < − ω belongs to the dual cone σ ∨ associated to the closure of T · x in Z . Since Z ( L ) / ( Z ( L ) ∩ H ) ≡ Z ( L ) · x is contained in Z , we have 1 = ( ω, − ω ∨ ) =( aω + ω , − α − α ) = − a −
1, so ω is − ω + ω . Thus Aut ( X ) is isomorphicto G by the Remark 3.1. Observe that the connected automorphism group ofthe closed orbit of e X is G , while the connected automorphism group of theclosed orbit of X is either SO or SO × SO . (cid:3) Proposition 3.5
Suppose that ( G, θ ) has type G (and that G/H is isomorphicto G / ( SL × SL ) ). Then Aut ( X ) has three orbits in X , is connected and is iso-morphic to G . Furthermore, X is intersection of hyperplane sections of G (7) .More precisely, X is the intersection in P ( V V ( ω )) of G (7) with P ( V ( ω )) .Moreover, X is the subvariety of G ( Im O ) parametrizing the subspaces W suchthat C ⊕ W is a subalgebra of O isomorphic to H .Proof of Proposition 3.5. The center of G is trivial, thus Aut ( X ) is con-tained in Aut alg ( G ) (see the observations after the Remark 3.1); such group isconnected, so also Aut ( X ) is connected.Now, we prove that X is ”contained” in G (7). There is an involution of SO that extends θ ; again, we denote it by θ . We have SO θ = S ( O × SO ),thus G / ( SL × SL ) is a closed subvariety of SO /N SO ( S ( O × SO )). Thereis a unique smooth completion of SO /N SO ( S ( O × SO )) with Picard numberone and is isomorphic to G (7) (see Theorem 1.1 and Proposition 4.3). We haveto introduce some notations. Let V be a 7-dimensional vector space and let { e − , e − , e − , e , e , e , e } be a basis of V . Let q be the symmetric bilinearform associated to the quadratic form ( e ∗ ) + e ∗ e ∗− + e ∗ e ∗− + e ∗ e ∗− and let ̟ be the trilinear form e ∗ ∧ e ∗ ∧ e ∗− + e ∗ ∧ e ∗ ∧ e ∗− + e ∗ ∧ e ∗ ∧ e ∗− + 2 e ∗ ∧ e ∗ ∧ e ∗ + 2 e ∗− ∧ e ∗− ∧ e ∗− ∈ V V ∗ . The subgroup G of SL ( V ) composed by thelinear transformations which preserve q and ̟ is the simple group of type G ;moreover, we can realize SO as SO ( V, q ). The vector space V is the standardrepresentation V ( ω ) of G and we can suppose that { e − , e − , e , e , e , e , e } is a basis of weight vector for an appropriate maximal torus T of G . Moreover,17e can choose a Borel subgroup B of G so that the weight of e i is a positive root(respectively is 0) if and only if i > i = 0). The Grassmannian G (7) is contained in P ( V V ) and V V is isomorphic to V ⊕ V (2 ω ) ⊕ C as G -representation. The subrepresentation of V V isomorphic to V has thefollowing basis of T -weights: { e − ∧ e − ∧ e − e ∧ e ∧ e − − e ∧ e ∧ e − , e ∧ e − ∧ e − − e ∧ e ∧ e + e ∧ e − ∧ e − , e ∧ e − ∧ e − − e ∧ e ∧ e − e ∧ e − ∧ e − , e ∧ e ∧ e − e − ∧ e − ∧ e − , e ∧ e ∧ e − − e ∧ e ∧ e − + e ∧ e − ∧ e − , − e ∧ e ∧ e − − e ∧ e ∧ e − − e ∧ e − ∧ e − , − e ∧ e − ∧ e − − e ∧ e ∧ e − e ∧ e − ∧ e − } . Let X ′′ bethe closure of G / ( SL × SL ) in G (7) and let X ′ be the intersection of G (7)with P ( V (2 ω ) ⊕ C ); observe that X ′′ is contained in X ′ because V = V ( ω ) isnot spherical (see [T2], Proposition 26.4). We claim that X is the normalizationof X ′′ . Indeed P ( V (2 ω ) ⊕ C ) has two closed G -orbits: one is isomorphic to G /P ( ω ) and the other one is the point P ( C ). Therefore X ′ contains one closed G -orbit, otherwise G (7) would contain a G -fixed point, in particular V wouldbe reducible as G -representation. Thus, the normalization of X ′′ is a simplevariety with closed orbit G/P ( ω ), so it is X .We want to prove that X ′′ coincides with X ′ and is smooth (so it coincidesalso with X ). Notice that X ′ is connected, because it is