The pseudo-index of horospherical Fano varieties
Boris Pasquier
∗September 24, 2009
Abstract
We prove a conjecture of L. Bonavero, C. Casagrande, O. Debarre and S. Druel, on the pseudo-index of smooth Fano varieties, in the special case of horospherical varieties.
Mathematics Subject Classification. 14J45 14L30 52B20
Keywords. Horospherical varieties, Picard number, Pseudo-index, Fano varieties.
Let X be a normal, complex, projective algebraic variety of dimension d. Assume that X is Fano, namely the anticanonical divisor −KX is Cartier (in other words, X is Gorenstein) and ample. The pseudo-indexιX is the positive integer defined by
ιX:= min{−KX·C|C rational curve inX}.
The aim of this paper is to prove the following result.
Theorem 1. Let X be a Q-factorial horospherical Fano variety of dimension d, Picard number ρX, and pseudo-indexιX. Then
(ιX−1)ρX ≤d.
Moreover, equality holds if and only if X is isomorphic to(PιX−1)ρX.
This inequality has been conjectured by L. Bonavero, C. Casagrande, O. Debarre and S. Druel for all smooth Fano varieties [1]. Moreover, they have proved Theorem 1 in the case of Fano varieties of dimension 3 and 4, all flag varieties, and also for some particular toric varieties. More generally, this result has been also proved by C. Casagrande for allQ-factorial toric Fano varieties [4]. Theorem 1 generalizes these results to the family of horospherical varieties which contains in particular the families of toric varieties and flag varieties.
We will use arguments similar to the ones used by C. Casagrande, and we will use also a few results on horospherical varieties from [8]. So, let us summarize the theory of Fano horospherical varieties (see [8]
or [7] for more details).
LetGbe a connected algebraic group overC. A homogeneous space is said to behorosphericalif it is a torus bundle over a flag variety. Then ahorospherical varietyis aG/H-embedding(i.e., a normalG-variety
∗Boris PASQUIER, Universit´e Montpellier 2, CC 051, Place Eug`ene Bataillon, 34095 Montpellier cedex, France. Haus- dorff Center for Mathematics, Universit¨at Bonn, Villa Maria, Endenicher Allee 62, 53115 Bonn, Germany. E-mail:
boris.pasquier@math.univ-montp2.fr
with an openG-orbit isomorphic to G/H) of a horospherical homogeneous spaceG/H. The dimension of the torus is called therank of the variety and is denoted byn. Toric varieties (whereG= (C∗)n) and flag varieties (wheren= 0) are the first examples of horospherical varieties.
Let G/H be a horospherical homogeneous space. Denote by P the normalizer of H in G. It is a parabolic subgroup ofG. There exists a Borel subgroupB of Gcontained in P such that H contains the unipotent radical ofB. We fix such a Borel subgroup. We denote bySthe set of simple roots, by (ωα)α∈S
the fundamental weights, and by (P(ωα))α∈S the associated maximal parabolic subgroups ofGcontaining B. Then the application from the set of subsets of S to the set of parabolic subgroups of Gcontaining B, defined byJ ⊂S 7−→ PJ :=∩α∈S\JP(ωα), is a bijection. We denote by I the subset ofS such that P =PI.
Let us define a latticeM as the set of characters ofP whose restrictions toH are trivial. LetN be the dual lattice ofM. LetMR:=M⊗ZRandNR:=N⊗ZR.
For allα∈S\I, we denote by ˇαM the restriction of the coroot ˇαtoM. We also defineaα:=h2ρP,αiˇ where 2ρP is the sum of all the positive roots not generated by the simple roots inI. (Remark thataα≥2.) Let X be a complete G/H-embedding. Denote by (Xi)i∈{1,...,m} the irreducible G-stable divisor of X and by (Dα)α∈S\I the irreducible B-stable but not G-stable divisors of X (in factDα:=Bw0sαP/H wherew0 is the longest element of the Weyl group andsαthe simple reflection associated withα). Then
−KX =Pm
i=1Xi+P
α∈S\IaαDα. We denote byl(−KX) the piecewise linear function associated to−KX
(see for example [2] for piecewise linear functions associated to divisors on spherical varieties). And we defineQ(X) as
Q(X) :={u∈NR|l(−KX)(u)≤1}.
WhenX is Fano,Q(X) is a G/H-reflexive polytope in the following sense.
Definition 2. LetG/H be a homogeneous horospherical space. A convex polytopeQinNRis said to be G/H-reflexiveif the following three conditions are satisfied:
(1) The vertices ofQare in N∪ {αˇaM
α |α∈S\I}, and the interiorQo ofQcontains 0.
