Stability of the Tangent Bundle of the

Wonderful Compactification of an Adjoint Group

Indranil Biswas and S. Senthamarai Kannan

Received: April 3, 2013

Communicated by Thomas Peternell

Abstract. Let G be a complex linear algebraic group which is simple of adjoint type. Let \overline{G} be the wonderful compactification of G. We prove that the tangent bundle of \overline{G} is stable with respect to every polarization on \overline{G}.

2010 Mathematics Subject Classification: 32Q26, 14M27, 14M17 Keywords and Phrases: Wonderful compactification, adjoint group, stability, irreducibility, equivariant bundle

1. Introduction

De Concini and Procesi constructed compactifications of complex simple groups of adjoint type, which are known as wonderful compactifications. These compactifications have turned out to be very useful objects. Our aim here is to investigate equivariant vector bundles on a wonderful compactification. One of the key concepts associated to a vector bundle on a projective variety is the notion of stability introduced by Mumford. We prove the following (see Theorem 4.1):

Theorem 1.1. Let \overline{G} be the wonderful compactification of a complex simple group G of adjoint type. Take any polarization L on \overline{G}. Then the tangent bundle of \overline{G} is stable with respect to L.

Theorem 1.1 is proved using a result proved here on equivariant vector bundles over \overline{G} which we will now explain. Take G as in Theorem 1.1. Let G̃ be the universal cover of G. The action of G × G on \overline{G} produces an action of G̃ × G̃ on \overline{G}. A holomorphic vector bundle on \overline{G} is called equivariant if it is equipped with a lift of the action of G̃ × G̃; see Definition 2.1 for the details. Let e_{0} ∈ G be the identity element. The group G̃ is the connected component, containing the identity element, of the isotropy group of e_{0} for the action of G̃ × G̃ on \overline{G}. If (E , γ) is an equivariant vector bundle on \overline{G}, then the action γ of G̃ × G̃ on E produces an action of G̃ on the fiber E_{e_{0}}. We prove the following (see Proposition 3.3):

Proposition 1.2. Let (E , γ) be an equivariant vector bundle of rank r on \overline{G} such that the action of G̃ on E_{e_{0}} is irreducible. Then either E is stable or there is a holomorphic line bundle ξ on \overline{G} such that E is holomorphically isomorphic to ξ^{⊕r}.

We show that the tangent bundle T \overline{G} is not isomorphic to ξ^{⊕d}, where ξ is some holomorphic line bundle on \overline{G}, and d = dim_{ℂ} G. In view of this result, Theorem 1.1 follows from Proposition 1.2. A stable vector bundle admits an irreducible Einstein–Hermitian connection. It would be very interesting to be able to describe the Einstein–Hermitian structure of the tangent bundle of \overline{G}. In [Ka], Kato has carried out a detailed investigation of the equivariant bundles on partial compactifications of reductive groups.

2. Equivariant vector bundles on \overline{G}

Let G be a connected linear algebraic group defined over ℂ such that the Lie algebra of G is simple and the center of G is trivial. In other words, G is simple of adjoint type. The group G×G acts on G: the action of any (g_{1} , g_{2}) ∈ G×G is the map y ⟼ g_{1}yg_{2}^{−1}. Let \overline{G} be the wonderful compactification of G [DP]. A key property of the wonderful compactification of G is that the above action of G × G on G extends to an action of G × G on \overline{G}. Let

(2.1)  π : G̃ ⟶ G

be the universal cover. Using the projection π in (2.1), the above mentioned action of G × G on \overline{G} produces an action of G̃ × G̃ on \overline{G}

(2.2)  β : G̃ × G̃ ⟶ Aut^{0}(\overline{G}) ,

where Aut^{0}(\overline{G}) is the connected component, containing the identity element, of the group of automorphisms of the variety \overline{G}.

Definition 2.1. An equivariant vector bundle on \overline{G} is a pair (E , γ), where E is a holomorphic vector bundle on \overline{G} and

γ : G̃ × G̃ × E ⟶ E

is a holomorphic action of G̃×G̃ on the total space of E, such that the following two conditions hold:

(1) the projection of E to \overline{G} intertwines the actions of G̃ × G̃ on E and \overline{G}, and

(2) the action of G̃ × G̃ on E preserves the linear structure of the fibers of E.