G -stable and containsa unique closed G -orbit. Hence, it is sufficient to prove that X ′ has dimension8 and is smooth in a neighborhood of a point belonging to G /P ( ω ). One canverify that e ∧ e − ∧ e − is a highest weight vector of V (2 ω ) ⊂ V V . Let A be the affine open subset of G (7) composed by the subspaces with basis { e j + P a i,j e i } j =1 , − , − i =2 , , , − , where ( a i,j ) varies in C . The subvariety A ∩ P ( V (2 ω ⊕ C )) = A ∩ X ′ of A has equations: a , = − a , + a , ,a , − = − a , + T (2 , , (2 , ,a , − = − a , + T (2 , , (3 , ,a , − = T (2 , , (2 , ,T (1 , , (2 , − T (2 , , (2 , − + T (1 , , (0 , − ,T (1 , , (2 , − T (2 , , (3 , − + T (1 , , (0 , − ,a , − + T (1 , , , (2 , , + T (1 , , (3 , − (where T ( h,k ) , ( n,m ) is the minor of ( a i,j ) extracted by the h -th and k -th row andby n -th and m -th column). The closed subset A ′ of A defined by the the firstfour equation is the graph of a polynomial map, thus it is smooth of dimension8. Hence the last three equation are identically verified on A ′ , because X ′′ hasdimension 8 (and A ∩ X ′′ ⊂ A ∩ X ′ ⊂ A ′ ). Therefore A ∩ X ′ coincides with A ′ and is smooth.Now, we prove the last statement of the proposition. Identify V with Im ( O )and define the associator [ · , · , · ] : V ( O ) → ( O ) as the linear map such that18 a, b, c ] = ( ab ) c − a ( bc ). This map is G -equivariant and has kernel V (2 ω ) ⊕ C .Thus X parametrizes the 3-dimensional subspaces W of Im ( O ) over which [ · , · , · ]is zero. Furthermore, [1 , · , · ] is identically zero. Let W be a subspace associatedto a point in X and let W be the subalgebra of O generated by W and 1. Itcan be either the entire algebra O or a subalgebra of dimension four. But, ifit is the whole algebra, then O is generated by four elements which associatesbetween them; a contradiction (recall that O and W are composition algebras).Thus, W is a composition algebra of dimension four, so it is isomorphic to H . (cid:3) Let ( V , q ) and ( V , q ) be copies respectively of ( V, q ) and ( V, − q ), where V is as in the previous proposition; we can suppose G × G ⊂ SO ( V ) × SO ( V ).Moreover, let W i be a maximal anisotropic subspace of V i for both the i . Let W be a maximal anisotropic subspace of V ⊕ V which contains W ⊕ W . Proposition 3.6
Suppose that
G/H is is isomorphic to the simple group oftype G . Then Aut ( X ) is G × G , while Aut ( X ) is generated by Aut ( X ) together with the flip ( g, h ) → ( h, g ) . In particular, Aut ( X ) has index twoin Aut ( X ) . Furthermore, X is intersection of hyperplane sections of IG (14) :more precisely, X is the intersection in P ( V even ( V ⊕ V ))) ∼ = P (( V ⊗ V ) ⊕ V ⊕ V ⊕ C ) of IG (14) with P (( V ⊗ V ) ⊕ C ) Proof of Proposition 3.6 . The group G has type G × G , G θ is the diagonaland X has dimension 14. The automorphism group of X is determined in [Br4],Example 2.4.5. Alternatively, one can study it in a very similar way to theProposition 3.5.Clearly the involution of G × G can be extended to an involution of SO ( V ) × SO ( V ) which we denote again by θ ; in particular we have G ∼ = G × G/ ( G × G ) θ ⊂ SO ( V ) × SO ( V ) / ( SO ( V ) × SO ( V )) θ ∼ = SO ( V, q ).Let X ′′ be the closure of G in the unique smooth completion of SO ( V, q )with Picard number one. This last variety is isomorphic to the spinorial variety S and the application Φ : SO ( V, q ) ֒ → S sends an element g ∈ SO ( V, q ) tothe graph graph ( g ) := { ( v, gv ) : v ∈ V } ⊂ V ⊕ V ≡ V ⊕ V of g (see alsoProposition 4.2). We claim that X is the normalization of X ′′ (one could showthat X ′′ is normal