Principal G-bundle Research Articles

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.

Poincaré families of G-bundles on a curve

Let G be a reductive group over an algebraically closed field k. Consider the moduli space of stable principal G-bundles on a smooth projective curve C over k. We give necessary and sufficient conditions for the existence of Poincare bundles over open subsets of this moduli space, and compute the orders of the corresponding obstruction classes. This generalizes the previous results of Newstead, Ramanan and Balaji–Biswas–Nagaraj–Newstead to all reductive groups, to all topological types of bundles, and also to all characteristics.

SEMISTABILITY AND NUMERICALLY EFFECTIVENESS IN POSITIVE CHARACTERISTIC

Let G be a connected reductive linear algebraic group defined over an algebraically closed field k of positive characteristic. Let Z(G) ⊂ G be the center, and [Formula: see text], where each Gi is simple with trivial center. For i ∈ [1, m], let ρi : G → Gi be the natural projection. Fix a proper parabolic subgroup P of G such that for each i ∈ [1, m], the image ρi(G) ⊂ Gi is a proper parabolic subgroup. Fix a strictly anti-dominant character χ of P such that χ is trivial on Z(G). Let M be a smooth projective variety, defined over k, equipped with a very ample line bundle ξ. Let EG → M be a principal G-bundle. We prove that the following six statements are equivalent: (1) The line bundle EG(χ) → EG/P associated to the principal P-bundle EG → EG/P for the character χ is numerically effective. (2) The sequence of principal G-bundles [Formula: see text] is bounded, where FM is the absolute Frobenius morphism of M. (3) The principal G-bundle EG is strongly semistable with respect to ξ, and c2( ad (EG)) is numerically equivalent to zero. (4) The principal G-bundle EG is strongly semistable with respect to ξ, and [c2( ad (EG))c1(ξ)d-2] = 0. (5) The adjoint vector bundle ad (EG) is numerically effective. (6) For every pair of the form (Y,ψ), where Y is an irreducible smooth projective curve and ψ : Y → M is a morphism, the principal G-bundle ψ*EG → Y is semistable.

Antiperfect morse stratification

For an equivariant Morse stratification that contains a unique open stratum, we introduce the notion of equivariant antiperfection, which means the difference of the equivariant Morse series and the equivariant Poincaré series achieves the maximal possible value (instead of the minimal possible value 0 in the equivariantly perfect case). We also introduce a weaker condition of local equivariant antiperfection. We prove that the Morse stratification of the Yang-Mills functional on the space of connections on a principal G-bundle over a connected, closed, nonorientable surface Σ is locally equivariantly \({\mathbb{Q}}\)-antiperfect when G = U(2), SU(2), U(3), SU(3); we propose that the Morse stratification is actually equivariantly \({\mathbb{Q}}\)-antiperfect in these cases. Our proposal yields formulas of Poincaré series \({P_t^G({\rm Hom}(\pi_1(\Sigma),G);\mathbb{Q})}\) when G = U(2), SU(2), U(3), SU(3). Our U(2), SU(2) formulas agree with formulas proved by T. Baird, who also verified our conjectural U(3) formula.

Splitting of gauge groups

Let G be a topological group and let P be a principal G-bundle over a based space B. We denote the gauge group of P by G(P) and the based gauge group of P by G 0 (p). Then the inclusion of the basepoint of B induces the exact sequence of topological groups 1 → G 0 (P) → G(P) → G → 1. We study the splitting of this exact sequence in the category of A n -spaces and A n -maps in connection with the triviality of the adjoint bundle of P and with the higher homotopy commutativity of G.

Principal Bundles on p-Adic Curves and Parallel Transport

We define functorial isomorphisms of parallel transport along étale paths for a class of principal G-bundles on a p-adic curve. Here G is a connected reductive algebraic group of finite presentation over the ring of integers of ℂp and the considered principal bundles are those with potential strongly semistable reduction of degree zero. The constructed isomorphisms yield continuous functors from the étale fundamental groupoid of the curve to the category of topological spaces with a simply transitive continuous right G(ℂp)-action. This generalizes a recent construction for vector bundles on a p-adic curve by Deninger and Werner. Our result can be viewed as a partial p-adic analogue of the classical theory by Ramanathan of principal bundles on compact Riemann surfaces, which generalizes the classical Narasimhan-Seshadri theory of vector bundles on compact Riemann surfaces.

