Semistability of Principal Bundles on a K\"ahler Manifold with a Non-Connected Structure Group

We investigate principal $G$-bundles on a compact K\"ahler manifold, where $G$ is a complex algebraic group such that the connected component of it containing the identity element is reductive. Defining (semi)stability of such bundles, it is shown that a principal $G$-bundle $E_G$ admits an Einstein-Hermitian connection if and only if $E_G$ is polystable. We give an equivalent formulation of the (semi)stability condition. A question is to compare this definition with that of math.AG/0506511.


Introduction
Let X be a compact connected Kähler manifold equipped with a Kähler form ω. Let G be a connected complex reductive group. The connected component, containing the identity element, of the center of G will be denoted by Z 0 (G). In the study of principal G-bundles on X, it is usually assumed that the group G is connected.
In [8], Ramanathan gave the following definition. A principal G-bundle E is stable (respectively, semistable) if for all proper parabolic subgroups P and all reductions of structure group E P ⊂ E| U on a big open subset U ⊂ X (complement of a Zariski closed subset of codimension at least two), and for all strictly dominant characters χ : P −→ C * , the associated line bundle E P (χ) over U satisfies the inequality deg(E P (χ)) < 0 (respectively, deg(E P (χ)) ≤ 0), where the degree is calculated with respect to the Kähler form ω. We recall that a strictly dominant character of P is a character of P trivial on Z 0 (G) such that the dual of the line bundle on G/P associated to the character is ample.
A reduction E P ⊂ E| U on a big open set U ⊂ X to a parabolic subgroup P is called admissible if for every character χ of P trivial on Z 0 (G), we have deg(E P (χ)) = 0. A semistable principal G-bundle on X is called polystable if there exists a reduction E L(P ) ⊂ E G to a Levi factor L(P ) of a parabolic subgroup P ⊂ G such that the following two conditions hold: • the principal L(P )-bundle E L(P ) is stable, and • the reduction of E G to P obtained by extending the structure group of E L(P ) to P using the inclusion L(P ) → P is admissible. Ramanathan used this notion in [9,10] to construct the moduli space of semistable principal bundles when dim X = 1. In [11] Ramanathan and Subramanian proved that a principal Gbundle admits an Einstein-Hermitian connection if and only if it is polystable.
Behrend defined stability for group schemes in [3]. A reductive group scheme G/X on X is stable (respectively, semistable) if for all parabolic subgroup schemes P/X, we have deg(P/X) := deg(ad(P)) < 0 (respectively, deg(ad(P)) ≤ 0), where ad(P) is the associated Lie algebra bundle on X. Then he defines a principal G-bundle on X to be stable (respectively, semistable) if the associated group scheme Ad(E) is semistable (respectively, semistable). If G is connected, then it is easy to see that Behrend's definition of stability coincides with Ramanathan's, because the normalizer N G (P ) of a parabolic subgroup P in G is equal to P when G is connected.
If G is reductive but not connected, the moduli space of principal G-bundles has been constructed in [5] for projective varieties. Given a 1-parameter subgroup λ : C * −→ [G, G] ⊂ G, a parabolic subgroup P (λ) is defined as follows [12] and the condition of stability is checked only with these parabolic subgroups. The construction in [5] is done using Geometric Invariant Theory, and one obtains a stability condition involving Hilbert polynomials. Looking only at the leading coefficients one obtains the associated "slope" stability, as usual, which only involves degrees. This is the stability we consider, since this is the one which is expected to correspond to the existence of Einstein-Hermitian connections.
Here we address the problem of finding a condition for the existence of an Einstein-Hermitian connection on a principal G-bundle on X when G is not connected.
If G 0 is the connected component of identity of G, then E −→ E/G 0 is a principal G 0 -bundle on Y := E/G 0 , and Y is a finiteétale cover of X. In Section 2 we define the principal G-bundle E to be polystable when the principal G 0 -bundle E −→ Y is polystable (cf. Definition 1), and prove that this is a necessary and sufficient condition for the existence of an Einstein-Hermitian connection on E. In Section 3 we show that this definition of polystability is equivalent to checking the usual condition for those parabolic subgroups of G of the form N G (p), where p is a parabolic subalgebra of the Lie algebra g of G. Note that N G (Lie(P )) = N G (P 0 ) for any parabolic subgroup P , where Lie(P ) is the Lie algebra and P 0 is the connected component of P containing the identity element.
We remark that, if G is connected, all parabolic subgroups are of this form because for a connected reductive group, N G (p) = N G (P ) = P .
It is natural to ask whether our condition is equivalent to the condition in [5] in terms of 1-parameter subgroups. For any parabolic subgroup P , it is P ⊂ N G (Lie(P )). Hence, a reduction of structure group to P gives reduction to N G (Lie(P )), and therefore if a principal G-bundle is (semi)stable in the sense of [5], then it is also (semi)stable in our sense.
The implication in the other direction is not clear, since there exist examples of non-connected groups G and 1-parameter subgroups λ such that N G (P (λ) 0 ) is strictly larger than P (λ). It would be interesting to be able to compare the two definitions.