(2) Q∗:={v∈MR| ∀u∈Q,hv, ui ≥ −1}is a lattice polytope (i.e., its vertices are in M).
(3) For allα∈S\I, αˇaM
α ∈Q.
Moreover, we have the following.
Proposition 3. Let G/H be a horospherical homogeneous space. Then the application X 7−→ Q(X) is a bijection between the set of isomorphism classes of Fano G/H-embeddings and the set ofG/H-reflexive polytopes.
There is also a classification ofG/H-embeddings in terms of colored fans, due to D. Luna and T. Vust [6]. Thecolored fanof a FanoG/H-embeddingX is the set of couples (CF,DF), whereCF is the cone of NRgenerated by a faceF ofQ(X) andDF :={α∈S\S| αˇaM
α ∈F}.
From now on, let us fix a horospherical homogeneous space G/H, a Fano G/H-embedding X, and the associated G/H-reflexive polytope Q. We define the colors of X to be the rootsα∈ S\I such that
ˇ αM
aα ∈Q\Q. Denote byo DX the set of colors ofX.
Moreover,XisQ-factorial if and only ifQis simplicial and the points αˇaM
α ∈Q\
o
Q, α∈ DXare disjoint vertices ofQ. From now on, suppose thatX isQ-factorial. Then the Picard number ofX is given by
ρX=r+](S\I)−](DX) (1)
wheren+ris the number of vertices ofQ.a And for all verticesuofQ, we can now define an integerau
to beaαif there existsα∈S\Isuch thatu= αˇaM
α, and 1 otherwise.
Let us denote by V(Q) andV(Q∗) the set of vertices ofQ andQ∗ respectively. For each v ∈V(Q∗), the set Fv := {u∈ Q | hv, ui = −1} is a facet of Q. Moreover the application v 7−→ Fv is a bijection between V(Q∗) and the set of facets of Q(and likewise between V(Q) and the set of facets of Q∗). By Luna-Vust theory,V(Q∗) (i.e., the set of facets ofQ) is in bijection with the set of closedG-orbits ofX. Moreover, in our case, we can describe explicitly the closedG-orbits ofX as follows. Letv ∈V(Q∗) and Jv :={α∈S\I| αˇaM
α ∈Fv}.Then theG-orbit associated withv is isomorphic toG/PI∪Jv. Let us now define
Q:= min{au(1 +hv, ui)|u∈V(Q), v∈V(Q∗), u6∈Fv}.
The pseudo-index ofX is related toQ as follows.
Proposition 4. Let X and Qas above. Then, (i) ιX≤Q.
(ii) ∀α∈(S\I)\DX,ιX≤aα.
To prove this result we need the following lemma.
Lemma 5. Let u∈V(Q)andv∈V(Q∗)be such thatQ =au(1 +hv, ui). Thenuis adjacent to Fv (i.e., Fv and some facet containing uintersect along an(n−2)-dimensional face).
Proof. Letv∈V(Q∗). Lete1, . . . , en be the vertices of the complexFv. For allj∈ {1, . . . , n}, denote by Fj the facet ofQ, distinct fromFv, containinge1, . . . , ej−1, ej+1, . . . , en and denote byuj the vertex ofFj distinct frome1, . . . , en.
Let (e∗1, . . . , e∗n) be the dual basis of (e1, . . . , en) inMR. Letj∈ {1, . . . , n}. Thenhe∗j, uji 6= 0 (otherwise, uj is in the hyperplane generated by the (ei)i6=j) and one can define
γj= −1− hv, uji he∗j, uji .
Moreover, the vertex ofQ∗ associated toFj isvj=v+γje∗j. Thenγj >0 becausehvj, eji>−1.
Letv∈V(Q∗) andu∈V(Q) be such thatQ =au(1 +hv, ui). Then au(1 +hvj, ui) =au(1 +hv, ui) +auγjhe∗j, ui, so that
he∗j, ui<0⇐⇒ hvj, ui=−1⇐⇒u∈Fj, or, in other words:
u6∈Fj ⇐⇒ he∗j, ui ≥0.
Now, ifu6=uj for allj ∈ {1, . . . , n}, thenhe∗j, ui ≥0 for all j ∈ {1, . . . , n} and thenuis in the cone generated bye1, . . . , en, which is a contradiction. So,uis one of the uj, i.e., uis adjacent toFv.
