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ON SIMPLICITY AND STABILITY OF TANGENT BUNDLES OF RATIONAL HOMOGENEOUS VARIETIES 9 0 ADA BORALEVI 0 2 n a J Abstract. GivenarationalhomogeneousvarietyG/P whereGiscom- 5 plex simple and of type ADE, we prove that all tangent bundles TG/P 1 aresimple,meaningthattheironlyendomorphismsarescalarmultiples oftheidentity. ThisresultcombinedwithHitchin-Kobayashicorrespon- ] dence implies stability of thesetangent bundleswith respect to the an- G ticanonical polarization. Our main tool is the equivalence of categories A between homogeneous vector bundles on G/P and finite dimensional . representations of a given quiverwith relations. h t a m 1. Introduction [ 1 In [Ram67] Ramanan proved that irreducible homogeneous bundles on v rational homogeneous varieties are stable, and hence in particular simple. 0 If the underlying variety is Hermitian symmetric then this result applies to 5 3 tangent bundles. For the general case, the Hitchin-Kobayashi correspon- 2 dence gives a weaker result for the tangent bundles, polystability. In this . 1 paper we show that in fact tangent bundles of any G/P are simple, where G 0 is complex, simpleand of typeADE. Simplicity and polystability combined 9 give stability. 0 : Our main tool is the equivalence of categories between homogeneous bun- v dles on G/P and finite dimensional representations of a given quiver with i X relations. Once the machinery of this equivalence of categories is set up, the r simplicity of tangent bundles turns out to be an immediate and surprisingly a easy consequence of it. Indeed one only needs to look at endomorphisms of the bundle as endomorphisms of the associated quiver representations. Homogeneous vector bundles have been classically studied using another equivalence of categories, namely that between homogeneous bundles on G/P and finite dimensional representations of the parabolic subgroup P. In [BK90] Bondal and Kapranov had the idea of associating to any ratio- nalhomogeneousvariety aquiverwithrelations. Byputtingtheappropriate Date: 2009/01/10. 2000 Mathematics Subject Classification. 14F05,14M17,14D20,16G20. Key words and phrases. Homogeneous Vector Bundle, Simplicity, Stability, Quiver Representation. 1 relations one gets the aforementioned equivalence of categories between G- homogeneous vector bundles on G/P and finite dimensional representations of the quiver. The relations were later refined by Hille in [Hil94]. In [ACGP03] A´lvarez-Co´nsul and Garc´ıa-Prada gave an equivalent con- struction, while in [OR06] Ottaviani and Rubei used the quiver for comput- ing cohomology, obtaining a generalization of the well-known Borel-Weil- Bott theorem holding on Hermitian symmetric varieties of ADE type. We describe both equivalence of categories and give details on the quiver, its relations and its representations in Section 2 and 3. Section 4 and 5 contain results on simplicity and stability. We use the quiver to prove that homogeneous vector bundles whose associated quiver representation has a particular configuration—we call such bundles multi- plicity free—are weakly simple, which means that their only G-invariant endomorphisms are scalar multiple of the identity. Our result holds on any G/P, where G is a simple group of type ADE: Proposition A. Let E be a multiplicity free homogeneous vector bundle of rank r on G/P. Let k be the number of connected components of the quiver . Then H0(EndE)C = Ck. In particular if is connected, E E Q| Q| then E is weakly simple. It turns out that all tangent bundles T are multiplicity free and