On some smooth projective two-orbits varieties with Picard number 1

0.34 MB·English

by  Boris Pasquier

On some smooth projective two-orbits varieties with
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0 Picard number 1
0
2

n
a Boris Pasquier
J

3
February 2, 2008
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]
G
Abstract
A
.
h We classify all smooth projective horospherical varieties with Picard number 1.
t
a We prove that the automorphism group of any such variety X acts with at most
m two orbits and that this group still acts with only two orbits on X blown up at the
[ closed orbit. We characterize all smooth projective two-orbits varieties with Picard

1 number 1 that satisfy this latter property.
v
4 Mathematics Subject Classification. 14J45 14L30 14M17
3
5
3 Keywords. Two-orbits varieties, horospherical varieties, spherical varieties.
.
1
0
8 Introduction
0
:
v
i Horospherical varieties are complex normal algebraic varieties where a connected complex
X
reductive algebraic group acts with an open orbit isomorphic to a torus bundle over a flag
r
a variety. The dimension of the torus is called the rank of the variety. Toric varieties and
flag varieties are the first examples of horospherical varieties (see [12] for more examples
and background).
It is well known that the only smooth projective toric varieties with Picard number 1
are the projective spaces. This is not the case for horospherical varieties: for example any
flag variety G/P with P a maximal parabolic subgroup of G is smooth, projective and
horospherical with Picard number 1.
Moreover, smooth projective horospherical varieties with Picard number 1 are not nec-
essarily homogeneous. For example, let ω be a skew-form of maximal rank on C2m+1. For
i 1,...,m , define the odd symplectic grassmannian Gr (i,2m + 1) as the variety of
ω
∈ { }
i-dimensional ω-isotropic subspaces of C2m+1. Odd symplectic grassmannians are horo-
spherical varieties (see Proposition 1.12) and, for i = m they have two orbits under the
6
1
action of their automorphism group which is a connected non-reductive linear algebraic
group (see [11] for more details).
OurfocusonsmoothhorosphericalvarietieswithPicardnumber1,thatgivesinteresting
examples of Fano varieties with Picard number 1, is also motivated by the main result of
[12], where Fano horospherical varieties are classified in terms of rational polytopes. Indeed
in [12, Th.0.1], the degree (i.e. the self-intersection number of the anticanonical bundle)
of smooth Fano horospherical varieties is bounded. Two different bounds are obtained in
the case of Picard number 1 and in the case of higher Picard number.
In Section 1, we classify all smooth projective horospherical varieties with Picard num-
ber 1. More precisely, we prove the following result.
Theorem 0.1. Let G be a connectedreductive algebraicgroup. Let X be a smooth projective
horospherical G-variety with Picard number 1.
Then we have the following alternative:
(i) X is homogeneous, or
(ii) X is horospherical of rank 1. Its automorphism group is a connected non-reductive
linear algebraic group, acting with exactly two orbits.
Moreover in the second case, X is uniquely determined by its two closed G-orbits Y and
Z, isomorphic to G/P and G/P respectively; and (G,P ,P ) is one of the triples of the
Y Z Y Z
following list.
1. (B ,P(ω ),P(ω )) with m 3
m m−1 m
≥
2. (B ,P(ω ),P(ω ))
3 1 3
3. (C ,P(ω ),P(ω )) with m 2 and i 1,...,m 1
m i i+1
≥ ∈ { − }
4. (F ,P(ω ),P(ω ))
4 2 3
5. (G ,P(ω ),P(ω ))
2 2 1
Here we denote by P(ω ) the maximal parabolic subgroup of G corresponding to the domi-
i
nant weight ω with the notation of Bourbaki [3].
i
Remark that Case 3 of Theorem 0.1 corresponds to odd symplectic grassmannians.
It would be natural to investigate other complete smooth spherical varieties with Picard
number 1 (A normal variety is spherical if it admits a dense orbit of a Borel subgroup,
for example horospherical varieties and symmetric varieties are spherical). A classification
hasbeen recently given inthespecial case of projective symmetric varieties byA Ruzzi[14].