The Atiyah bundle and connections on a principal bundle

Let M be a C∞ manifold and G a Lie a group. Let EG be a C∞ principal G-bundle over M. There is a fiber bundle C(EG) over M whose smooth sections correspond to the connections on EG. The pull back of EG to C(EG) has a tautological connection. We investigate the curvature of this tautological connection.

Bundles on non-proper schemes: representability

Let X be a proper scheme over a field k which satisfies Serre’s condition S 2 and G a reductive group over k. We prove that the functor of principal G-bundles, defined away from a non-fixed closed subset in X of codimension at least 3, is an algebraic stack in the sense of Artin.

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).

Note on mod p decompositions of gauge groups

We give fibrewise mod p decompositions of the adjoint bundle of a principal G-bundle P when the topological group G has mod p decompositions by automorphisms as in [5], which imply mod p decompositions of the gauge group of P.

Equivariant bundles and isotropy representations

We introduce a new construction, the isotropy groupoid , to organize the orbit data for split Γ-spaces. We show that equivariant principal G-bundles over split Γ-CW complexes X can be effectively classified by means of representations of their isotropy groupoids. For instance, if the quotient complex A = Γ \ X is a graph, with all edge stabilizers toral subgroups of Γ, we obtain a purely combinatorial classification of bundles with structural group G a compact connected Lie group. If G is abelian, our approach gives combinatorial and geometric descriptions of some results of Lashof–May–Segal [18] and Goresky–Kottwitz–MacPherson [10].

Poincaré families and automorphisms of principal bundles on a curve

Let C be a smooth projective curve, and let G be a reductive algebraic group. We give a necessary condition, in terms of automorphism groups of principal G-bundles on C, for the existence of Poincaré families parameterized by Zariski-open parts of their coarse moduli schemes. Applications are given for the moduli spaces of orthogonal and symplectic bundles. To cite this article: I. Biswas, N. Hoffmann, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

On principal bundles over a projective variety defined over a finite field

AbstractLet M be a geometrically irreducible smooth projective variety, defined over a finite field k, such that M admits a k-rational point x0. Let (M,x0/ denote the corresponding fundamental group-scheme introduced by Nori. Let EG be a principal G-bundle over M, where G is a reduced reductive linear algebraic group defined over the field k. Fix a polarization ξ on M. We prove that the following three statements are equivalent:1. The principal G-bundle EG over M is given by a homomorphism (M,x0)→G.2. There are integers b > a ≥ 1, such that the principal G-bundle (FbM)* EG is isomorphic to (FaM) * EG where FM is the absolute Frobenius morphism of M.3. The principal G-bundle EG is strongly semistable, the degree(c2(ad(EG))c1 (ξ)d−2 = 0, where d = dimM, and the degree(c1(EG(χ))c1(ξ)d−1) = 0 for every character χ of G, where EG(χ) is the line bundle over M associated to EG for χ.In [16], the equivalence between the first statement and the third statement was proved under the extra assumption that dimM = 1 and G is semisimple.

Principal bundles over the projective line

Let G be a connected reductive linear algebraic group defined over a field k and E G a principal G-bundle over the projective line P k 1 satisfying the condition that E G is trivial over some k-rational point of P k 1 . If the field k is algebraically closed, then it is known that the principal G-bundle E G admits a reduction of structure group to the multiplicative group G m . We prove this for arbitrary k. This extends the results of Harder (1968) [10] and Mehta and Subramanian (2002) [14].

Conjugacy classes in affine Kac–Moody groups and principal G-bundles over elliptic curves

For a simple complex Lie group G the connected components of the moduli space of semistable G-bundles over an elliptic curve are weighted projective spaces or quotients of weighted projective spaces by a finite group action. In this note we will provide a new proof of this result using the invariant theory of affine Kac–Moody groups, in particular the action of the (twisted) Coxeter element on the root system of G.

TOTAL SPACE OF ABELIAN GERBES

We present a generalization of the construction of a principal G-bundle from a one Čech cocycle to the case of higher abelian gerbes. We prove that the sheaf of local sections of the associated bundle to a higher abelian gerbe is isomorphic to the sheaf of sections of the gerbe itself. Our main result states that equivalence classes of higher abelian gerbes are in bijection with isomorphism classes of the corresponding bundles. We also present topological characterization of those bundles. In the last section, we show that the usual notion of Ehresmann connection leads to the gerbe connection for higher ℂ*-gerbes.