Note that the first condition in Definition 2.1 implies that the action of any g ∈ G̃×G̃ sends a fiber E_{x} to the fiber E_{β(g)(x)}, where β is the homomorphism in (2.2). The second condition in Definition 2.1 implies that the self-map of E defined by v ⟼ γ(g , v) is a holomorphic isomorphism of the vector bundle E with the pullback β(g^{−1})^{∗}E. Therefore, if (E , γ) is an equivariant vector bundle on \overline{G}, then for every g ∈ G̃ × G̃, the pulled back holomorphic vector bundle β(g)^{∗}E is holomorphically isomorphic to E. The following proposition is a converse statement of it.

Proposition 2.2. Let E be a holomorphic vector bundle on \overline{G} such that for every g ∈ G̃ × G̃, the pulled back holomorphic vector bundle β(g)^{∗}E is holomorphically isomorphic to E. Then there is a holomorphic action γ of G̃ × G̃ on E such that the pair (E , γ) is an equivariant vector bundle on \overline{G}.

Proof. Let Aut(E) denote the group of holomorphic automorphisms of the vector bundle E over the identity map of \overline{G}. This set Aut(E) is the Zariski open subset of the affine space H^{0}(\overline{G}, E ⊗E^{∨}) defined by the locus of invertible endomorphisms of E. Therefore, Aut(E) is a connected complex algebraic group. Let Ãut(E) denote the set of all pairs of the form (g , f), where g ∈ G̃ × G̃ and

f : β(g^{−1})^{∗}E ⟶ E

is a holomorphic isomorphism of vector bundles. This set Ãut(E) has a tautological structure of a group

(g_{2} , f_{2}) · (g_{1} , f_{1}) = (g_{2}g_{1} , f_{2} ∘ f_{1}) .

We will show that it is a connected complex algebraic group. Let p_{1} : G̃ × G̃ × \overline{G} ⟶ \overline{G} be the projection to the last factor. Let

β̂ : G̃ × G̃ × \overline{G} ⟶ \overline{G}

be the algebraic morphism defined by (g , y) ⟼ β(g^{−1})(y), where g ∈ G̃ × G̃ and y ∈ \overline{G}. Let

q : G̃ × G̃ × \overline{G} ⟶ G̃ × G̃

be the projection to the first two factors. Now consider the direct image

E := q_{∗}((p_{1}^{∗}E) ⊗ (β̂^{∗}E)^{∨}) ⟶ G̃ × G̃ .

It is locally free. The set Ãut(E) is a Zariski open subset of the total space of the algebraic vector bundle E. Therefore, Ãut(E) is a connected complex algebraic group. The Lie algebra of G will be denoted by 𝔤. The Lie algebra of Ãut(E) will be denoted by A(E). We have a short exact sequence of groups

(2.3)  e ⟶ Aut(E) ⟶ Ãut(E) ⟶^{ρ} G̃ × G̃ ⟶ e ,

where ρ sends any (g , f) to g. Let

(2.4)  ρ′ : A(E) ⟶ 𝔤 ⊕ 𝔤

be the homomorphism of Lie algebras corresponding to ρ in (2.3). Since 𝔤 ⊕ 𝔤 is semisimple, there is a homomorphism of Lie algebras

τ : 𝔤 ⊕ 𝔤 ⟶ A(E)

such that

(2.5)  ρ′ ∘ τ = Id_{𝔤⊕𝔤}

[Bo, p. 91, Corollaire 3]. Fix such a homomorphism τ satisfying (2.5). Since the group G̃×G̃ is simply connected, there is a unique holomorphic homomorphism

τ̃ : G̃ × G̃ ⟶ Ãut(E)

such that the corresponding homomorphism of Lie algebras coincides with τ. From (2.5) it follows immediately that ρ ∘ τ̃ = Id_{G̃×G̃}. We now note that τ̃ defines an action of G̃ × G̃ on E. The pair (E , τ̃) is an equivariant vector bundle. □