We need also to define two different families of rational curves inX. The first family is very similar to the family of the curves stable under the action of the torus in the case of toric varieties. And the second one is a family of Schubert curves in the closedG-orbits of X.
aNote that, as in the toric case, the Picard number of aQ-factorial horospherical Fano variety is bounded by twice its dimension [8, Theorem 1.2].
Denote byx0the point ofX with isotropy groupH. Recall thatP/H'(C∗)n. Then the closureX0 of theP-orbit ofx0 in X is a toric variety associated to the fan consisting of the cones (CFv)v∈Q∗ and their faces, i.e., the fan ofX without colors.
Letµbe an (n−2)-dimensional face ofQ(or, which is the same an (n−1)-dimensional colored cone of the colored fan ofX, or also an (n−1)-dimensional cone of the fan ofX0). Then there exists a uniqueP- stable curve ofX0⊂X containing twoP-fixed points lying in the two closedG-orbits ofX corresponding to the two facets ofQcontainingµ. This curve is rational; we denoted it byCµ.
Letv∈V(Q∗) andα∈(S\I)\DX such that ˇαM ∈ CFv. Recall thatvcorresponds to a closedG-orbitY ofX. Moreover, sinceαis not a color ofX,Y =G/PY, wherePY is a parabolic subgroup ofGcontaining B such thatωαis a character ofPY. Then we can define the rationalB-stable curveCα,v in X to be the Schubert curveBsαPY/PY wheresαis the simple reflection associated to the simple rootα.
It is not difficult to check that the curve Cµ is the one defined in [3, Prop.3.3]. Moreover, the curve Cα,v is the one defined in [3, Prop.3.6]. Indeed,Cα,v is contracted by the extremal contraction described in [3, Ch.3.4], which consists in ”adding” the color α. In fact, the curves Cµ and Cα,v are exactly the irreducibleB-stable curves in X and they generate the cone of effective 1-cycles.
Now, using the definition of the polytopeQand the intersection formulas given in [3, Ch.3.2], we prove the following lemma.
Lemma 6. (i) Letµ an(n−2)-dimensional face of Q, and choosev∈V(Q∗)andu∈V(Q) such that Fv and some facet ofQcontaininguintersect alongµ. Letχµ the primitive element ofM such that hχµ, xi= 0 for allx∈µandhχµ, ui>0. Then we have
−KX·Cµ=au(1 +hv, ui) hχµ, auui .
(ii) Let v∈V(Q∗)andα∈(S\I)\DX be such thatαˇM ∈ CFv. Then we have
−KX·Cα,v =aα+hv,αˇMi.
Proof. In both cases, let us consider and explain the formulas given in [3, Ch.3.2].
(i) The formula we will use is the following (with the notations of [3]):
δ·Cµ= l(δ)|µ+−l(δ)|µ−
χµ .
In our case, let us choose µ+ to be the cone generated by the facet ofQ containing uand µ, and µ− to be the cone generated by the facet Fv. The divisorδis−KX so that, by the definition of the polytopesQandQ∗, we have
l(δ)|µ+:x7−→ −hv0, xi and l(δ)|µ− :x7−→ −hv, xi, where v0 is the vertex of Q∗ such thatµ+ is the cone generated byFv0.
Now we have
l(δ)|µ+−l(δ)|µ−
χµ (u) =1 +hv, ui hχµ, ui . (ii) Here we use the following formula of [3]:
δ·CD,Y =nD−vD(lY).
In our case,δ=−KX,D=Dα,nD=aα,Y corresponds tovandvD(lY) =l(δ)|Fv( ˇαM) =−hv,αˇMi, which induces the wanted formula.
We are now able to prove Proposition 4.
Proof of Proposition 4. (i) Letu ∈V(Q) and v ∈ V(Q∗) be such that Q =au(1 +hv, ui). By Lemma 5, uis adjacent to Fv. This means that u lies in a facet F of Q such thatF and Fv intersect along an (n−2)-dimensional faceµ. Then, by Lemma 6, we have
−KX·Cµ= au(1 +hv, ui) hχµ, auui , whereχµ∈M andhχµ, auuiis a positive integer. Then we have
ιX ≤ −KX·Cµ≤au(1 +hv, ui) =Q.
(ii) Letα∈(S\I)\DX. Let us choosev∈V(Q∗) such that ˇαM ∈ CFv. Then, by Lemma 6, ιX ≤ −KX·Cα,v=aα+hv,αˇMi ≤aα
becausehv,αˇMiis non-positive by the choice ofv.