2
In the second part of this paper, we focus on another special feature of the non-
homogeneous varieties classified by Theorem 0.1: the fact that they have two orbits even
when they are blown up at their closed orbit. Two-orbits varieties (i.e. normal vari-
eties where a linear algebraic group acts with two orbits) have already be studied by
D. Akhiezer and S. Cupit-Foutou. In [1], D. Akhiezer classified those whose closed orbit
is of codimension 1 and proved in particular that they are horospherical when the group
is not semi-simple. In [7], S. Cupit-Foutou classified two-orbits varieties when the group
is semi-simple, and she also proved that they are spherical. In section 2, we define two
smooth projective two-orbits varieties X1 and X2 with Picard number one (see Definitions
2.11 and 2.12) and we prove the following:
Theorem 0.2. Let X be a smooth projective variety with Picard number 1 and put G :=
Aut0(X).
Assume that X has two orbits under the action of G and denote by Z the closed orbit.
Then the codimension of Z is at least 2.
Assume furthermore that the blow-up of Z in X still has two orbits under the action of
G. Then, one of the following happens:
G is not semi-simple and X is one of the two-orbits varieties classified by Theorem
•
0.1;
G = F4 and X = X1;
•
G = G2 PSL(2) and X = X2.
• ×
The varieties in Theorem 0.2 are spherical of rank one [5]. Remark also that odd sym-
plectic grassmannians have been studied in detail by I.A. Mihai in [11]. In particular, he
proved that an odd symplectic grassmannian is a linear section of a grassmannian [11,
Prop.2.3.15]. It could be interesting to obtain a similar description also for the varieties of
Theorem 0.2.
The paper is organized as follows.
In Section 1.1, we recall some results on horospherical homogeneous spaces and horo-
spherical varieties, which we will use throughout Section 1. In particular we briefly sum-
marize the Luna-Vust theory [10] in the case of horospherical homogeneous spaces.
In Section 1.2, we prove that any horospherical homogeneous space admits at most
one smooth equivariant compactification with Picard number 1. Then we give the list of
horospherical homogeneous spaces that admit a smooth compactification not isomorphic
to a projective space and with Picard number 1. We obtain a list of 8 cases (Theorem 1.7).
In Section 1.3, we prove that in 3 of these cases, the smooth compactification is homo-
geneous (under the action of a larger group).
3
InSection1.4,westudythe5remainingcases(theyarelistedinTheorem0.1). Wecom-
pute the automorphism group of the corresponding smooth compactification with Picard
number 1. We prove that this variety has two orbits under the action of its automorphism
group and that the latter is connected and not reductive.
In Section 1.5, we prove Theorem 0.2 in the case where the automorphism group is not
semi-simple. This gives another characterization of the varieties obtained in Section 1.4.
The aim of Section 2 is to prove Theorem 0.2 when G is semi-simple.
Definition 0.3. AprojectiveG-varietyX satisfies(*),ifitissmoothwithPicardnumber1,
has two orbits under the action of G such that its closed orbit Z has codimension at least 2,
and the blowing-up of X along Z has also two orbits under the action of G.
In Section 2.1, we prove the first part of Theorem 0.2 and we reveal two general cases.
In Sections 2.2 and 2.3, we study these two cases respectively. First, we reformulate
part of the classification of two-orbits varieties with closed orbit of codimension one due
to D. N. Akhiezer, in order to give a complete and precise list of possible cases. Then
we study separately all these possible cases. We prove that the two varieties X1 and X2
satisfy (*) and are non-homogeneous, and that in all other cases, the varieties satisfying
(*) are homogeneous.