by [T1], Proposition 9).Let ϕ : S → P ( V even W ) be the Spin -equivariant embedding of S in theprojectivitation of a half-spin representation of Spin . Write V i = W i ⊕ f W i ⊕ C i , V = V ⊕ V = W ⊕ f W where W i , f W i , f W are maximal anisotropic subspacessuch that f W ⊕ f W ⊂ f W . The representation V even W is isomorphic to V • W ⊗ V • W ∼ = V • W ⊗ ( V • W ) ∗ as ( Spin × Spin )-representation (see the highestweights and the dimensions). Moreover V • W i is isomorphic to V i ⊕ C i as G -representation, so V • W ⊗ V • W is isomorphic to ( V ⊗ V ) ⊕ V ⊕ V ⊕ C as G -representation. Let P be the projective subspace of P ( V even W ) isomorphicto P (( V ⊗ V ) ⊕ C ). Observe that X ′′ is contained in X ′ := S ∩ P ⊂ P ( V even W )because V ⊕ V does not contain a line fixed by G θ (see [T2], Proposition 26.4).19bserve that X ′ contains one closed G -orbit. Indeed P contains two closed G -orbits: one isomorphic to G/P ( ω ) and the other one isomorphic to the G -stable point P ( C ). On the other hand, there is not a G -stable maximalisotropic subspace of V , so P ( C ) is not contained in S . Thus X ′ is connected.Moreover, the normalization of X ′′ is the simple symmetric variety with closedorbit G/P ( ω ), so it is X . We want to prove that X ′′ is smooth and coincideswith X ′ ; it is sufficient to prove that X ′ is smooth of dimension 12; in this case X ′ is irreducible, so it coincides with X ′′ (and X ). Moreover, it is sufficient tostudy X ′ in a open neighborhood of an arbitrarily fixed point of G/P ( ω ), forexample x = [ e ∧ e − ∧ e − ∧ e + f ∧ f ∧ f ∧ f − ].Let { e − , e − , e − , e , e , e , e } be a basis of V as before and let { f − , f − , f ,f , f , f , f } be the corresponding basis of V . Let u be e + f ; we can supposethat W is generated by e , e , e , f , f , f and u . The trivial subrepresentation C of V • W is spanned by 2 √− W i + e ∧ e ∧ e ; moreover, W ⊕ V W is contained in the G -stable subspace of V • W isomorphic to V . An openneighborhood of x in S is given by U - · x , where U - is the unipotent radicalof standard parabolic subgroup opposite to Stab
Spin ( x ) (notice that, as al-gebraic variety, U - · x is isomorphic to Lie ( U - ) by the exponential map). Thecoordinates of exp ( p ) · x are the pfaffians of the diagonal minors of p . Let x i,j be the coordinates of the space M of matrices of order 14 with respect to thebasis { e , e , e , u, f , f , f , e − , e − , e − , ( e − f ) , f − , f − , f − } (notice that Lie ( Spin ) ⊂ M ). We claim that a open neighborhood of x in X ′ ∩ ( U - · x )is the graph of a polynomial map. Given an skew-symmetric matrix of order2 n , let [ i , i , ..., i k ] be the Pfaffian of the principal minor extracted from therows and the columns of indices i < i < ... < i k .The three equations corresponding to the vectors 2 √− e i ∧ e j + e i ∧ e j ∧ u ∧ f ∧ f ∧ f ( ∈ V ) are, respectively, 0 = x i,j − [ i, j, , , ,
7] = x i,j − ( x , p i,j + q i,j ), where the p i,j , q i,j are homogeneous polynomials in the x h,k such that : 1) ( h, k ) = (4 , h > k >
3. Finally, we consider theequation 0 = x , − [1 , , ,
