Principal bundles on compact complex manifolds with trivial tangent bundle
aa r X i v : . [ m a t h . DG ] A p r PRINCIPAL BUNDLES ON COMPACT COMPLEX MANIFOLDSWITH TRIVIAL TANGENT BUNDLE
INDRANIL BISWAS
Abstract.
Let G be a connected complex Lie group and Γ ⊂ G a cocompact lattice.Let H be a complex Lie group. We prove that a holomorphic principal H –bundle E H over G/ Γ admits a holomorphic connection if and only if E H is invariant. If G is simplyconnected, we show that a holomorphic principal H –bundle E H over G/ Γ admits a flatholomorphic connection if and only if E H is homogeneous. Introduction
Let T = C n / Γ be a complex torus, so Γ is a lattice of C n of maximal rank. For any x ∈ T , let τ x : T −→ T be the holomorphic automorphism defined by z z + x .Let H be a connected linear algebraic group defined over C . A holomorphic principal H –bundle E H over T admits a holomorphic connection if and only if τ ∗ x E H is holomorphicallyisomorphic to E H for every x ∈ T ; also, if E H admits a holomorphic connection, then itadmits a flat holomorphic connection [3, p. 41, Theorem 4.1].If Γ is a cocompact lattice in a connected complex Lie group G , then G/ Γ is clearly acompact connected complex manifold with trivial tangent bundle. Let M be a connectedcompact complex manifold such that the holomorphic tangent bundle T M is holomor-phically trivial. Then there is a connected complex Lie group G and a cocompact latticeΓ ⊂ G such that G/ Γ is biholomorphic to M . The manifold M is K¨ahler if and only if M is a torus. Our aim here is to investigate principal bundles on M admitting a (flat)holomorphic connection.Let G be a connected complex Lie group and Γ ⊂ G a cocompact lattice. For any g ∈ G , let β g : M := G/ Γ −→ G/ Γbe the automorphism defined by x gx . Let H be a connected complex Lie group.A holomorphic principal H –bundle E H over M is called invariant if for each g ∈ G ,the pulled back bundle β ∗ g E H is isomorphic to E H . A homogeneous holomorphic principal H –bundle on M is a pair ( E H , ρ ), where f : E H −→ M is a holomorphic principal H –bundle, and ρ : G × E H −→ E H is a holomorphic left–action on the total space of E H , such that the following two condi-tions hold: Mathematics Subject Classification.
Key words and phrases.
Homogeneous bundle, invariant bundle, holomorphic connection. (1) ( f ◦ ρ )( g , z ) = β g ( f ( z )) for all ( g , z ) ∈ G × E H , and(2) the actions of G and H on E G commute.If ( E H , ρ ) is homogeneous, then E H is invariant.We prove the following theorem (see Theorem 3.1 and Proposition 3.2): Theorem 1.1.
A holomorphic principal H –bundle E H over M admits a holomorphicconnection if and only if E H is invariant.Assume that the group G is simply connected. A holomorphic principal H –bundle E H over M admits a flat holomorphic connection if and only if E H is homogeneous. If Lie( G ) is semisimple and G is simply connected, then we prove that given any in-variant principal H –bundle E H , there is a holomorphic action ρ : G × E H −→ E H suchthat ( E H , ρ ) is homogeneous (see Lemma 4.1). This gives the following corollary (seeCorollary 4.2): Corollary 1.2.
Assume that
Lie( G ) is semisimple and G is simply connected. If a holo-morphic principal H –bundle E H −→ M admits a holomorphic connection, then it admitsa flat holomorphic connection. The compact complex manifolds G/ Γ of the above type with G non-commutative arethe key examples of non–K¨ahler comapct complex manifolds with trivial canonical bundle.Recently, these manifolds have started to play important role in string theory of theoreticalphysics (see [5], [2], [6]). They have also become a topic of investigation in complexdifferential geometry (see [8], [7]).2. Homogeneous bundles and holomorphic connection
Holomorphic connection.
Let G be a connected complex Lie group. LetΓ ⊂ G be a cocompact lattice. So(1) M := G/ Γis a compact complex manifold. Let g be the Lie algebra of G . Using the right–invariantvector fields, the holomorphic tangent bundle T M is identified with the trivial vectorbundle M × g with fiber g , so(2) T M = M × g −→ M .