Holonomy and parallel transport in the differential geometry of loop spaces and generalized gauge transformations

The motivation of this paper stems from the need to construct explicit isomorphisms of (possibly nontrivial) principal G-bundles on the space of loops or, more generally, of paths in some manifold M, over which we consider a fixed principal bundle P; the aforementioned bundles are then pull backs of P with respect to evaluation maps at different points. The explicit construction of these isomorphisms between pulled-back bundles relies on the notion of parallel transport. We introduce and discuss extensively at this point the notion of generalized gauge transformation between (a priori) different principal G-bundles over the same base M, and we see immediately that the parallel transport is a generalized gauge transformation for two special kinds of bundles on the space of loops or paths; at this point, we may try and generalize the previous arguments for more general pulled-back bundles. Finally, we discuss how flatness of the reference connection, with respect to which we consider parallel transport, is related to horizontality of the associated generalized gauge transformation.

Universal moduli spaces of surfaces with flat bundles and cobordism theory

For a compact, connected Lie group G, we study the moduli of pairs ( Σ , E ) , where Σ is a genus g Riemann surface and E → Σ is a flat G-bundle. Varying both the Riemann surface Σ and the flat bundle leads to a moduli space M g G , parametrizing families Riemann surfaces with flat G-bundles. We show that there is a stable range in which the homology of M g G is independent of g. The stable range depends on the genus of the surface. We then identify the homology of this moduli space in the stable range, in terms of the homology of an explicit infinite loop space. Rationally, the stable cohomology of this moduli space is generated by the Mumford–Morita–Miller κ-classes, and the ring of characteristic classes of principal G-bundles, H ∗ ( BG ) . Equivalently, our theorem calculates the homology of the moduli space of semi-stable holomorphic bundles on Riemann surfaces. We then identify the homotopy type of the category of one-manifolds and surface cobordisms, each equipped with a flat G-bundle. Our methods combine the classical techniques of Atiyah and Bott, with the new techniques coming out of Madsen and Weiss's proof of Mumford's conjecture on the stable cohomology of the moduli space of Riemann surfaces.

PRINCIPAL BUNDLES ON RATIONALLY CONNECTED FIBRATIONS OVER ABELIAN VARIETIES

Let f : M → A be a smooth surjective algebraic morphism, where M is a connected complex projective manifold and A a complex abelian variety, such that all the fibers of f are rationally connected. We show that an algebraic principal G-bundle EG over M admits a flat holomorphic connection if EG admits a holomorphic connection; here G is any connected reductive linear algebraic group defined over ℂ. We also show that EG admits a holomorphic connection if and only if any of the following three statements holds. (1) The principal G-bundle EG is semistable, c2( ad (EG)) = 0, and all the line bundles associated to EG for the characters of G have vanishing rational first Chern class. (2) There is an algebraic principal G-bundle E'G on A such that f*E'G = EG, and all the translations of E'G by elements of A are isomorphic to E'G itself. (3) There is a finite étale Galois cover [Formula: see text] and a reduction of structure group [Formula: see text] to a Borel subgroup B ⊂ G such that all the line bundles associated to ÊB for the characters of B have vanishing rational first Chern class. In particular, the above three statements are equivalent.

Degeneration of the strange duality map for symplectic bundles

Global sections of the line bundles on a moduli space of vector bundles (or, more generally, principal G-bundles) are called generalized theta functions. The dimension of the vector spaces of generalized theta functions is given by the celebrated Verlinde formula. If you compute several Verlinde numbers, you will find that some of them coincide unexpectedly. Behind the coincidence, there is often geometric meaning. Let SU(r) be the moduli space of rank r bundles with trivial determinant on a smooth projective curve C of genus g, and let L be the ample generator of PicSU(r). Let U(n) be the moduli space of rank n bundles of slope g − 1 on C, and let U(n) ⊃ Θ be the locus of E ∈ U(n) such that H(C,E) 6= 0. Then by the Verlinde formula, you see that the dimensions of the vector spaces H0(SU(r),L⊗n) and H0(U(n),O(Θ)⊗r) are equal. There is a strange duality map (cf. [B1]) H0(SU(r),L⊗n)∗ → H0(U(n),O(Θ)⊗r),