4] = x , − ( x , x , − x , x , + x , x , ) associatedto f ∧ u + f ∧ e ∧ e ∧ e ( ∈ V ). Substituting the first three equation in thelast one, we obtain x , (1 + x , f , + x , f , + x , f , ) = g where the f i,j and g are polynomial in the x ,j with j < A of x in U - · x where the previousfour equation became x , = h , , x , = h , , x , = h , and x , = h , (herethe h i,j are polynomials in the coordinates different from x , , x , , x , and x , ). Observe that the previous equation are independent; on the other hand X ′ has dimension at least 12, because it contains X ′′ . Therefore, the subvariety A ′ of A obtained imposing the previous four equation is smooth with dimension12, thus it is equal to A ∩ X ′′ . Hence X ′ = X ′′ (= X ). (cid:3) .4 Restricted root system of type A (with H = G θ ) Now, we consider the symmetric varieties such that: 1) the restricted rootsystem has type A ; 2) H = G θ . We prove that they are hyperplane sectionsof the varieties of the subexceptional serie of subadjoint varieties (these lastvarieties constitute the third line of the geometric version of Freudenthal’s magicsquare and are Legendrian varieties). For the completion of SL this resultis due to Buczy´nski (see [Bu]). These varieties are contained nested in eachother. First, we prove that such varieties are hyperplane sections of Legendrianvarieties. Then, we study their connected automorphism group; in particularwe show that these varieties are not homogeneous. Finally, we study theirautomorphism group.Recall that J ( A ) is the space of Hermitian matrices of order three overthe complex composition algebra A . Moreover SL ( A ) is the subgroup of GL C ( J ( A )) of complex linear transformations preserving the determinant, while SO ( A ) is the subgroup of complex linear transformations preserving the Jordanmultiplication. The space Z ( A ) := C ⊕ J ( A ) ⊕ J ( A ) ∗ ⊕ C ∗ is an irreducible Sp ( A )-representation. The closed Sp ( A )-orbit in P ( Z ( A )) is LG ( A , A ) andis the image of the Sp ( A )-equivariant rational map: φ : P ( C ⊕ J ( A )) P ( Z ( A ))( x, P ) → ( x , x P, xcom P, det P ) . The quotient SL ( A ) /SO ( A ) is a symmetric variety isomorphic to the imageof SL ( A ) · [1 , I ] by φ ; in particular it is contained in the hyperplane section X ′ := { [ x , x , x , x ] ∈ LG ( A , A ) ⊂ P ( C ⊕ J ( A ) ⊕ J ( A ) ∗ ⊕ C ∗ ) : x = x } .Such section is irreducible because it is the image by φ of { [ x, P ] : x − det ( P ) } ;moreover, it has the same dimension of SL ( A ) /SO ( A ), thus it is the closureof SL ( A ) /SO ( A ) in P ( Z ( A )). Lemma 3.3
The variety X ′ is isomorphic to the unique smooth completion X of SL ( A ) /SO ( A ) with Picard number one.Proof of Lemma 3.3 . The variety X ′ and its normalization X ′′ have the samenumber of orbits by [T1], Proposition 1; moreover, each orbit of X ′′ have thesame dimension of its image in X ′ . Observe that no orbits of X ′′ have dimension0; thus X ′ has two closed orbits, namely G/P ( ω ) and G/P ( ω ). One caneasily show that X ′′ has a unique orbit O of codimension one (otherwise, by thetheory of spherical embeddings, X ′ would contain a closed orbit isomorphic to G/P ( ω + ω )). Thus X ′′ = G/H ∪ O ∪
G/P ( ω ) ∪ G/P ( ω ). By Proposition8.2 in [LM], we know the possible dimensions for the singular locus of X ′ .Because it is SL ( A )-stable, one can easily see that X ′ is smooth (and coincideswith X ′′ ). Studying the colored fan of X ′′ , one can show easily that X ′′ hasPicard number one, so it is isomorphic to X . (cid:3) Observe that we have proved the following proposition.21 roposition 3.7
The smooth completion of SL ( A ) /SO ( A ) with Picard num-ber one is contained in the smooth completion of SL ( A ′ ) /SO ( A ′ ) with Picardnumber one and A ′ ⊂ A . Moreover, we have a commutative diagram: SL /SO (cid:31) (cid:127) / / (cid:127) _ (cid:15) (cid:15) SL (cid:31) (cid:127) / / (cid:127) _ (cid:15) (cid:15) SL /Sp (cid:31) (cid:127) / / (cid:127) _ (cid:15) (cid:15) E /F (cid:127) _ (cid:15) (cid:15) LG (6) (cid:31) (cid:127) / / G (6) (cid:31) (cid:127) / / S (cid:31) (cid:127) / / G ( O ) Lemma 3.4