Connections on principal bundles 2.1 Semistable and polystable bundles
Let G be a reductive linear algebraic group defined over C. We do not assume that G is connected. Let G 0 ⊂ G be the connected component containing the identity element. We note that G 0 is a normal subgroup of G. The quotient group parametrizes the connected components of G. Let X be a compact connected Kähler manifold equipped with a Kähler form ω. Let The natural projection is a unramified Galois covering map with Galois group Γ (defined in (2)). The pulled back form f * ω is a Kähler form on Y . It should be clarified that Y need not be connected.
is semistable (respectively, polystable). The same criterion holds for stability.
Proof . Take two connected components Y 1 and Y 2 of Y . Since the covering map f in (4) is Galois, there is an element g of the Galois group such that the automorphism g of Y takes Y 1 to Y 2 . Let On the other hand, a principal G 0 -bundle is semistable (respectively, polystable) if and only if the corresponding adjoint vector bundle is semistable (respectively, polystable); see [1, p This isomorphism in (5) is compatible with the Lie algebra structure of the fibers of the two adjoint bundles. We recall that a principal G 0 -bundle F G 0 is stable if for every parabolic subalgebra bundle p ⊂ ad(F G 0 ), we have degree( p) < 0 (see [3]). Therefore, if E 1 is stable, then E 2 is also stable.

Einstein-Hermitian connections
Any two maximal compact subgroups of G differ by an inner automorphism of G. Fix a maximal compact subgroup K ⊂ G. The quotient G/K is a contractible manifold, in particular, G/K is connected.
Take a holomorphic principal G-bundle E G over X. A Hermitian structure on E G is a C ∞ reduction of structure group of E G to K. Since G/K is contractible, and any C ∞ fiber bundle with a contractible fiber is trivial, it follows immediately that E G admits Hermitian structures.
Any two connections on the principal G-bundle E G differ by a smooth 1-form with values in ad(E G ). Two C ∞ connections ∇ 1 and ∇ 2 on the principal G-bundle E G are called equivalent if ∇ 1 − ∇ 2 is of type (1, 0) [7, p. 87]. The complex structure on the total space of E G defines an equivalence class of connections on E G [7, p. 87, Proposition 2].
Let E K ⊂ E G be a Hermitian structure. Then there is a unique connection ∇ on the principal K-bundle E K such that the connection ∇ on E G induced by ∇ lies in the equivalence class of connections defined by the complex structure on E G [2,Proposition 5]. This ∇ is called the Chern connection corresponding to E K .
Let ad(E K ) −→ X be the adjoint vector bundle for the principal K-bundle E K . Let be the curvature of the above connection ∇ on E K . The Hermitian structure E K is called Einstein-Hermitian if the section is given by some element of the center of the Lie algebra of K; here Λ ω is the adjoint of multiplication of differential forms by the Kähler form ω.