3. Irreducible representations and stability

Fix a very ample class L ∈ NS(\overline{G}), where NS(\overline{G}) is the Néron–Severi group of \overline{G}. The degree of a torsionfree coherent sheaf F on \overline{G} is defined to be

degree(F) := (c_{1}(F) ∪ c_{1}(L)^{d−1}) ∩ [\overline{G}] ∈ ℤ ,

where d = dim_{ℂ} G. If rank(F) ≥ 1, then

μ(F) := \frac{degree(F)}{rank(F)} ∈ ℚ

is called the slope of F. A holomorphic vector bundle F over \overline{G} is called stable (respectively, semistable) if for every nonzero coherent subsheaf F′ ⊂ F with rank(F′) < rank(F), the inequality

μ(F′) < μ(F) (respectively, μ(F′) ≤ μ(F))

holds. A holomorphic vector bundle on \overline{G} is called polystable if it is a direct sum of stable vector bundles of same slope. Let

e_{0} ∈ G ⊂ \overline{G}

be the identity element. Let Iso_{e_{0}} ⊂ G̃ × G̃ be the isotropy subgroup of e_{0} for the action of G̃ × G̃ on \overline{G}. The connected component of Iso_{e_{0}} containing the identity element is G̃. If (E , γ) is an equivariant vector bundle on \overline{G}, then γ gives an action of Iso_{e_{0}} on the fiber E_{e_{0}}. In particular, we get an action of G̃ on E_{e_{0}}.

Lemma 3.1. Let (E , γ) be an equivariant vector bundle on \overline{G} such that the above action of G̃ on E_{e_{0}} is irreducible. Then the vector bundle E is polystable.

Proof. Let

(3.1)  E_{1} ⊂ ⋯ ⊂ E_{n} = E

be the Harder–Narasimhan filtration of E [HL, p. 16, Theorem 1.3.4]. Since G̃ × G̃ is connected, the action of G̃ × G̃ on \overline{G} preserves the Néron–Severi class L. Therefore, the filtration in (3.1) is preserved by the action of G̃ × G̃ on E. Note that (E_{1})_{e_{0}} ≠ 0 because in that case E_{1}|_{G} = 0 by the equivariance of E, which in turn implies that E_{1} = 0. Now, from the irreducibility of the action of G̃ on E_{e_{0}} we conclude that (E_{1})_{e_{0}} = E_{e_{0}}. In particular, rank(E_{1}) = rank(E). This implies that E_{1} = E. Hence E is semistable. Let

F ⊂ E

be the unique maximal polystable subsheaf of the semistable vector bundle E [HL, p. 23, Theorem 1.5.9]. From the uniqueness of F and the connectivity of G̃ × G̃ we conclude that F is preserved by the action of G̃ × G̃ on E. Just as done above, using the irreducibility of the action of G̃ on E_{e_{0}} we conclude that F_{e_{0}} = E_{e_{0}}. Hence F = E, implying that E is polystable. □

The following lemma is well-known.

Lemma 3.2. Let V_{1} and V_{2} be two finite dimensional irreducible complex G̃– modules such that both V_{1} and V_{2} are nontrivial. Then the G̃–module V_{1} ⊗ V_{2} is not irreducible.

Lemma 3.2 is a very special case of the PRV conjecture, [PRV], which is now proved. We also note that Lemma 3.2 is an immediate consequence of [Ra, p. 683, Theorem 1]. Since (V_{1} ⊗ V_{2}) ⊗ (V_{1} ⊗ V_{2})^{∗} = End(V_{1}) ⊗ End(V_{2}), and End(V_{i}) contains ℂ⊕𝔤, the G̃–module End(V_{1})⊗End(V_{2}) contains ℂ⊕(𝔤⊗𝔤), and hence contains ℂ^{⊕2} as 𝔤 ⊗ 𝔤 contains ℂ. So V_{1} ⊗ V_{2} is not irreducible.