Remark 7. In (ii), if ˇαM 6= 0, we have in fact the strict inequalityιX< aα.
Proof of the inequality of Theorem 1. Since the origin lies in the interior ofQ∗, there exists an equation
m1v1+· · ·+mhvh= 0 (2)
where h >0,v1, . . . , vh are vertices ofQ∗, andm1, . . . , mh are positive integers. Let I :={1, . . . , h} and M :=P
i∈Imi. For anyu∈V(Q), we define
A(u) ={i∈I| hvi, ui=−1}.
Now, we have
0 = X
i∈I
mihvi, ui
= − X
i∈A(u)
mi+ X
i6∈A(u)
mihvi, ui
≥ − X
i∈A(u)
mi+ X
i6∈A(u)
mi(Q
au −1) = (Q
au −1)M−Q
au X
i∈A(u)
mi (3)
so that
Q−au
Q
M ≤ X
i∈A(u)
mi.
Denote byrthe positive integer such that the number of vertices ofQisn+r. Let us sum the preceding inequalities over all verticesuofQ. We obtain
(n+r)M− P
u∈V(Q)au
Q
M ≤ X
u∈V(Q)
X
i∈A(u)
mi=nM and Qr≤ X
u∈V(Q)
au.
(The last equality comes from the fact that X is Q-factorial, so that each facet of Q has exactly n vertices.)
Then, by Proposition 4 (i), we have
ιXr≤Qr≤ X
u∈V(Q)
au. (4)
But recall thatρX=r+](S\I)−](DX), hence X
u∈V(Q)
au= X
α∈DX
aα+ρX+n−](S\I).
Thus, we have
ιX(ρX−](S\I) +](DX))≤ X
α∈DX
aα+ρX+n−](S\I) and, by Proposition 4, we obtain that
ρX(ιX−1) ≤ X
α∈DX
aα+ X
α∈(S\I)\DX
ιX−](S\I) +n
≤ X
α∈DX
aα+ X
α∈(S\I)\DX
aα−](S\I) +n= X
α∈S\I
(aα−1) +n. (5)
This yields the inequality of Theorem 1 by [8, Lemmma 4.8] which asserts that X
α∈S\I
(aα−1) +n≤d. (6)
Let us study now the cases of equality. In view of the above proof, one can remark that (ιX−1)ρX=d if and only if equality holds in inequalities 3, 4, 5 and 6, i.e., all of the following conditions are satisfied:
(a) ∀u∈V(Q),∀i∈ {1, . . . , h},au(1 +hvi, ui) equals either 0 orQ; (b) ιX=Q;
(c) ∀α∈(S\I)\DX,ιX =aα; (d) P
α∈S\I(aα−1) +n=d.
Let us now interpret these conditions.
Lemma 8. (i) Condition (a) is equivalent to
(a’) ∀u∈V(Q),∀v∈V(Q∗),au(1 +hv, ui)equals either 0 or Q. (ii) Conditions (a’) and (b) imply thatX is locally factorial.
(iii) Condition (c) implies that every simple rootα∈S\I such thatαˇM 6= 0 is a color ofX. (iv) Condition (d) is equivalent to
(d’) The decomposition Γ = tγi=1Γi into connected components of the Dynkin diagram Γ of G satisfies: ∀i∈ {1, . . . , γ}, either no vertex ofΓi corresponds to a simple root ofS\I or only one vertex ofΓi corresponds to a simple rootαi of S\I. In the latter case,Γi is of typeAm orCm, andαi corresponds to a simple end ofΓi (”simple” means not adjacent to a double edge).
(v) Conditions (a’), (b) and (d’) imply that X is smooth.
Proof. (i) Just choose Equation 2 such that it involves all vertices ofQ∗.
(ii) Let us suppose thatX is not locally factorial (butQ-factorial). Then there exists a facet ofQwith vertices u1, . . . , un such that (au1u1, . . . , aunun) is a basis of NRbut not ofN (just compare locally factoriality andQ-factoriality criteria given respectively in [8, Propositions 4.4 and 4.7]). This means that there exist a non-zero element u0 = Pn
i=1λiauiui of N such that ∀i ∈ {1, . . . , n}, λi ∈ Q, 0≤λi ≤1 and somej satisfies 0< λj <1. Letµbe the wall generated byu1, . . . , uj−1, uj+1, . . . , un and let χµ be the associated element of M as in the proof of Proposition 4 (i) (with χµ(uj)>0).