Suppose that R G,θ has type A and that H = G θ . Then the con-nected automorphism group Aut ( X ) is isomorphic to G , up to isogeny.Proof of Lemma 3.4. The character group of χ ( S ) is the lattice generatedby spherical weights. Let X ′ be the simple variety corresponding to the coloredcone ( σ ( α ∨ , − ω ∨ − ω ∨ ) , { D α ∨ } ); its closed orbit is isomorphic to G/P ( ω ). Let Z be as § ω = a ω + a ω be the highest weight of Z (with respectto L ). The simple restricted root of R [ L,L ] ,θ is α , so a = ( ω, α ∨ ) = 1 (see § Z ( L ) / ( Z ( L ) ∩ H ) is the one-parameter subgroup of T /T ∩ H corresponding to − ω ∨ ; indeed ( − ω , ω ∨ ) < − ω belongs to the dual cone σ ∨ associated to the closure of T · x in Z . Since Z ( L ) / ( Z ( L ) ∩ H ) ≡ Z ( L ) · x is contained in Z , we have 1 = ( ω, − ω ∨ ) = − − a , so ω is ω − ω . We canstudy he simple variety corresponding to the colored cone ( σ ( α ∨ , − ω ∨ − ω ∨ ) , { D α ∨ } ) in a similar way. Thus Aut ( X ) is isomorphic to G by the Remark 3.1. (cid:3) Proposition 3.8
Suppose that
G/H is SL /SO (and the type of ( G, θ ) is AI ). Then X is an hyperplane section of LG (6) . The full automorphism group Aut ( X ) is generated by SL ( = Aut ( X ) ) and by θ ; moreover Aut ( X ) has threeorbits in X . Proposition 3.9 (see [Bu], Theorem 1.4) Suppose that
G/H is SL . Then X is an hyperplane section of G (6) . Moreover Aut ( X ) is isomorphic to thequotient of SL × SL by the intersection of the center with the diagonal. Thefull automorphism group Aut ( X ) is generated by Aut ( X ) , by θ and by ( ϕ, ϕ ) where ϕ is the automorphism of SL corresponding to the automorphism ofthe Dynkin diagram; in particular Aut ( X ) has index 4 in Aut ( X ) . Moreover Aut ( X ) has three orbits in X . Proposition 3.10
Suppose that
G/H is SL /Sp (and the involution’s type is AII ). Then X is an hyperplane section of S . The full automorphism group Aut ( X ) is generated by SL / {± id } ( = Aut ( X ) ) and by θ ; thus Aut ( X ) hasindex 2 in Aut ( X ) . Moreover Aut ( X ) has three orbits in X . Proposition 3.11
Suppose that
G/H is E /F (and the involution’s type is AII ). Then X is an hyperplane section of E /P ≡ LG ( A , A ) . Moreover Aut ( X ) is isomorphic to the simply-connected group of type E , while the fullautomorphism group Aut ( X ) is generated by E and by θ ; in particular Aut ( X ) has index 2 in Aut ( X ) . Besides Aut ( X ) has three orbits in X . X . By the Lemma 3.4, we knowthat Lie ( Aut ( X )) is Lie ( G ). Notice that the center of G acts no-trivially over X . Moreover, if G is different from SL × SL and SL , then the center of G is a simple group, thus Aut ◦ ( X ) is G . If G is SL × SL , then the center is C × C , where C n is the group of n -th roots of the unit. Thus the intersectionof the center of G with G θ is the diagonal of C × C . If G is SL then the centeris C and its intersection with Sp (= G θ ) is {± id } . Observe that N G ( H ) /H issimple because the fundamental group of R G,θ is A .Now, we determine K (see § K ′ of E . Let ̟ be the longest element of W and let ̟ be the longest elementof the Weyl group of R G (here we consider the root system of G with respectto T ). Then θ ′ := ωω θ fixes T and B . One can easily show that θ ′ exchange ω with ω (see [T2], table 5.9 and [dCP1], § E with E (recall that the restriction of the valuation of E i to C ( X ) ( B ) / C ∗ is − ω i ).Hence, also θ exchanges the previous two divisors; in particular θ exchangesthe two closed orbits of e X . Notice that θ belongs to K ′ and induces the non-trivial automorphism of the Dynkin diagram of G , with respect to T ′ and B ′ (see page 493 in [LM] and §