Let(3) β : G × M −→ M be the left translation action. The map β is holomorphic. For any g ∈ G , let(4) β g : M −→ M be the automorphism defined by x β ( g , x ). ANIFOLDS WITH TRIVIAL TANGENT BUNDLE 3
Let H be a connected complex Lie group. The Lie algebra of H will be denoted by h .We recall that a holomorphic principal H –bundle over M is a complex manifold E H , asurjective holomorphic submersion f : E H −→ M and a right holomorphic action of H on E H ϕ : E H × H −→ E H (so ϕ is a holomorphic map), such that the following two conditions hold:(1) f ◦ ϕ = f ◦ p , where p is the projection of E H × H to the first factor, and(2) the action of H on each fiber of f is free and transitive.Let f : E H −→ M be a holomorphic principal H –bundle. Let T E H be the holomor-phic tangent bundle of E H . The group H acts on the direct image f ∗ T E H . The invariantpart At( E H ) := ( f ∗ T E H ) H ⊂ f ∗ T E H defines a holomorphic vector bundle on M , which is called the Atiyah bundle for E H . Let K := kernel( df )be the kernel of the differential df : T E H −→ f ∗ T M of f . The invariant direct image( f ∗ K ) H coincides with the sheaf of sections of the adjoint vector bundle ad( E H ). Werecall that ad( E H ) −→ M is the vector bundle associated to E H for the adjoint actionof H on h . Using the inclusion of ( f ∗ K ) H in ( f ∗ T E H ) H , we get a short exact sequence ofholomorphic vector bundles on M (5) 0 −→ ad( E H ) −→ At( E H ) df −→ T M −→ . It is known as the
Atiyah exact sequence for E H . (See [1].)A holomorphic connection on E H is a holomorphic splitting of the short exact sequencein (5). In other words, a holomorphic connection on E H is a holomorphic homomorphism D : T M −→ At( E H )such that ( df ) ◦ D = Id T M , where df is the homomorphism in (5) (see [1]).Let D be a holomorphic connection connection on E H . The curvature of D is theobstruction of D to be Lie algebra structure preserving (the Lie algebra structure ofsheaves of sections of T M and At( E H ) is given by the Lie bracket of vector fields). Thecurvature of D is a holomorphic section of ad( E H ) ⊗ V ( T M ) ∗ . (See [1] for the details.)A flat holomorphic connection is a holomorphic connection whose curvature vanishesidentically.2.2. Invariant and homogeneous bundles.
We will now define invariant holomorphicprincipal bundles and homogeneous principal bundles.
Definition 2.1.
A holomorphic principal H –bundle E H over M will be called invariant if for each g ∈ G , the pulled back holomorphic principal H –bundle β ∗ g E H is isomorphicto E H , where β g is the map in (4). Definition 2.2. A homogeneous holomorphic principal H –bundle on M is defined to bea pair ( E H , ρ ), where I. BISWAS • f : E H −→ M is a holomorphic principal H –bundle, and • ρ : G × E H −→ E H is a holomorphic left–action on the total space of E H ,such that the following two conditions hold:(1) ( f ◦ ρ )( g , z ) = β g ( f ( z )) for all ( g , z ) ∈ G × E H , where β g is defined in (4), and(2) the actions of G and H on E G commute.If ( E H , ρ ) is a homogeneous holomorphic principal H –bundle, then E H is invariant.Indeed, for any g ∈ G , the automorphism of E H defined by z ρ ( g , z ) produces anisomorphism of E H with β ∗ g E H .3. Automorphisms of principal bundles
We continue with the notation of the previous section. We will give a criterion for theexistence of a (flat) holomorphic connection on a holomorphic principal H –bundle over M . Theorem 3.1.
A holomorphic principal H –bundle E H over M admits a holomorphicconnection if and only if E H is invariant.Proof. Let f : E H −→ M be a holomorphic principal H –bundle. Let A denote the spaceof all pairs of the form ( g , φ ), where g ∈ G , and φ : E H −→ E H is a biholomorphism satisfying the following two conditions:(1) φ commutes with the action of H on E H , and(2) f ◦ φ = β g ◦ f , where β g is defined in (4).So φ gives a holomorphic isomorphism of the principal H –bundle β ∗ g E H with E H .We note that A is a group using the composition rule( g , φ ) · ( g , φ ) = ( g g , φ ◦ φ ) . In fact, A is a complex Lie group, and the Lie algebra a := Lie( A )is identified with H ( M, At( E H )). Note that from (2) it follows that all holomorphicvector fields on M are given by the Lie algebra of G using the action β in (3). Thecompactness of M ensures that A is of finite dimension. As noted before, using the Liebracket of vector fields, the sheaf of holomorphic sections of At( E H ) has the structure ofa Lie algebra. Hence the space H ( M, At( E H )) of global holomorphic sections is a Liealgebra.Let(6) q : A −→ G be the projection defined by ( g , φ ) g . The homomorphism of Lie algebras(7) dq : a = H ( M, At( E H )) −→ g := Lie( G ) = H ( M, T M ) ANIFOLDS WITH TRIVIAL TANGENT BUNDLE 5 associated to q in (6) coincides with the one given by the homomorphism df in (5).First assume that E H admits a holomorphic connection. Recall that a holomorphic con-nection on E H is a holomorphic splitting of the exact sequence in (5). Using a holomorphicconnection, the vector bundle At( E H ) gets identified with the direct sum ad( E H ) ⊕ T M .In particular, the homomorphism H ( M, At( E H )) −→ H ( M, T M )induced by df in (5) is surjective. Hence the homomorphism dq in (7) is surjective. Since G is connected, this implies that the homomorphism q in (6) is surjective. This immediatelyimplies that E H is invariant (recall that for any ( g , φ ) ∈ A , the map φ is a holomorphicisomorphism of the principal H –bundle β ∗ g E H with E H ).To prove the converse, assume that E H is invariant. Therefore, the homomorphism q in(6) is surjective. Hence the homomorphism dq in (7) is surjective. Let d be the dimensionof G . Since dq is surjective, and T M is the trivial vector bundle of rank d , there are d sections σ , · · · , σ d ∈ H ( M, At( E H ))such that H ( M, T M ) = g is generated by { dq ( σ ) , · · · , dq ( σ d ) } .We have a holomorphic homomorphism D : T M −→ At( E H )defined by P di =1 c i · dq ( σ i )( x ) P di =1 c i · σ i ( x ), where x ∈ M , and c i ∈ C . It isstraight–forward to check that ( df ) ◦ D = Id T M , where df is the homomorphism in (5).Hence D defines a holomorphic connection on E H . (cid:3) Proposition 3.2.