con- G/P nected, and that moreover the isotypical component H0(EndT )C co- G/P incides with the whole space H0(EndT ), or in other words that these G/P bundles are simple. Theorem B. Let T the tangent bundle on a rational homogeneous G/P variety G/P, where G is a complex simple Lie group of type ADE and P one of its parabolic subgroups. Then T is simple. G/P If algebraic geometry, representation theory and quiver representations give us simplicity, for stability differential geometry also joins the team. A homogeneous variety G/P is in fact also a Ka¨hler manifold, and as such it admits a Hermite-Einstein structure. In virtue of the Hitchin-Kobayashi correspondence this is equivalent to the polistability of its tangent bundle. This together with Theorem B gives: Theorem C. Let T the tangent bundle on a rational homogeneous G/P variety G/P, where G is a complex simple Lie group of type ADE, and P one of its parabolic subgroups. Then T is stable with respect to the anti- G/P canonical polarization K induced by the Hermite-Einstein structure. G/P − 2 In the case where G/P is a flag manifold point-hyperplane in Pn, we obtained a complete understanding of stability of the tangent bundle with respect to different polarizations: Proposition D. Let F = F(0,n 1,n) be the flag manifold of type − SL /P(α ,α ), and set: n+1 1 n n+n√4n2+4n 3 m(n) = − − . 2(n2+n 1) − Then the tangent bundle TF is stable with respect to the polarization F(a,b) O if and only if it is semistable if and only if m(n)a b m(n)−1a. ≤ ≤ We also show similar computations for SL /B. 4 InthelastSection 6wedealwith modulispaces. We quoteandgeneralize the results from [OR06], where the authors showed that King’s notion of semistability [Kin94] for a representation [E] of the quiver is in fact G/P Q equivalent to the Mumford-Takemoto semistability of the associated bundle E on G/P, when the latter is a Hermitian symmetric variety. We can thus construct moduli spaces of G-homogeneous semistable bundles with fixed grE on any homogenous variety G/P of ADE type. Acknowledgements. This paper is part of my PhD thesis. I am very grateful to my advisor Professor Giorgio Ottaviani for the patience with which he followed this work very closely and for always transmitting melots of encouragement and mathematical enthusiasm. I would also like to thank ProfessorJean-PierreDemailly forinvitingmetoGrenobleandforhiswarm hospitality. 2. Preliminaries 2.1. Notations and first fundamental equivalence of categories. Let G be a complex semisimple Lie group. We make a choice ∆ = α ,...,α 1 n { } of simple roots of g = LieG, and we call Φ+ (respectively Φ−) the set of positive (negative) roots. We denote by h g the Cartan subalgebra so ⊂ that g decomposes as: g = h g g . α α ⊕ ⊕ αM∈Φ+ αM∈Φ− A parabolic subgroup P G is a subgroup P = P(Σ), where: ≤ Lie(P(Σ)) = h g g , ⊕ α⊕ α α∈Φ+ α∈Φ−(Σ) M M for a subset Σ ∆ that induces Φ−(Σ)= α Φ− α = p α . ⊂ { ∈ | αi∈/Σ i i} If Σ= ∆, then P(∆)= B is the Borel subgroup. P A rational homogeneous variety is a quotient G/P. 3 Avector bundleE onG/P is called (G)-homogeneous ifthereisanaction of G on E such that the following diagram commutes: G E // E × (cid:15)(cid:15) (cid:15)(cid:15) G G/P // G/P × where the bottom row is just the natural action of G on the cosets G/P. Note that the tangent bundle T on any rational homogeneous variety G/P G/P is obviously a G-homogeneous bundle. The category of G-homogeneous vector bundle on G/P is equivalent to thecategory P-modofrepresentations ofP, andalsotothecategory p-mod, where p = LieP, see for example [BK90]. More in detail, the group