Thenχµ(aujuj) = λ1
jχµ(u0)>1 becauseχµ(u0)∈Z>0. This implies, by the proof of Proposition 4 (i) together with Condition (a’), that ιX< Q.
(iii) It is Remark 7.
(iv) Without loss of generality, we assume that Γ is connected and we also suppose that S\I6=∅.
Let us consider [8, Lemma 4.8], where the inequality P
α∈S\I(aα−1) +n≤dis proved in 4 steps.
Then we have to look at equality cases in Steps 2, 3 and 4.
In Step 2, we reduced to the case where the subdiagram ΓI of Γ, with edges the simple roots inI and all possible arrows, is connected. One can check that equality occurs in that step if and only if ΓI is already connected.
In Step 3, we reduced to the case where S\I consists of one point. Here also, it is not difficult to check that equality occurs if and only ifS\I already consists of a point.
In Step 4, we computed explicitly both terms of the desired inequality in all possible cases. Then, there are only two cases where equality holds:
Γ is of typeAm+1 and ΓI is of typeAmwithm≥0;
Γ is of typeCm+1and ΓI is of typeCmwithm≥1.
(v) By (ii)X is locally factorial, and the smoothness criterion of horospherical varieties [7, Theorem 2.6]
then tells us that Condition (d’) implies thatX is smooth.
We may now complete the proof of Theorem 1.
Proof of the case of equality of Theorem 1. Step 1: we can suppose thatGis the direct product of a semi- simple groupG0 and a torus (for example, one can replaceGby the product of the semi-simple part ofG by the torusP/H, see [8, proof of Proposition 3.5]). Then one can also suppose thatG0 is a direct product of simple groups and a torus. Then, because of Condition (d’), we have
X =X0× Y
α∈S\I,ˇαM=0
G/P(ωα),
where X0 is a horospherical variety. Moreover, for allα∈S\I such that ˇαM = 0, the varietyG/P(ωα) is isomorphic to Paα−1 by condition (d’), and hence to PιX−1 by Condition (c). Also, X0 has the same combinatorial data asX, except for the colors with zero image.
Then we have to prove the result in the case where there is no α∈ S\I such that ˇαM = 0. In that case, we haveDX=S\I.
Step 2: Let v be a vertex of Q∗ and let e1, . . . , en be the vertices ofFv (which satisfy hv, eii =−1).
Let us denote byf1, . . . , fr the remaining vertices ofQ. LetK:={1, . . . , n} andJ :={1, . . . , r}. For any
k∈K, the face of Qwith verticese1, . . . , ek−1, ek+1, . . . , en lies on exactly two facets, which are Fv and another one we denote byFk. Thus, there exists a unique φ(k)∈J such that fφ(k)∈Fk. This defines a functionφ:K−→J.
Let (e∗1, . . . , e∗n) the dual basis of (e1, . . . , en) inMR. We havev =−e∗1− · · · −e∗n. Fixk∈K and let vk be the vertex ofQ∗ corresponding to the facetFk ofQ. We have, by Condition (a’),
∀i∈K\{k}, aei(1 +hvk, eii) = 0 and aek(1 +hvk, eki) =Q, so thatvk =v+aQ
eke∗k.
Now for any j∈J, using condition (a’) foru=fj, we have he∗k, fji= aek
Qhvk−v, fji=
( −aaek
fj ifφ(k) =j, 0 otherwise.
And then, for anyj∈J,
afjfj+ X
k∈φ−1(j)
aekek= 0. (7)
Let us defineMj to be the intersection ofM with the supspace ofMRgenerated by{e∗k |k∈φ−1({j})}.
Then M =L
j∈JMj (remark that Mj 6={0} because of Condition (a’) and Lemma 5). By Step 1 and Condition (d’), we can assume that G=G1× · · · ×Gr and Mj =M ∩ΛGj where ΛGj is the character group of a maximal torus ofGj. ThenG/H is isomorphic toG1/H1× · · · ×Gr/Hrfor some horospherical subgroups Hj of Gj. And, by Equation 7, the colored fan of X is the same as the colored fan of the productX1× · · · ×Xr whereXj is theGj/Hj-embedding associated to the reflexive polytope defined as the convex hull offj and allek such thatφ(k) =j.
Moreover, for all i∈ {1, . . . , r}, with formula 1, we compute that the Picard number ofXiis 1. But a smooth projective horospherical varietyXwith Picard number one such that two simple roots inDX ⊂S\I are not in the same connected component of the Dynkin diagram ofG, is a projective space [9, Theorem 1.4]. This completes the proof of Theorem 1.
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