Assume that the group G is simply connected. A holomorphic prin-cipal H –bundle E H over M admits a flat holomorphic connection if and only if E H ishomogeneous.Proof. Let f : E H −→ M be a holomorphic principal H –bundle equipped with a flatholomorphic connection D . Therefore, E H is given by a homomorphism from the funda-mental group π ( M, ( e , e Γ)) to H . We will construct an action of G on E H .Let p : G × M −→ M be the projection to the second factor. Since G is simplyconnected, the homomorphisms p ∗ , β ∗ : π ( G × M, ( e , e Γ)) −→ π ( M, ( e , e Γ))induced by p and β (see (3)) coincide. Therefore, there is a canonical isomorphism offlat principal H –bundles µ : p ∗ E H −→ β ∗ E H which is the identity map over { e } × M .This map µ defines an action ρ : G × E H −→ E H ;for any ( g , x ) ∈ G × M , the isomorphism ρ g,x : ( E H ) x −→ ( E H ) β g ( x ) I. BISWAS is the restriction of µ to ( g , x ), where β g is the map in (4). This action ρ makes E H ahomogeneous bundle.To prove the converse, take a homogeneous holomorphic principal H –bundle ( E H , ρ ).For any g ∈ G , let ρ g : E H −→ E H be the map defined by z ρ ( g , z ). Consider the group A constructed in the proof ofTheorem 3.1. Let(8) δ : G −→ A be the homomorphism defined by g ( g , ρ g ). It is easy to see that(9) q ◦ δ = Id G , where q is the homomorphism in (6).Let(10) dδ : g −→ Lie( A ) = H ( M, At( E H ))be the homomorphism of Lie algebras associated to the homomorphism δ in (8). From(9) it follows that(11) ( dq ) ◦ dδ = Id g , where dq is the homomorphism in (7) (it is the homomorphism of Lie algebras correspond-ing to q ).Since T M is the trivial vector bundle with fiber g , the homomorphism dδ in (10)produces a homomorphism of vector bundles e dδ : T M −→ At( E H ) ; e dδ ( x , v ) = ( dδ )( v )( x ), where v ∈ g , and x ∈ M . Combining (11) and the fact thatthe homomorphism dq coincides with the one given by the homomorphism df in (5), weconclude that ( df ) ◦ e dδ = Id T M . Therefore, e dδ defines a holomorphic connection on E H .The curvature of this holomorphic connection vanishes identically because the linear map dδ is Lie algebra structure preserving. (cid:3) Remark 3.3.
In Proposition 3.2, it is essential to assume that G is simply connected.To give examples, take G to be an elliptic curve and Γ to be the trivial group e . Take H = C ∗ , and take E H = L to be a nontrivial holomorphic line bundle of degree zero.Then L admits a flat holomorphic connection because L is topologically trivial, while L does not admit any homogeneous structure because L is not trivial.4. The semisimple case
In this section we assume that g is semisimple, and G is simply connected.Let f : E H −→ M := G/ Γ be an invariant holomorphic principal H –bundle. ANIFOLDS WITH TRIVIAL TANGENT BUNDLE 7
Lemma 4.1.
There is a holomorphic left-action of Gρ : G × E H −→ E H such that ( E H , ρ ) is homogeneous.Proof. Consider the homomorphism dq : a := Lie( A ) −→ g in (7). It is surjective because E H is invariant implying that q is surjective. Since g issemisimple, there is a Lie algebra homomorphism θ : g −→ a such that ( dq ) ◦ θ = Id g (see [4, p. 91, Corollaire 3]). Fix such a homomorphism θ .Since G is simply connected, there is a unique homomorphism of Lie groupsΘ : G −→ A such that θ is the Lie algebra homomorphism corresponding to Θ. Now we have an action ρ : G × E H −→ E H defined by Θ( g ) = ( g , { z ρ ( g , z ) } ). The pair ( E H , ρ ) is a homogeneous holomorphicprincipal H –bundle. (cid:3) Combining Lemma 4.1 with Theorem 3.1 and Proposition 3.2, we get the following:
Corollary 4.2.
If a holomorphic principal H –bundle E H −→ G/ Γ admits a holomorphicconnection, then it admits a flat holomorphic connection. References [1] M. F. Atiyah, Complex analytic connections in fibre bundles,
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