G is a principal bundle over G/P with fiber P. Any G-homogeneous vector bundle E of rank r is induced by this principal bundleviaarepresentationρ: P GL(r). WedenoteE = E . Indeed,E of ρ → rank r over G/P is homogeneous if and only if there exists a representation ρ : P GL(r) such that E E , and this entails the aforementioned ρ → ≃ equivalence of categories. ForanyweightλwedenotebyE thehomogeneousbundlecorresponding λ totheirreduciblerepresentationofP withmaximalweightλ. Hereλbelongs to the fundamental Weyl chamber of the reductive part of P. Indeed, P decomposes as P = R N into a reductive part R and a nilpotent part N. · At the level of Lie algebras this decomposition entails a splitting p = r n, ⊕ with the obvious notation r= LieR and n = LieN. Moreover from a result by Ise [Ise60] we learn that a representation of p is completely reducible if and only if it is trivial on n, hence it is completely determined by its restriction to r. Thewell-known Borel-Weil-Bott theorem [Bot57] computes the cohomology ofsuchE ’sbyusingpurelyLiealgebratools, namelytheaction oftheWeyl λ groupontheweightλ. Inparticularthetheoremstates thatifλisdominant then H0(E ) = Σ (the irreducible representation of highest weight λ) and λ λ all higher cohomology vanishes. 3. The quiver G/P Q 3.1. Definition of the quiver and its representations. Other G/P Q than looking at homogeneous bundles as P-modules, it is useful to try a different point of view and look at these same bundles as representation of a given quiver with relations. For basics on quiver theory we refer the reader to [DW05] or [Kin94]. To any rational homogeneous variety G/P we can associate a quiver with relations, thatwedenoteby . Theideaistoexploit alltheinformation G/P Q given by the choice of the parabolic subgroup P, with its decomposition P = R N. · 4 Let Λ be the fundamental Weyl chamber of G, and let Λ+ be the Weyl chamber of the reductive part R. Then we can give the following: Definition 3.1. Let G/P be any rational homogeneous variety. The quiver is constructed as follows. The set of vertices is: G/P Q = Λ+ = λ λ dominant weight for R . 0 Q { | } There is an arrow connecting the vertices λ and µ if and only if the vector space Hom(n E ,E )G is non-zero. λ µ ⊗ Remark 3.1. Definition3.1ispreciselytheoriginaloneofBondalandKapra- nov [BK90], later usedalso by Alvarez-Co´nsul andGarc´ıa-Prada [ACGP03]. Arrows correspond to weights of the nilpotent algebra n, considered as an r-module with the adjoint action. In fact one could obtain an equivalent theory by considering the same ver- tices with a smaller number of arrows, i.e. by taking only weights of the quotient n/[n,n]. This is for example the choice made by Hille (see [Hil98] and [Hil96]). Note thatvertices λ of correspondtoirreduciblehomogeneous bun- G/P Q dles E on G/P. λ The relations on the quiver will be defined in Section 3.2. G/P Q Let now E be an homogeneous vector bundle over G/P: we want to associate to it a representation [E] of the quiver . The bundle E G/P Q comes with a filtration: (3.1) 0 E E ... E = E, 1 2 k ⊂ ⊂ ⊂ ⊂ where each E /E is completely reducible. Of course the filtration does i i−1 not split in general. We define grE = E /E for any filtration (3.1). i i i−1 ⊕ The graded bundlegrE does not depend on the filtration: in fact it is given by looking at our p-module E as a module over r, so that it decomposes as a direct sum of irreducibles: (3.2) grE = E V , λ λ ⊗ λ M with multiplicity spaces V Ck, where k Z is the number of times E λ ≥0 λ ≃ ∈ occurs. The representation [E] associates to the vertex λ of the quiver G/P Q precisely the multiplicity space V in the decomposition (3.2). λ Goingonwiththedefinitionoftherepresentation[E],givenanyλ,µ 0 ∈ Q such that there is an arrow λ µ , we now need to define a linear map 1 → ∈ Q V V . This information is given by the nilpotent part n. More precisely, λ µ → it is given by the natural action of n over grE, both viewed as r-modules: θ :n grE grE. ⊗ → The morphism θ encodes all the information we need, including that on the relations of the quiver. 