Proposition 3.3. Let (E , γ) be an equivariant vector bundle of rank r on \overline{G} such that the action of G̃ on E_{e_{0}} is irreducible. Then either E is stable or there is a holomorphic line bundle ξ on \overline{G} such that E is holomorphically isomorphic to ξ^{⊕r}.

Proof. From Lemma 3.1 we know that E is polystable. Therefore, there are distinct stable vector bundles F_{1} , ⋯, F_{ℓ} and positive integers n_{1} , ⋯, n_{ℓ}, such that μ(F_{i}) = μ(E) for every i and

(3.2)  E = ⨁_{i=1}^{ℓ} F_{i}^{⊕n_{i}} .

We emphasize that F_{i} ≠ F_{j} if i ≠ j. The vector bundles F_{1} , ⋯, F_{ℓ} are uniquely determined by E up to a permutation of {1 , ⋯, ℓ} [At, p. 315, Theorem 2]. Fix a holomorphic isomorphism between the two vector bundles in the two sides of (3.2). Take F_{i} and F_{j} with i ≠ j. Since they are nonisomorphic stable vector bundles of same slope, we have

H^{0}(\overline{G}, F_{j} ⊗ F_{i}^{∨}) = 0 = H^{0}(\overline{G}, F_{i} ⊗ F_{j}^{∨}) .

Consequently, for every i ∈ {1 , ⋯, ℓ}, there is a unique subbundle of E which is isomorphic to F_{i}^{⊕n_{i}}. Using this it follows that for any g ∈ G̃ × G̃, and any j ∈ {1 , ⋯, ℓ}, there is a k ∈ {1 , ⋯, ℓ} such that the action of g on E takes the subbundle F_{j}^{⊕n_{j}} to F_{k}^{⊕n_{k}}. Since G̃ × G̃ is connected, this implies that the action γ of G̃×G̃ on E preserves the subbundle F_{i}^{⊕n_{i}} for every i ∈ {1 , ⋯, ℓ}. Now from the irreducibility of the action of G̃ on E_{e_{0}} we conclude that ℓ = 1. We will denote F_{1} and n_{1} by F and n respectively. So

(3.3)  F = F^{⊕n} .

Since for every g ∈ G̃ × G̃, the pulled back holomorphic vector bundle β(g)^{∗}(F^{⊕n}) is holomorphically isomorphic to F^{⊕n}, using [At, p. 315, Theorem 2] and the fact that F is indecomposable (recall that F is stable), we conclude that the pulled back holomorphic vector bundle β(g)^{∗}F is holomorphically isomorphic to F for every g ∈ G̃ × G̃. Therefore, from Proposition 2.2 we know that there is an action δ of G̃ × G̃ on F such that (F , δ) is an equivariant vector bundle on \overline{G}. The actions γ and δ together define an action of G̃ × G̃ on the vector bundle

Hom(F , E) = E ⊗ F^{∨} .

This action of G̃×G̃ on Hom(F , E) produces an action of G̃×G̃ on the vector space H^{0}(\overline{G}, Hom(F , E)). In view of (3.3), we have a canonical isomorphism

(3.4)  E = F ⊗_{ℂ} H^{0}(\overline{G}, Hom(F , E)) .

This isomorphism sends any (v , σ) ∈ (F_{x} , H^{0}(\overline{G}, Hom(F , E))) to the evaluation σ_{x}(v) ∈ E_{x}. The isomorphism in (3.4) is G̃ × G̃–equivariant. Since the action of G̃ on E_{e_{0}} is irreducible, from Lemma 3.2 we conclude that either rank(F) = 1 or

dim H^{0}(\overline{G}, Hom(F , E)) = 1 .

If dim H^{0}(\overline{G}, Hom(F , E)) = 1, then from (3.4) and the fact that F is stable it follows immediately that E is stable. If rank(F) = 1, then from (3.4) it follows that

E = F^{⊕r} ,

where r = rank(E). □

Remark 3.4. Let V be any irreducible G–module. Consider the trivial right action of G̃ on V as well as the left action of G̃ on V given by the combination of the action of G on V and the projection in (2.1). Therefore, we get the diagonal action of G̃ × G̃ on the trivial vector bundle \overline{G} × V over \overline{G}. Consequently, the trivial vector bundle \overline{G} × V gets the structure on an equivariant vector bundle.