1 Smooth projective horospherical varieties with Pi-
card number 1
1.1 Notation
Let G be a reductive and connected algebraic group over C, let B be a Borel subgroup of
G, let T be a maximal torus of B and let U be the unipotent radical of B. Denote by C
the center of G and by G′ the semi-simple part of G (so that G = C.G′). Denote by S
the set of simple roots of (G,B,T), and by Λ (respectively Λ+) the group of characters of
B (respectively the set of dominant characters). Denote by W the Weyl group of (G,T)
and, when I S, denote by W the subgroup of W generated by the reflections associated
I
⊂
to the simple roots of I. If α is a simple root, we denote by αˇ its coroot, and by ω the
α
fundamental weight corresponding to α (when the roots are α ,...α , we will write ω
1 n i
instead of ω ). Denote by P(ω ) the maximal parabolic subgroup containing B such that
αi α
ω is a character of P(ω ). Let Γ be the Dynkin diagram of G. When I S, we denote by
α α
⊂
Γ the full subgraph of Γ with vertices the elements of I. For λ Λ+, we denote by V(λ)
I
∈
the irreducible G-module of highest weight λ and by v a highest weight vector of V(λ). If
λ
G is simple, we index the simple roots as in [3].
4
A closed subgroup H of G is said to be horospherical if it contains the unipotent radical
of a Borel subgroup of G. In that case we also say that the homogeneous space G/H is
horospherical. Up to conjugation, one can assume that H contains U. Denote by P the
normalizer N (H) of H in G. Then P is a parabolic subgroup of G such that P/H is a
G
torus. Thus G/H is a torus bundle over the flag variety G/P. The dimension of the torus
is called the rank of G/H and denoted by n.
A normal variety X with an action of G is said to be a horospherical variety if G has
an open orbit isomorphic to G/H for some horospherical subgroup H. In that case, X is
also said to be a G/H-embedding. The classification of G/H-embeddings (due to D. Luna
et Th. Vust [10] in the more general situation of spherical homogeneous spaces) is detailed
in [12, Chap.1].
Let us summarize here the principal points of this theory. Let G/H be a fixed horo-
spherical homogeneous space of rank n. This defines a set of simple roots
I := α S ω is not a character of P
α
{ ∈ | }
where P is the unique parabolic subgroup associated to H as above. We also introduce a
lattice M of rank n as the sublattice of Λ consisting of all characters χ of P such that the
restriction of χ to H is trivial. Denote by N the dual lattice to M.
In this paper, we call colors the elements of S I. For any color α, we denote by αˇ
M
\
the element of N defined as the restriction to M of the coroot αˇ : Λ Z. The point αˇ
M
−→
is called the image of the color α. See [12, Chap.1] to understand the link between colors
and the geometry of G/H.
Definition 1.1. A colored cone of NR := N Z R is an ordered pair ( , ) where is a
⊗ C F C
convex cone of NR and is a set of colors (called the set of colors of the colored cone),
F
such that
(i) is generated by finitely many elements of N and contains the image of the colors of
C
,
F
(ii) does not contain any line and the image of any color of is not zero.
C F
One defines a colored fan as a set of colored cones such that any two of them intersect
in a common colored face (see [12, def.1.14] for the precise definition).
Then G/H-embeddings are classified in terms of colored fans. Define a simple G/H-
embedding of X as one containing a unique closed G-orbit. Let X be a G/H-embedding
and F its colored fan. Then X is covered by its simple subembeddings, and each of them
corresponds to a colored cone of the colored fan of X. (See [6] or [9] for the general theory
of spherical embeddings.)
5
1.2 Classification of smooth projective embeddings with Picard
number 1
The Picard number ρ of a smooth projective G/H-embedding X satisfies
X
ρ = r +♯(S I) ♯( )
X X X
\ − D
where denotes the set of simple roots in S I which correspond to colors of F and r
X X
D \
is the number of rays of the colored fan of X minus the rank n [12, (4.5.1)]. Since X is
projective, its colored fan is complete (i.e. it covers NR) and hence rX 1. Moreover
≥
S I, so ρ = 1 if and only if r = 1 and = S I. In particular the colored fan
X X X X
D ⊂ \ D \
of X has exactly n+1 rays.
Lemma 1.2. Let G/H be a horospherical homogeneous space. Up to isomorphism of
varieties, there exists at most one smooth projective G/H-embedding with Picard number 1.