Theorem 1.
A holomorphic principal G-bundle E G on X admits an Einstein-Hermitian structure if and only if E G is polystable. Given a polystable principal G-bundle E G over X, the Chern connection on E G corresponding to a Einstein-Hermitian structure on E G is independent of the choice of Einstein-Hermitian structure.
Proof . First assume that E G −→ X admits an Einstein-Hermitian structure. The connection on the adjoint vector bundle ad(E G ) induced by an Einstein-Hermitian connection on E G is also Einstein-Hermitian. Therefore, ad(E G ) is polystable. Take any connected To prove the converse, assume that the principal G 0 -bundle E G −→ X is polystable. Take a connected component Y of Y . As before, f is the restriction of f to Y , and E −→ Y is the principal G 0 -bundle obtained by restricting E G −→ Y to Y . The adjoint vector bundle ad(E ) is polystable because E is polystable. Since f is anétale covering map, an Einstein-Hermitian connection on ad(E ) produces an Einstein-Hermitian connection on the direct image f * ad(E ). We note that this uses the fact that the Kähler form on Y is the pullback of the Kähler form on X.
Since f * ad(E ) admits an Einstein-Hermitian connection, it follows that f * ad(E ) is polystable. Since ad(E G ) is a direct summand of f * ad(E ), we conclude that ad(E G ) is polystable.
The Einstein-Hermitian connection on f * ad(E ) preserves ad(E G ), because ad(E G ) is a direct summand of f * ad(E ). The connection ∇ on ad(E G ) obtained this way is Einstein-Hermitian.
Take the Einstein-Hermitian connection on ad(E ) to be one given by an Einstein-Hermitian connection on E . Therefore, the Einstein-Hermitian connection on ad(E ) is compatible with the Lie algebra structure of the fibers of ad(E ). This implies that the above connection ∇ on ad(E G ) is compatible with the Lie algebra structure of the fibers of ad(E G ). Therefore, ∇ defines a connection on the principal Aut(Lie(G))-bundle associated to E G for the homomorphism G −→ Aut(Lie(G)) given by the adjoint action of G on Lie(G). This connection on E Aut(Lie(G)) given by ∇ will be denoted by ∇ 0 . This connection ∇ 0 is Einstein-Hermitian because ∇ is so. Define The quotient homomorphism G −→ G ab will be denoted by q. Let G ab 0 ⊂ G ab be the connected component containing the identity element. Let β : G ab −→ G ab be the homomorphism defined by z −→ z n , where n is the order of the quotient group G ab /G ab 0 . Note that β(G ab 0 ) = G ab 0 , and the homomorphism given by β is the trivial homomorphism. Hence β(G ab ) = G ab 0 . Define Since G ab 0 is a torus, the principal G ab 0 -bundle E G ab 0 on X obtained by extending the structure group of E G using γ has a unique Einstein-Hermitian connection. We will denote this Einstein-Hermitian connection on E G ab 0 by ∇ . The connection ∇ 0 (respectively, ∇ ) is a 1-form on the total space E Aut(Lie(G)) (respectively, E G ab 0 ) with values in the Lie algebra Lie(Aut(Lie(G))) (respectively, Lie(G ab 0 )). Using the natural map E G −→ E Aut(Lie(G)) (respectively, E G −→ E G ab 0 ), the 1-form ∇ 0 (respectively, ∇ ) pulls back to a 1-form on E G with values in Lie(Aut(Lie(G))) (respectively, Lie(G ab 0 )); this 1-form on E G will be denoted by ∇ (respectively, ∇ ). Note that Lie(G) = Lie(Aut(Lie(G))) ⊕ Lie G ab 0 .
Therefore, ∇ + ∇ is a 1-form on E G with values in the Lie algebra Lie(G). It is straightforward to check that this Lie(G)-valued 1-form ∇ + ∇ defines a connection on E G . This connection on E G will be denoted by ∇. Fix a point x 0 ∈ X. Let Aut((E G ) x 0 ) denote the group of automorphisms of (E G ) x 0 that commute with the action of G on (E G ) x 0 . Note that Aut((E G ) x 0 ) is identified with the fiber of the adjoint bundle Ad(E G ) x 0 , and it is isomorphic to G. Consider parallel translations of the fiber (E G ) x 0 , along loops based at x 0 , with respect to the above connection ∇. These together produce a subgroup of Aut((E G ) x 0 ). It can be shown that this subgroup is contained in a compact subgroup of Aut((E G ) x 0 ). Indeed, this follows from the fact that the holonomies of both ∇ 0 and ∇ are compact. Therefore, possibly taking an extension of structure group, we get a Hermitian structure on E G . (The above subgroup of Aut((E G ) x 0 ) is a conjugate of a subgroup of K.) This Hermitian structure is Einstein-Hermitian because both ∇ 0 and ∇ are so.
An Einstein-Hermitian connection on the principal G-bundle E G −→ X pulls back to an Einstein-Hermitian connection on the principal G 0 -bundle E −→ Y . Therefore, the uniqueness of the Einstein-Hermitian connection on E G follows from the uniqueness of the Einstein-Hermitian connection on E . To explain this, from [11, p. 24, Theorem 1] we know that a stable bundle has a unique Einstein-Hermitian connection, and from [6,p. 111,Theorem 3.27] we know that for any decomposition of a polystable vector bundle F into a direct sum of stable vector bundles, each direct summand is preserved by any Einstein-Hermitian connection on F . Now apply this to the adjoint vector bundle ad(E ) and the principal G 0 /[G 0 , G 0 ]-bundle associated to E .