We note that the action of G̃ ⊂ Iso_{e_{0}} on the fiber of \overline{G} × V over the point e_{0} is irreducible because the G–module V is irreducible.

4. The tangent bundle

Theorem 4.1. Let L ∈ NS(\overline{G}) be any ample class. The tangent bundle of \overline{G} is stable with respect to L.

Proof. Since G is simple, the adjoint action of G̃ on the Lie algebra 𝔤 of G is irreducible. In view of Proposition 3.3, it suffices to show that the tangent bundle T \overline{G} is not of the form ξ^{⊕d}, where ξ is a holomorphic line bundle on \overline{G} and d = dim_{ℂ} G. Assume that

(4.1)  T \overline{G} = ξ^{⊕d} ,

where ξ is a holomorphic line bundle on \overline{G}. Since the variety G is unirational (cf. [Gi, p. 4, Theorem 3.1]), the compactification \overline{G} is also unirational. Hence \overline{G} is simply connected [Se, p. 483, Proposition 1]. As T \overline{G} holomorphically splits into a direct sum of line bundles (see (4.1)) and \overline{G} is simply connected, it follows that

\overline{G} = (ℂℙ^{1})^{d}

[BPT, p. 242, Theorem 1.2]. But the tangent bundle of (ℂℙ^{1})^{d} is not of the form ξ^{⊕d} (see (4.1)); although the tangent bundle of (ℂℙ^{1})^{d} is a direct sum of line bundles, the line bundles in its decomposition are not isomorphic. Therefore, T \overline{G} is not of the form ξ^{⊕d}. This completes the proof. □

Acknowledgements

We thank the referee and D. S. Nagaraj for some comments. This work was carried out when both the authors were visiting KSOM, Calicut. We thank KSOM for hospitality. The first-named author acknowledges the support of the J. C. Bose Fellowship.

References

[At] M. F. Atiyah, On the Krull–Schmidt theorem with application to sheaves, Bull. Soc. Math. Fr. 84 (1956), 307–317.

[Bo] N. Bourbaki, Éléments de mathématique. XXVI. Groupes et algèbres de Lie. Chapitre 1: Algèbres de Lie, Actualités Sci. Ind. No. 1285, Hermann, Paris, 1960.

[BPT] M. Brunella, J. V. Pereira and F. Touzet, Kähler manifolds with split tangent bundle, Bull. Soc. Math. Fr. 134 (2006), 241–252.

[DP] C. De Concini and C. Procesi, Complete symmetric varieties, Invariant theory (Montecatini, 1982), 1–44, Lecture Notes in Math., 996, Springer, Berlin, 1983.

[Gi] P. Gille, Rationality properties of linear algebraic groups and Galois cohomology, (lecture notes), www.dma.ens.fr/∼gille/prenotes/mcm.pdf.

[HL] D. Huybrechts and M. Lehn, The geometry of moduli spaces of sheaves, Aspects of Mathematics, E31, Friedr. Vieweg & Sohn, Braunschweig, 1997.

[Ka] S. Kato, Equivariant vector bundles on group completions, Jour. Reine Angew. Math. 581 (2005), 71–116.

[PRV] K. R. Parthasarathy R. Ranga Rao and V. S. Varadarajan, Representations of complex semi-simple Lie groups and Lie algebras, Ann. of Math. 85 (1967), 383–429.

[Ra] C. S. Rajan, Unique decomposition of tensor products of irreducible representations of simple algebraic groups, Ann. of Math. 160 (2004), 683–704.

[Se] J.-P. Serre, On the fundamental group of a unirational variety, Jour. Lond. Math. Soc. 34 (1959), 481–484.

Indranil Biswas School of Mathematics Tata Institute of Fundamental Research Homi Bhabha Road Bombay 400005 India indranil@math.tifr.res.in S. Senthamarai Kannan Chennai Mathematical Institute H1 SIPCOT IT Park Siruseri Kelambakkam 603103 India kannan@cmi.ac.in