26 in [T2]). If G is different from SL × SL , then E/T ′ contains exactly two elements. Thus K ′ /N T ′ ( H ) coincides with E/ T ′ . Inparticular, Aut ( G/H ) is generated by
Aut ( G/H ) and θ . Moreover, Aut ( G/H )coincides with
Aut ( X ) by Remark 3.1.Finally, suppose G = SL × SL . Let ˙ T be a maximal torus of SL , let˙ B be a Borel subgroup of SL and let ϕ be the equivariant automorphism of SL associated to the non-trivial automorphism of the Dynkin diagram (withrespect to ˙ T and ˙ B ). We can set T = T ′ = ˙ T × ˙ T , B = ˙ B × ˙ B - and B ′ = ˙ B × ˙ B .One can easily see that K ′ is generated by N T ′ ( H ), θ and ( ϕ, ϕ ). (Notice that E/T ′ has eight elements: id , θ , ( ϕ, id ), ( id, ϕ ), ( ϕ, ϕ ), θ ◦ ( ϕ, ϕ ), θ ◦ ( ϕ, id ) and θ ◦ ( id, ϕ )). Notice that ( ϕ, ϕ ) stabilizes both B and B - . We have ω i = ̟ i − ̟ i where { ̟ j , ̟ j } are the fundamental weights of the j -th copy of SL in G (withrespect to ˙ T and ˙ B ). Thus ( ϕ, ϕ ) exchange ω with ω , so ( ϕ, ϕ ) exchange E with E . Furthermore, ( ϕ, ϕ ) induces a non-trivial automorphism of the Dynkindiagram of G (with respect to T ′ and B ′ ). Therefore, Aut ( X ) coincides with Aut ( G/H ) by Remark 3.1.
In the following we always suppose that the rank of
G/H is at least two.
Inthis section we consider the symmetric varieties which belongs to an infinitefamily; in particular, we consider all the symmetric varieties of rank at leastthree. We consider also completions of the following varieties of rank two:1) the symmetric variety
P SL ; 2) the symmetric variety P SL /P SO oftype AI ; 3) the symmetric variety P SL /P Sp of type AII ; 4) the sym-metric variety SO ; 5) the symmetric variety Spin ≡ Sp ; 6) the symmet-23ic varieties SO n /N SO n ( SO × SO m − ) of type BI and DI ; 7) the symmet-ric varieties Sp n /N Sp n ( Sp × Sp n − ) of type CII ; 8) the symmetric variety SO /N SO ( GL ) of type DIII .Given a linear endomorphism ϕ of a vector space V , let graph ( ϕ ) be thesubspace { ( v, ϕ ( v )) : v ∈ V } of V ⊕ V . Observe that graph ( ϕ ) has the samedimension of V . If we have fixed a (skew-)symmetric bilinear form on V , thenwe define a (skew-)symmetric bilinear q ′ form on V ⊕ V such that q ′ ( v, w ) = q ( v ) − q ( w ) for each ( v, w ) ∈ V ⊕ V . A l Now, we consider the symmetric varieties such that: 1) H = N G ( G θ ); 2) therestricted root system has type A l (with l ≥ Proposition 4.1
Let X be a smooth projective symmetric variety with Picardnumber one and rank at least two. Suppose that H = N G ( G θ ) and that therestricted root system has type A l , then X is isomorphic to the projectivizationof an irreducible G -representation.Proof. It is sufficient to show an irreducible spherical representation withdimension equal to dim
G/H + 1. We have already considered the case of E /N E ( F ) in Proposition 3.2.1) If G/H is isomorphic to
P GL l +1 , then X is isomorphic to P ( M l +1 ) as( P GL l +1 × P GL l +1 )-variety (here M l +1 is the space of matrices of order l + 1).Indeed P GL l +1 × P GL l +1 acts on M l +1 and the stabilizer of C is the diagonal,namely G θ . Moreover P ( M l +1 ) has dimension equal to l + l .2) If G/H is isomorphic to SL l +1 /N SL l +1 ( SO l +1 ) and ( G, θ ) has type AI ,then X is isomorphic to P ( Sym ( C l +1 )) (see [CM], Theorem 2.3 or [T2], Propo-sition 26.4).3) If G/H is isomorphic to SL l +1 /N SL l +1 ( Sp l +1 ) and ( G, θ ) has type