Proof. Let X and X′ be two smooth projective G/H-embeddings with respective colored
fansFandF′ andbothwithPicardnumber1. Denotebye ,...,e theprimitiveelements
1 n+1
of the n+1 rays of F. By the smoothness criterion of [12, Chap.2], (e ,...,e ) is a basis
1 n
of N, e = e e and the images in N of the colors are distinct and contained
n+1 1 n
− −···−
in e ,...,e . The same happens for F′. Then there exists an automorphism φ of the
1 n+1
{ }
lattice N which stabilizes the imageof each color and satisfies F = φ(F′). Thus the varieties
X and X′ are isomorphic [12, Prop. 3.10].
If it exists, we denote by X1 the unique smooth projective G/H-embedding with Picard
number 1 and we say that G/H is “special”.
Remark 1.3. By the preceding proof, we have ♯( ) n+1.
X
D ≤
1.2.1 Projective space
We first give a necessary condition for the embedding X1 of a special homogeneous space
not to be isomorphic to a projective space. In particular we must have n = 1, so that X1
has three orbits under the action of G: two closed ones and G/H.
Theorem 1.4. Let G/H be a “special” homogeneous space. Then X1 is isomorphic to a
projective space in the following cases:
(i) ♯( X1) n,
D ≤
(ii) n 2,
≥
(iii) n = 1, ♯( X1) = 2 and the two simple roots of X1 are not in the same connected
D D
component of the Dynkin diagram Γ.
6
Proof. (i) In that case, there exists a maximal colored cone of the colored fan of X which
contains all colors. Then the corresponding simple G/H-embedding of X1, whose closed
orbitisapoint[12,Lem.2.8],isaffine[9,th.3.1]andsmooth. Soitisnecessarilyahorospher-
ical G-module V [12, Lem.2.10]. Thus P(C V) is a smooth projective G/H-embedding
⊕
with Picard number 1. Then by Lemma 1.2, X1 is isomorphic to P(C V).
⊕
(ii) We may assume that ♯( X1) = n+1. Denote by α1,...,αn+1 the elements of S I
D \
and by Γ the Dynkin diagram Γ . The smoothness criterion of horospherical varieties
i S\{αi}
[12, Chap.2] applied to X1 tells us two things.
Firstly, foralli 1,...,n+1 and forall j = i, α isa simple end (”simple” means not
j
∈ { } 6 j
adjacent to a double edge) of a connected component Γ of Γ of type A or C . Moreover
i i m m
j
the Γ are distinct, in other words, each connected component of Γ has at most one vertex
i i
among the (α ) .
i i∈{1,...,n+1}
Secondly, (αˇ ) is a basis of N and αˇ = αˇ αˇ . Thus a basis
iM i∈{1,...,n} (n+1)M 1M nM
− −···−
of M (dual of N) is of the form
(ω ω +χ )
i n+1 i i∈{1,...,n}
−
where χ is a character of the center C of G, for all i 1,...,n .
i
∈ { }
Let us prove that a connected component of Γ contains at most one vertex among the
(α ) . Suppose the contrary: there exist i,j 1,...,n + 1 , i = j such that
i i∈{1,...,n+1}
∈ { } 6
α and α are vertices of a connected component of Γ. One can choose i and j such that
i j
there is no vertex among the (α ) between α and α . Since n 2, there exists
k k∈{1,...,n+1} i j
≥
an integer k 1,...,n + 1 different from i and j. Then we observe that Γ does not
k
∈ { }
satisfy the condition that each of its connected component has at most one vertex among
the (α ) (because Γi = Γj).
i i∈{1,...,n+1} k k
Thus we have proved that
n+1
Γ = G Γj (1.4.1)
j=0
such that for all j 1,...,n+1 , Γj is a connected component of Γ of type A or C in
m m
∈ { }
which α is a simple end.
j
For all λ Λ+, denote by by V(λ) the simple G-module of weight λ. Then Equation
∈
1.4.1 tells us that the projective space
P(V(ω ) V(ω +χ ) V(ω +χ ))
n+1 1 1 n n
⊕ ⊕···⊕
is a smooth projective G/H-embedding with Picard number 1. Thus X1 is isomorphic to
this projective space.