Equivalence of stability conditions
For a parabolic subalgebra p of Lie(G), by N G (p) we will denote the subgroup of G that preserves p by the adjoint action. In this section, by a parabolic subgroup of G we will mean a group of the form N G (p) for some parabolic subalgebra p. As before, by Z 0 (G) we will denote the connected component of the center of G containing the identity element.

Definition 2.
A principal G-bundle is called adjoint semistable (respectively, adjoint stable) if for all reductions to a proper parabolic subgroup P , and all reductions of structure group E P ⊂ E| U on a big open subset U ⊂ X, and for all strictly dominant characters χ : P −→ C * , the associated line bundle E P (χ) satisfies where the degree is calculated with respect to the Kähler form ω.
A character of P is called dominant if the restriction to P 0 := P G 0 is dominant. Recall that a reduction E P ⊂ E| U on a big open set U ⊂ X to a parabolic subgroup P is called admissible if for every nontrivial character χ of P trivial on Z 0 (G), we have deg(E P (χ)) = 0.
A semistable principal G-bundle on X is called adjoint polystable if there exists a reduction E L(P ) ⊂ E G to a Levi factor L(P ) of a parabolic subgroup P ⊂ G such that the following two conditions hold: • the principal L(P )-bundle E L(P ) is stable, and • the reduction of E G to P obtained by extending the structure group of E L(P ) to P is admissible.

Lemma 2.
A principal G-bundle E on X is adjoint semistable if and only if it is semistable in the sense of Definition 1.
Proof . Recall that a principal G-bundle E on X induces a principal G 0 -bundle E 0 on Y =E/G 0 . By Definition 1, a principal G-bundle E is semistable if and only if the restriction of E 0 to a connected component Y of Y is semistable. This is equivalent to ad(E 0 ) being semistable (cf. [1, p. 214, Proposition 2.10]). Note that ad(E 0 ) is isomorphic to f * ad(E), which is Γ-equivariant. If ad(E 0 ) is semistable, the it is also equivariantly semistable. On the other hand, suppose that it is unstable. Its Harder-Narasimhan filtration is unique, so it will be equivariant, and hence ad(E 0 ) will be equivariantly unstable. Therefore, ad(E 0 ) is semistable if and only if it is equivariantly semistable.
Taking the quotient by Γ, this is equivalent to the vector bundle ad(E) on X being semistable. We remark that the proof of [1, p. 214, Proposition 2.10] also works for disconnected groups if we use Definition 2. So ad(E) is semistable if and only if E is adjoint semistable. Proof . If E is polystable in the sense of Definition 1, then from Theorem 1 it follows that the adjoint vector bundle ad(E) is polystable. Conversely, if ad(E) is polystable, then f * ad(E) is polystable because an Einstein-Hermitian connection on ad(E) pulls back to an Einstein-Hermitian connection on f * ad(E). If f * ad(E) is polystable, from [1, p. 224, Corollary 3.8] we know that E is polystable in the sense of Definition 1.
On the other hand, ad(E) is polystable if and only if E is adjoint polystable; its proof is identical to the proof of [1, p. 224, Corollary 3.8].