Holomorphic Principal G-bundle Research Articles

Isomonodromic deformations of logarithmic connections and stability

Let \(X_0\) be a compact connected Riemann surface of genus g with \(D_0 \subset X_0\) an ordered subset of cardinality n, and let \(E_G\) be a holomorphic principal G-bundle on \(X_0\), where G is a reductive affine algebraic group defined over \(\mathbb C\), that is equipped with a logarithmic connection \(\nabla _0\) with polar divisor \(D_0\). Let \((\mathcal {E}_G , \nabla )\) be the universal isomonodromic deformation of \((E_G ,\nabla _0)\) over the universal Teichmuller curve \((\mathcal {X}, \mathcal {D})\,{\longrightarrow }\, \text {Teich}_{g,n}\), where \(\text {Teich}_{g,n}\) is the Teichmuller space for genus g Riemann surfaces with n–marked points. We prove the following (see Sect. 5): (1) Assume that \(g \ge 2\) and \(n= 0\). Then there is a closed complex analytic subset \(\mathcal {Y} \subset \text {Teich}_{g,n}\), of codimension at least g, such that for any \(t\in \text {Teich}_{g,n} {\setminus } \mathcal {Y}\), the principal G-bundle \(\mathcal {E}_G\vert _{{\mathcal X}_t}\) is semistable, where \({\mathcal X}_t\) is the compact Riemann surface over t. (2) Assume that \(g\ge 1\), and if \(g= 1\), then \(n > 0\). Also, assume that the monodromy representation for \(\nabla _0\) does not factor through some proper parabolic subgroup of G. Then there is a closed complex analytic subset \(\mathcal {Y}' \subset \text {Teich}_{g,n}\), of codimension at least g, such that for any \(t\in \text {Teich}_{g,n} {\setminus } \mathcal {Y}'\), the principal G-bundle \(\mathcal {E}_G\vert _{{\mathcal X}_t}\) is semistable. (3) Assume that \(g\ge 2\). Assume that the monodromy representation for \(\nabla _0\) does not factor through some proper parabolic subgroup of G. Then there is a closed complex analytic subset \(\mathcal {Y}'' \subset \text {Teich}_{g,n}\), of codimension at least \(g-1\), such that for any \(t\in \text {Teich}_{g,n} {\setminus } \mathcal {Y}'\), the principal G-bundle \(\mathcal {E}_G\vert _{{\mathcal X}_t}\) is stable. In [12], the second-named author proved the above results for \(G= \text {GL}(2,{\mathbb C})\).

A criterion for flatness of sections of adjoint bundle of a holomorphic principal bundle over a Riemann surface

Let EG be a holomorphic principal G-bundle over a com- pact connected Riemann surface, where G is a connected reductive affine algebraic group defined overC, such that EG admits a holo- morphic connection. Take any β ∈ H 0 (X, ad(EG)), where ad(EG) is the adjoint vector bundle for EG, such that the conjugacy class β(x) ∈ g/G, x ∈ X, is independent of x. We give a sufficient condi- tion for the existence of a holomorphic connection on EG such that β is flat with respect to the induced connection on ad(EG).

Principal bundles over a real algebraic curve

Let X be a compact connected Riemann surface equipped with an anti-holomorphic involution \sigma. Let G be a connected complex reductive affine algebraic group, and let \sigma_G be a real form of G. We consider holomorphic principal G-bundles on X satisfying compatibility conditions with respect to \sigma and \sigma_G. We prove that the points defined over $\mathbb R$ of the smooth locus of a moduli space of principal G-bundles on X are precisely these objects, under the assumption that {\rm genus}(X) > 2. Stable, semistable and polystable bundles are defined in this context. Relationship between any of these properties and the corresponding property of the underlying holomorphic principal G-bundle is explored. A bijective correspondence between unitary representations and polystable objects is established.

A criterion for holomorphic extension of principal bundles

Let G be a complex affine algebraic group, and let EG be a holomorphic principal G-bundle on the complement M$\backslash$S, where S is a closed complex analytic subset, of complex codimension at least two, of a connected complex manifold M. We give a criterion for EG to extend to M as a holomorphic principal G-bundle. Two applications of this criterion are given.

Homogeneous principal bundles over the upper half-plane

Let G be a connected complex reductive linear algebraic group, and let K⊂G be a maximal compact subgroup. The Lie algebra of K is denoted by k. A holomorphic Hermitian principal G-bundle is a pair of the form (EG,EK), where EG is a holomorphic principal G-bundle and EK⊂EG is a C∞-reduction of structure group to K. Two holomorphic Hermitian principal G-bundles (EG,EK) and (E′G,E′K) are called holomorphically isometric if there is a holomorphic isomorphism of the principal G-bundle EG with E′G which takes EK to E′K. We consider all holomorphic Hermitian principal G-bundles (EG,EK) over the upper half-plane H such that the pullback of (EG,EK) by each holomorphic automorphism of H is holomorphically isometric to (EG,EK) itself. We prove that the isomorphism classes of such pairs are parameterized by the equivalence classes of pairs of the form (χ,A), where χ:R→K is a homomorphism, and A∈k⊗RC such that [A,dχ(1)]=2-1⋅A. (Here dχ:R→k is the homomorphism of Lie algebras associated to χ.) Two such pairs (χ,A) and (χ′,A′) are called equivalent if there is an element g0∈K such that χ′=Ad(g0)◦χ and A′=Ad(g0)(A).