(iii) As in case (ii), one checks that X1 is isomorphic to P(V(ω ) V(ω +χ )) for some
2 1 1
⊕
character χ of C.
1
7
1.2.2 When X1 is not isomorphic to a projective space
According to Theorem 1.4 we have to consider the case where the rank of G/H is 1 and
where there are two colors corresponding to simple roots α and β in the same connected
component of Γ. As we have seen in the proof of Theorem 1.4, the lattice M (here of
rank 1) is generated by ω ω +χ where χ is a character of the center C of G. Moreover,
α β
−
H is the kernel of the character ω ω +χ : P(ω ) P(ω ) C∗.
α β α β
− ∩ −→
We may further reduce to the case where G is semi-simple (recall that G′ denotes the
semi-simple part of G).
Proposition 1.5. Let H′ = G′ H. Then G/H is isomorphic to G′/H′.
∩
Proof. We are going to prove that G/H and G′/H′ are both isomorphic to a horospherical
homogeneous space under (G′ C∗). In fact G/H is isomorphic to (G′ P/H)/H˜ [12,
× ×
Proof of Prop.3.10], where
H˜ = (g,pH) G′ P/H gp H .
{ ∈ × | ∈ }
Similarly, G′/H′ is isomorphic to (G′ P′/H′)/H˜′ where P′ = P G′ and H˜′ defined as
the same way as H˜. Moreover the mor×phisms ∩
P/H C∗ and P′/H′ C∗
−→ −→
pH (ω ω +χ)(p) p′H′ (ω ω )(p′)
α β α β
7−→ − 7−→ −
are isomorphisms. Then
H˜ = (p′,c) P′ C∗ (ω ω +χ)(p′) = c−1
α β
{ ∈ × | − }
= (p′,c) P′ C∗ (ω ω )(p′) = c−1
α β
{ ∈ × | − }
= H˜′.
This completes the proof.
Remark 1.6. In fact P/H C∗ acts on G/H by right multiplication, so it acts on the
≃
C∗-bundle G/H G/P by multiplication on fibers. Moreover, this action extends to X1
−→
(where C∗ acts trivially on the two closed G-orbits).
So we may assume that G is semi-simple. Let G ,...,G the simple normal subgroups
1 k
of G, so that G is the quotient of the product G G by a central finite group C .
1 k 0
We can suppose that C is trivial, because G/H ×·G˜··/×H˜ where G˜ = G G and
0 1 k
≃ × ···×
H˜ is the preimage of H in G˜. If α and β are simple roots of the connected component
corresponding to G , denote by H is the kernel of the character ω ω of the parabolic
i i α β
−
subgroup P(ω ) P(ω ) of G . Then
α β i
∩
H = G G H G G
1 i−1 i i+1 k
×···× × × ×···×
and G/H = G /H .
i i
So from now on, without loss of generality, we suppose that G is simple.
8
Theorem 1.7. With the assumptions above, G/H is ”special” if and only if (Γ,α,β)
appears in the following list (up to exchanging α and β).
1. (A ,α ,α ), with m 2;
m 1 m
≥
2. (A ,α ,α ), with m 3 and i 1,...,m 1
m i i+1
≥ ∈ { − }
3. (B ,α ,α ), with m 3;
m m−1 m
≥
4. (B ,α ,α )
3 1 3
5. (C ,α ,α ) with m 2 and i 1,...,m 1
m i+1 i
≥ ∈ { − }
6. (D ,α ,α ), with m 4;
m m−1 m
≥
7. (F ,α ,α )
4 2 3
8. (G ,α ,α )
2 2 1
Proof. The Dynkin diagrams Γ and Γ are respectively of type A or C by the
S\{α} S\{β} m m
smoothness criterion [12, Chap.2]. And for the same reason, α and β are simple ends of
Γ and Γ respectively.