5 Obviously if we have a vector bundleE we then have the graded grE and the morphism θ. Viceversa, if we have an r-module and a morphism that behaves “just like θ”, we can reconstruct a p-module and hence a vector bundle. More in detail, let us state and prove the following generalization of [OR06, Theorem 3.1]: Theorem 3.1. Consider n as an r-module with the adjoint action. (1) Given a p-module E on X, the action of n over E induces a mor- phism of r-modules: θ :n grE grE. ⊗ → The morphism θ θ : 2n grE grE ∧ ∧ ⊗ → defined by θ θ((n n ) f):= n (n f) n (n f) satisfies 1 2 1 2 2 1 ∧ ∧ ⊗ · · − · · the equality θ θ = θϕ in Hom( 2n grE,grE), where ϕ is given ∧ ∧ ⊗ by: ϕ: 2n grE n grE ∧ ⊗ → ⊗ (n n ) e [n ,n ] e. 1 2 1 2 ∧ ⊗ 7→ ⊗ (2) Conversely, given an r-module F on X and a morphism of r-modules θ : n F F ⊗ → such that θ θ = θϕ, we have that θ extends uniquely to an action ∧ of p over F, giving a bundle E such that grE = F. Proof. (1) Obviously θ is r-equivariant, almost by definition. Take n and n in n. We have that: 1 2 θ θ((n n ) f)= n (n f) n (n f)= [n ,n ] f, 1 2 1 2 2 1 1 2 ∧ ∧ ⊗ · · − · · · which means exactly that θ θ =θϕ. ∧ (2) For any f F and any r+n p = r n we set: ∈ ∈ ⊕ (3.3) (r+n) f := r f +θ(n f). · · ⊗ We need to show that given any p ,p p, the action (3.3) respects 1 2 ∈ the bracket, i.e. that for every f F: ∈ [p ,p ] f = p (p f) p (p f). 1 2 1 2 2 1 · · · − · · Now if both p ,p r, then there is nothing to prove. 1 2 ∈ If both p ,p n, then from the equality θ θ = θϕ, we get: 1 2 ∈ ∧ [p ,p ] f = θ([p ,p ] f)= θϕ((p p ) f)= 1 2 1 2 1 2 · ⊗ ∧ ⊗ = θ θ((p p ) f)= p (p f) p (p f). 1 2 1 2 2 1 ∧ ∧ ⊗ · · − · · Finally, in case p r and p n, then [p ,p ] n and we have: 1 2 1 2 ∈ ∈ ∈ p (p f)= θ(p (p f))= θ(([p ,p ] f +p (p f)) = 1 2 1 2 1 2 2 1 · · · ⊗ ⊗ ⊗ · = [p ,p ] f +θ(p (p f)) = [p ,p ] f +p (p f). 1 2 2 1 1 2 2 1 · ⊗ · · · · (cid:3) 6 Remark that (3.2) entails that we have a decomposition: (3.4) θ Hom(n grE,grE) = Hom(V ,V ) Hom(n E ,E ). λ µ λ µ ∈ ⊗ ⊗ ⊗ λ,Mµ∈Q0 Lemma3.2. [BK90,Proposition2]IntheADE casedimHom(n E ,E )G λ µ ⊗ is either 0 or 1 for every λ,µ Λ+. ∈ Remark 3.2. From now on G will thus always denote a complex Lie group of ADE type. Nevertheless, the construction of the quiver with its relation can be done for any type of Lie group, like in [ACGP03]. We can now conclude the construction of the representation of the quiver [E] associated to the bundle E. For any λ r-dominant weight fix a maximal vector v (it is unique up to λ constants). For any α weight of n, fix an eigenvector n n. α ∈ Now suppose that there is an arrow λ µ in the quiver. Then the vec- → tor space Hom(n E ,E )G is non-zero, and in particular is 1-dimensional. λ µ ⊗ Notice that by definition, being given by the action of n, the arrow will send the weight λ into a weight µ = λ+α, for some α Φ+ root of n (for we ∈ have g W W ). α λ λ+α · ⊆ Then fix the generator f of Hom(n E ,E )G that takes n v v . λµ λ µ α λ µ ⊗ ⊗ 7→ Once all the generators are fixed, from (3.4) we write the map θ uniquely as: (3.5) θ = g f , λµ λµ λ,µ X and thus we can associate to the arrow λ fλµ µ exactly the element g in λµ −−→ Hom(V ,V ). All in all: λ µ Definition 3.2. Toanyhomogeneous vector bundleE onG/P weassociate a representation [E] of the quiver as follows. G/P Q To any vertex λ we associate the vector space V from the decompo- 0 λ ∈ Q sition (3.2). To any arrow λ µ we associate the element g Hom(V ,V ) from the λµ λ µ → ∈ decomposition (3.4). Noticethatadifferentchoiceofgeneratorswouldhaveledtoanequivalent construction. 