AII ,then X is isomorphic to P ( V l − ( C l )). (cid:3) B l or D l Now we consider the symmetric varieties whose restricted root system has type D l or B l . In the second case we suppose also H = N G ( G θ ). Explicitly, weconsider the following cases: 1) the symmetric varieties SO n ; 2) the symmetricvarieties SO n / ( SO l × SO n − l ) of type BI and DI . We consider also the symmet-ric variety SO /GL of type DIII because it is isomorphic to SO / ( SO × SO ). Proposition 4.2
The smooth projective embedding of SO n with Picard numberone is isomorphic to S n .Proof. We have an inclusion of SO n into IG n (2 n ) given by the map g → graph ( ϕ ). It is easy to show that this map is compatible with the action of SO n × SO n on SO n and IG n (2 n ). Thus it is sufficient to observe that SO n and IG n (2 n ) have the same dimension. (cid:3) roposition 4.3 The smooth projective embedding of SO n / ( SO l × SO n − l ) withPicard number one is isomorphic to G l ( n ) .Proof. Let W be a l -dimensional subspace of C n over which the quadraticform is nondegenerate. We denote by W ⊥ the orthogonal complement of W .The stabilizer of W contains SO ( W ) × SO ( W ⊥ ). Thus G l ( n ) is a smoothcompletion with Picard number one of the symmetric variety SO n /Stab SO n ( W )(see [dCP1], Lemma 1.7). This fact implies, by Theorem 1.1, that H is SO ( W ) × SO ( W ⊥ ) (notice that G/H cannot be
Spin n / ( Spin × Spin n − ) because thislast variety is Hermitian and its restricted root system has type B ). (cid:3) Proposition 4.4
The smooth projective embedding of SO /GL with Picardnumber one is isomorphic to G (8) .Proof. Observe that SO /GL is isomorphic to SO /S ( O × O ). Indeed theinvolution of SO of type DIII (and rank two) is conjugated by an equivariantautomorphism of SO to the involution of SO of type DI and (and rank two).Indeed, these varieties have the same Satake diagrams up to an automorphism ofthe Dynkin diagram of SO . Now, it is sufficient to observe that a homogeneoussymmetric variety has at most one smooth completion with Picard number one. (cid:3) BC l or C l Now, we consider the symmetric varieties whose restricted root system hastype BC l or C l . We have the following two cases: 1) the symmetric vari-eties Sp n / ( Sp l × Sp n − l ) of type CII ; 2) the symmetric varieties Sp l . Weconsider also Spin , because it is isomorphic to Sp . Proposition 4.5
The smooth projective embedding of Sp n /Sp l × Sp n − l withPicard number one is isomorphic to G l (2 n ) .Proof. Let W be a 2 l -dimensional subspace of C n over which the standardbilinear skew-symmetric form is nondegenerate and let W ⊥ be its orthogonalcomplement. The Sp n -orbit of W in G l (2 n ) is isomorphic to Sp n /Sp ( W ) × Sp ( W ⊥ ) (see [dCP1], Lemma 1.7), thus is dense in G l (2 n ). (cid:3) Proposition 4.6
The smooth projective embedding of Sp (2 l ) with Picard num-ber one is isomorphic to IG l (4 l ) .Proof. We have an inclusion of Sp (2 l ) into IG l (4 l ) given by the map g → graph ( ϕ ). It is easy to show that this map is compatible with the action of Sp (2 l ) × Sp (2 l ) over Sp (2 l ) and IG l (4 l ). Furthermore, Sp (2 l ) and IG l (4 l )have the same dimension. (cid:3) Proposition 4.7
The smooth projective embedding of
Spin with Picard num-ber one is isomorphic to LG (8) .Proof. Observe that
Spin × Spin is isomorphic to Sp × Sp . (cid:3) cknowledgments We would like to thank M. Brion for the continued support and for many veryhelpful suggestions. I would also thank L. Manivel for many enlightening sug-gestion. The author has been partially supported by C.N.R.S.’S grant in col-laboration with the Liegrits.
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