S\{β} S\{α}
Suppose Γ is of type A . If α equals α then, looking at Γ , we remark that β must
m 1 S\{α}
be α or α . So we are in Case 1 or 2. If α equals α the argument is similar. Now if α
2 m m
is not an end of Γ, in other words if α = α for some i 2,...,m 1 then, looking at
i
∈ { − }
Γ , we see that β can be α , α , α or α . The cases where β equals α or α are
S\{α} 1 i−1 i+1 m 1 m
already done and the case where β equals α or α is Case 2.
i−1 i+1
The study of the remaining cases is analogous and left to the reader.
In the next two sections we are going to study the variety X1 for each case of this
theorem. In particular we will see that X1 is never isomorphic to a projective space.
1.3 Homogeneous varieties
In this section, with the notation of Section 1.2.2, we are going to prove that X1 is homo-
geneous in Cases 1, 2 and 6.
In all cases (1 to 8), there are exactly 4 projective G/H-embeddings and they are all
smooth; they correspond to the 4 colored fans consisting of the two half-lines of R, without
color, with one of the two colors and with the two colors, respectively (see [12, Ex.1.19] for
a similar example).
Let us realize X1 in a projective space as follows. The homogeneous space G/H is
isomorphic to the orbit of the point [v + v ] in P(V(ω ) V(ω )), where v and v
ωβ ωα β ⊕ α ωα ωβ
9
are highest weight vectors of V(ω ) and V(ω ) respectively. Then X1 is the closure of this
α β
orbit in P(V(ω ) V(ω )), because both have the same colored cone (i.e. that with two
β α
⊕
colors)1.
We will describe the other G/H-embeddings in the proof of Lemma 1.17.
Proposition 1.8. In Case 1, X1 is isomorphic to the quadric Q2m = SO /P(ω ).
2m+2 1
Proof. Here, the fundamental G-modules V(ω ) and V(ω ) are the simple SL -modules
α β m+1
Cm+1 and its dual (Cm+1)∗, respectively. Let denote by Q the quadratic form on Cm+1
⊕
(Cm+1)∗ defined by Q(u,u∗) = u∗,u . Then Q is invariant under the action of SL .
m+1
h i
Moreover Q(v +v ) = 0, so that X1 is a subvariety of the quadric (Q = 0) in P(Cm+1
ωα ωβ ⊕
(Cm+1)∗) = P(C2m+2).
We complete the proof by computing the dimension of X1:
dim X1 = dim G/H = 1+dim G/P = 1+♯(R+ R+) (1.8.1)
\ I
where R+ is the set of positive roots of (G,B) and R+ is the set of positive roots generated
I
by simple roots of I. So dim X1 = dim Q2m = 2m and X1 = Q2m.
Proposition 1.9. In Case 2, X1 is isomorphic to the grassmannian Gr(i+1,m+2).
Proof. The fundamental SL -modules are exactly the
m+1
V(ω ) = iCm+1
i V
and a highest weight vector of V(ω ) is e e where e ,...,e is a basis of Cm+1.
i 1 i 1 m+1
∧···∧
We have
X (cid:31)(cid:127) // P( iCm+1 i+1Cm+1)
OO V ⊕OO V
(cid:31)? (cid:31)?
G/H G.[e e +e e ]
1 i 1 i+1
∧···∧ ∧···∧
Complete (e ,...,e ) to obtain a basis (e ,...,e ) of Cm+2, then the morphism
1 m+1 0 m+1
iCm+1 i+1Cm+1 i+1Cm+2
V V V
⊕ −→
x+y x e +y
0
7−→ ∧
is an isomorphism. Then X1 is a subvariety of the grassmannian
Gr(i+1,m+2) SL .[e e (e +e )] P( i+1Cm+1).
m+2 1 i 0 i+1 V
≃ ∧···∧ ∧ ⊂
We conclude by proving that they have the same dimension using Formula 1.8.1.
1
See [12, Chap.1] for the construction of the colored fan of a G/H- embedding.
10