3.2. Secondfundamentalequivalenceofcategories. Weintroducehere the equivalence of categories between homogeneous vector bundles on G/P and finite dimensional representations of the quiver . G/P Q From Proposition 3.1 it is clear that by putting the appropriate relations on the quiver, namely the equality θ θ = θϕ, we can get the following: ∧ Theorem 3.3. [BK90, Hil94, ACGP03] Let G/P a rational homogeneous variety of ADE type. The category of finite dimensional representations of 7 the Lie algebra p is equivalent to the category of finite dimensional represen- tations of the quiver with certain relations , and it is equivalent to G/P Q R the category of G-homogeneous bundles on G/P. Weshowherehowonecanderivetherelations. Fordetails,see[ACGP03]. Let λ,µ,ν . We start by defining the morphism φ : 0 λµν ∈ Q Hom(n E ,E ) Hom(n E ,E ) φλµν Hom( 2n E ,E ) λ µ µ ν λ ν ⊗ ⊗ ⊗ −−−→ ∧ ⊗ by setting φ (a a′) : (n n′) x z, where a : n x y and λµν ⊗ ∧ ⊗ 7→ ⊗ 7→ a′ : n′ y z. ⊗ 7→ Obviously then the image of the G-invariant part: φ (Hom(n E ,E )G Hom(n E ,E )G) Hom( 2n E ,E )G. λµν λ µ µ ν λ ν ⊗ ⊗ ⊗ ⊆ ∧ ⊗ In particular recall once the choice of constants has been made, there are fixed generators f , where f : n v v , and α = λ µ. Then if we λµ λµ α λ µ ⊗ 7→ − set β = µ ν, all in all: − φ (f f ) : (n n ) v v . λµν λµ µν α β λ ν ⊗ ∧ ⊗ 7→ Now consider the natural morphism 2n n sending n n′ [n,n′]. ∧ → ∧ 7→ It induces a morphism φ : λν Hom(n E ,E ) φλν Hom( 2n E ,E ). λ ν λ ν ⊗ −−→ ∧ ⊗ Once again it is clear that the invariant part φ (Hom(n E ,E )G) is λν λ ν ⊗ contained in Hom( 2n E ,E )G. λ ν ∧ ⊗ Theorem3.1togetherwiththesplitting(3.5)entailthatwehaveanequal- ity in Hom( 2n E ,E )G: λ ν ∧ ⊗ (3.6) φ (f f )(g g )+φ ([f ,f ])g = 0. λµν λµ µν λµ µν λν λµ µν λν ⊗ Xλ,ν Xµ (cid:16) (cid:17) Finally, let ck be a basis of the vector space Hom( 2n E ,E )G, for { λν} ∧ ⊗ λ ν k = 1,...,m , m = dim(Hom( 2n E ,E )G). Expand: λν λν λ ν ∧ ⊗ mλν φ (f f ) = xk ck λµν λµ⊗ µν λµν λν k=1 X mλν φ ([f ,f ]) = yk ck , λν λµ µν λν λν k=1 X Then for every couple of vertices λ,ν, equality (3.6) gives us a system of m equations that the maps g satisfy: λν γδ (3.7) λν : xk g g +yk g = 0 Rk λµν λµ µν λν λν µX∈Q0 8 Definition 3.3. We define the relations on the quiver as the ideal G/P R Q generated by all the equations (that with a slight abuse of notation we keep calling ): k R λν : xk f f +yk f = 0, Rk λµν λµ µν λν λν µX∈Q0 for k = 1,...,m and for any couple of weights λ,ν . λν 0 ∈ Q 4. Simplicity 4.1. Simplicity of multiplicity free bundles. In this section we work on rational homogeneous varieties G/P, where G is complex, simple of type ADE. Let E bea rankr homogeneous vector bundleon G/B, and let [E] bethe associated representation of the quiver . Denote by the subquiver G/P E Q Q| of given byallvertices where[E]isnon-zeroandallarrowsconnecting G/P Q any two such vertices. Clearly the support of has at most r vertices. E Q| Notice also that the representation [E] of Q induces a representation of G/P the subquiver . E Q| Call the vertices of λ ,...,λ , with n r. Then the usual decompo- E 1 n Q| { } ≤ sition for the graded of E can be written: n (4.1) grE = E V . λi ⊗ i i=1 M Definition 4.1. A homogeneous vector bundle E of rank r is multiplicity free if dimV = 1 for every i= 1,...,n in (4.1). i Soletnow E bemultiplicity free. Justbylooking attheassociated quiver representation [E], we will show that the only G-invariant endomorphisms that such a bundle can have are scalar multiple of the identity, i.e. that the isotypical component H0(EndE)C = C. If this holds we call the bundle weakly simple. Proposition 4.1. Let E be a multiplicity free homogeneous vector bundle of rank r on G/P. Let k be the number of connected components of the quiver . Then H0(EndE)C = Ck. In particular if is connected, then E is E E Q| Q| weakly simple. Proof. Suppose first that k = 1, so that the subquiver is connected. E Any element ϕ H0(EndE)C is a G-invariant endomoQrp|hism ϕ : E E. ∈ → In particular we can look at ϕ as a morphism [E] [E] between represen- → tations of the same quiver. This means that we can look at ϕ as a family of morphisms ϕ :V V i+1,...,n . ThehypothesisthatE ismultiplicity i i i { → | } free entails that in particular each ϕ = k Id and hence ϕ = (k ,...,k ) is i i 1 n in fact just an element of Cr. By definition of morphism of quiver representations, every time that there 9 is an arrow λ λ in the quiver , there is a commutative diagram: i j E → Q| Vi // Vj ki kj (cid:15)(cid:15) (cid:15)(cid:15) Vi // Vj Remark that the two horizontal arrows are the same. This means that if we fix the first constant k of ϕ then all ϕ is completely determined thanks to 1 connectedness. So this proves that H0(EndE)C C. On the other hand, notice we have homotheties≤, hence C H0(EndE)C, ⊆ and the thesis follows for the case k =1. The same argument applies for each connected component, and this com- pletes the proof. (cid:3) 4.2. Simplicity of tangent bundles. The results of the previous section can be applied to a “special” multiplicity free homogeneous bundle: the tangent bundle T , with G simple of ADE type. G/P Let T be the tangent bundle on G/P. Recall that any parabolic G/P subgroup P = P(Σ) is given by a subset Σ ∆ of simple roots. Define also ⊆ the subset of negative roots Φ− = Φ− Φ−(Σ). Notice that when P = B is P \ the Borel, Σ = ∆ and Φ− = Φ−. B The bundle T is a homogeneous bundle of rank r = dimG/P = Φ− , G/P | P| whose weights are exactly the elements of Φ−. P Remark 4.1. It is convenient for us to take into account all the weights of the tangent bundle, and not only the highest weights. Obviously in the case of the Borel it doesn’t make any difference. If P is any other parabolic subgroup of G, this means that instead of the tangent bundle T we are G/P considering its pull-back π∗T via the (flat!) projection G/P π : G/B G/P. → The obvious vanishing Riπ = 0 for i > 0, together with the Projection ∗ O formula (see [Har77], II.5) guarantee that: H0(G/B,End(π∗T )) = H0(G/B,π∗(EndT )) = H0(G/P,EndT ). G/P G/P G/P Hence we are allowed to work on π∗T instead of T . To simplify the G/P G/P notation we write := π∗T . G/P G/P T Notice that the representation associated to any , P B, is a subrep- G/P T ⊇ resentation of that associated to = T . G/B G/B T Now we make the easy but fundamental remark that for every homoge- neous variety G/P, the rank (that is, the dimension of G/P) coincides G/P T with the number of weights of the associated representation and with the number of vertices in the quiver representation [ ]. Hence they must G/P T all have multiplicity one. Moreover, notice that in the ADE case whenever α= β we have [g ,g ]= g . All in all: α β α+β 6 − 10

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