On the finite principal bundles

Let G be a connected linear algebraic group defined over \({\mathbb C}\). Fix a finite dimensional faithful G-module V0. A holomorphic principal G-bundle EG over a compact connected Kahler manifold X is called finite if for each subquotient W of the G-module V0, the holomorphic vector bundle EG(W) over X associated to EG for W is finite. Given a holomorphic principal G-bundle EG over X, we prove that the following four statements are equivalent: (1) The principal G-bundle EG admits a flat holomorphic connection whose monodromy group is finite. (2) There is a finite etale Galois covering \({f: Y \longrightarrow X}\) such that the pullback f*EG is a holomorphically trivializable principal G-bundle over Y. (3) For any finite dimensional complex G-module W, the holomorphic vector bundle EG(W) = E × GW over X, associated to the principal G-bundle EG for the G-module W, is finite. (4) The principal G-bundle EG is finite.

Numerically effectiveness and principal bundles on Kähler manifolds

Let EG be a holomorphic principal G-bundle over a compact connected Kahler manifold, where G is a connected complex reductive linear algebraic group. Consider a line bundle over EG/P corresponding to a character of P, where P is a parabolic subgroup of G. We give conditions for this holomorphic line bundle to be numerically effective.

On Semistable Principal Bundles over a Complex Projective Manifold

Let G be a simple linear algebraic group defined over the field of complex numbers. Fix a proper parabolic subgroup P of G, and also fix a nontrivial antidominant character χ of P. We prove that a holomorphic principal G-bundle E G over a connected complex projective manifold M is semistable satisfying the condition that the second Chern class vanishes if and only if the line bundle over E G /P defined by χ is numerically effective. Also, a principal G-bundle E G over M is semistable with if and only if for every pair of the form (Y, ψ), where ψ is a holomorphic map to M from a compact connected Riemann surface Y, and for every holomorphic reduction of structure group E P ⊂ ψ*E G to the subgroup P, the line bundle over Y associated with the principal P-bundle E P for χ is of nonnegative degree. Therefore, E G is semistable with if and only if for each pair (Y, ψ) of the above type the G-bundle ψ*E G over Y is semistable. Similar results remain valid for principal bundles over M with a reductive linear algebraic group as the structure group. These generalize an earlier work of Miyaoka [12], where he gave a characterization of semistable vector bundles over a smooth projective curve. Using these characterizations, one can also produce similar criteria for the semistability of parabolic principal bundles over a compact Riemann surface.

Connections and Higgs fields on a principal bundle

Let M be a compact connected Kahler manifold and G a connected linear algebraic group defined over \({\mathbb{C}}\) . A Higgs field on a holomorphic principal G-bundle eG over M is a holomorphic section θ of \(\text{ad}(\epsilon_{G})\otimes {\Omega}^{1}_{M}\) such that θ∧ θ = 0. Let L(G) be the Levi quotient of G and (eG(L(G)), θl) the Higgs L(G)-bundle associated with (eG, θ). The Higgs bundle (eG, θ) will be called semistable (respectively, stable) if (eG(L(G)), θl) is semistable (respectively, stable). A semistable Higgs G-bundle (eG, θ) will be called pseudostable if the adjoint vector bundle ad(eG(L(G))) admits a filtration by subbundles, compatible with θ, such that the associated graded object is a polystable Higgs vector bundle. We construct an equivalence of categories between the category of flat G-bundles over M and the category of pseudostable Higgs G-bundles over M with vanishing characteristic classes of degree one and degree two. This equivalence is actually constructed in the more general equivariant set-up where a finite group acts on the Kahler manifold. As an application, we give various equivalent conditions for a holomorphic G-bundle over a complex torus to admit a flat holomorphic connection.

Holomorphic principal bundles over a compact Kähler manifold

Abstract Let ρ:G→H be a homomorphism between connected reductive algebraic groups over C such that the center of the Lie algebra g is sent to the center of h . If EG is a holomorphic principal G-bundle over a compact connected Kahler manifold M, and EG is semistable (resp. polystable), then the principal H-bundle EG×GH is also semistable (resp. polystable). A G-bundle over M is polystable if and only if it admits an Einstein–Hermitian connection; this is an analog of a theorem of Uhlenbeck and Yau for G-bundles. Two different formulations of the G-bundle analog of the Harder–Narasimhan reduction have been established. The equivalence of the two formulations is a consequence of a group theoretic result.

Moduli of parabolic 𝐺-bundles

1. Let T be a maximal torus of G and let L{T) be its Lie algebra. Let L(G) denote the Lie algebra of G. Let E be a holomorphic principal Gbundle on a compact connected Riemann surface X. Let E(L(G)) denote the bundle with fibre L(G) associated to E via the adjoint representation of G. DEFINITION 1.1. Fix points x\,...,xn in X. For each i, a parabolic structure on E at xt is an element T,G L(T) in E(L(G))Xj. The bundle E together with the parabolic structures {T,}/ is called a parabolic principal G-bundle. DEFINITION 1.2. A parabolic G-bundle E with parabolic structures {T,-},is semistable (respectively stable) if for every reduction a: X —• E/P, P being a maximal parabolic subgroup of G, we have