Topological and analytical properties of Sobolev bundles,I: The critical case

We study the interrelationship between topological and analytical properties of Sobolev bundles and describe some of their applications to variational problems on principal bundles. We in particular show that the category of Sobolev principal G-bundles of class W ^2,m/2 defined over M ^m is equivalent to the category of smooth principal G-bundles on M and give a characterization of the weak sequential closure of smooth principal G-bundles with prescribed isomorphism class. We also prove a topological compactness result for minimizing sequences of a conformally invariant Yang-Mills functional.      

Chern–Weil map for principal bundles over groupoids

The theory of principal G-bundles over a Lie groupoid is an important one unifying various types of principal G-bundles, including those over manifolds, those over orbifolds, as well as equivariant principal G-bundles. In this paper, we study differential geometry of these objects, including connections and holonomy maps. We also introduce a Chern–Weil map for these principal bundles and prove that the characteristic classes obtained coincide with the universal characteristic classes. As an application, we recover the equivariant Chern–Weil map of Bott–Tu. We also obtain an explicit chain map between the Weil model and the simplicial model of equivariant cohomology which reduces to the Bott–Shulman map when the manifold is a point. P. Xu Research partially supported by NSF grant DMS-03-06665.     

On the Exceptional Principal Bundles Over a Surface

Let G be a simple simply connected complex Lie group. Some criteriaare given for the nonexistence of exceptional principal G-bundlesover a complex projective surface. As an application, it isshown that there are no exceptional G-bundles over a surfacewhose arithmetic genus is zero or one. It is also shown thatthere are no stable exceptional G-bundles over an abelian surface.2000 Mathematics Subject Classification 32L20, 14J60.     

Principal bundles on the projective line

We classify principalG-bundles on the projective line over an arbitrary fieldk of characteristic ≠ 2 or 3, whereG is a reductive group. If such a bundle is trivial at ak-rational point, then the structure group can be reduced to a maximal torus.     

The cohomology of the complex ofG-invariant forms onG-manifolds

The complex ofG-invariant forms and its cohomology for arbitraryG-manifolds and especially for a certain class ofG-manifolds, which are locally trivial fiber bundles over the orbit space, are considered. The transgression in the differential graded algebra of basic elements for tensor product of two identical Weil algebras of a reductive Lie groupG is calculated. This is used to get two convenient differential graded algebras with the same minimal models as the differential algebra of differential forms on the cross product of two principalG-bundles overG and ofG-invariant forms onG-manifolds of the above class. In particular, for compactG the generalization of the Cartan theorem on the cohomology of a homogeneous space is proved.Partially supported by the grant of the AMS's fSU Aid Fund     

Equivariant holomorphic Hermitian principal bundles over a generalized flag manifold

Let G be a simple and simply connected complex linear algebraic group. Fix a maximal compact subgroup K(G) ì G{K(G) \subset G}, and let P be a parabolic subgroup of G. Let H be any connected reductive complex linear algebraic group. We classify the K(G)-equivariant holomorphic Hermitian principal H-bundles over G/P.     

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.     

The motive of the moduli stack of <Emphasis Type="Italic">G</Emphasis>-bundles over the universal curve

We define relative motives in the sense of André. After associating a complex in the derived category of motives to an algebraic stack we study this complex in the case of the moduli of G-bundles varying over the moduli of curves.     

Poincaré families of <Emphasis Type="Italic">G</Emphasis>-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 Poincaré 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.     

Classification of real Nash fibre bundles

Let G be a compact subgroup of an orthogonal group and X an affine, real, semialgebraic Nash variety. A principal Nash G-bundle over X is said to be strongly Nash if it is induced, up to Nash equivalences, of some universal bundle under a Nash map. Not all Nash bundles are strongly Nash and we denote by S(X, G) the class of strongly Nash G-bundles over X. The principal aim of this paper is to prove the following classification theorem: two bundles of S(X, G) are Nash equivalent if and only if they are topologically equivalent; more,there exists a bijection between the family of the classes of Nash equivalent bundles of S(X, G) and , where is the sheaf of germs of the continous maps from X to G. This result leads to find the largest class of principal Nash G-bundles over X in which the topological equivalence always implies the Nash one. Well, we prove that this class is exactly S(X, G). Research partially supported by M.I.U.R.     

Riemann�CHilbert for tame complex parahoric connections

A local Riemann–Hilbert correspondence for tame meromorphic connections on a curve compatible with a parahoric level structure will be established. Special cases include logarithmic connections on G-bundles and on parabolic G-bundles. The corresponding Betti data involves pairs (M, P) consisting of the local monodromy M ∈ G and a (weighted) parabolic subgroup P ⊂ G such that M ∈ P, as in the multiplicative Brieskorn–Grothendieck–Springer resolution (extended to the parabolic case). The natural quasi-Hamiltonian structures that arise on such spaces of enriched monodromy data will also be constructed.     

Twisted Bundles and Admissible Covers

Abstract

We study the structure of the stacks of twisted stable maps to the classifying stack of a finite group G—which we call the stack of twisted G-covers, or twisted G-bundles. For a suitable group Gwe show that the substack corresponding to admissible G-covers is a smooth projective fine moduli space.     

Twisted vector bundles on pointed nodal curves

Motivated by the quest for a good compactification of the moduli space ofG-bundles on a nodal curve we establish a striking relationship between Abramovich’s and Vistoli’s twisted bundles and Gieseker vector bundles.     

A model for embedding finite coverings defined by principal bundles into bundles of manifolds

For any finite group G we construct a canonical model for embedding a principal G-bundle fibrewise into a given locally trivial fibration with a connected manifold M of dimension n⩾2 as fibre. The construction uses configuration spaces. We apply the construction to obtain a canonical model for the class of principal G-bundles which are polynomial when considered as covering maps. Finally, we give an algebraic characterization of the polynomial principal G-bundles in terms of homomorphisms into braid groups.     

A Generalisation of Nagata's Theorem on Ruled Surfaces

We prove a generalisation of a theorem of Nagata on ruled surface to the case of the fiber bundle E/P X, associated to a principal G-bundle E. Using this we prove boundedness for the isomorphism classes of semi-stable G-bundles in all characteristics.     

Borel-Weil-Bott theory on the moduli stack of G-bundles over a curve

Let G be a semi-simple group and M the moduli stack of G-bundles over a smooth, complex, projective curve. Using representation-theoretic methods, I prove the pure-dimensionality of sheaf cohomology for certain “evaluation vector bundles” over M, twisted by powers of the fundamental line bundle. This result is used to prove a Borel-Weil-Bott theorem, conjectured by G. Segal, for certain generalized flag varieties of loop groups. Along the way, the homotopy type of the group of algebraic maps from an affine curve to G, and the homotopy type, the Hodge theory and the Picard group of M are described. One auxiliary result, in Appendix A, is the Alexander cohomology theorem conjectured in [Gro2]. A self-contained account of the “uniformization theorem” of [LS] for the stack M is given, including a proof of a key result of Drinfeld and Simpson (in characteristic 0). A basic survey of the simplicial theory of stacks is outlined in Appendix B. Oblatum 17-XII-1996 & 26 VI-1997     

Kac–Moody $${\widetilde{E}_k}$$-bundles over elliptic curves and del Pezzo surfaces with singularities of type A

Given an elliptic curve Σ, flat E _k -bundles over Σ are in one-to-one correspondence with smooth del Pezzo surfaces of degree 9 − k containing Σ as an anti-canonical curve. This correspondence was generalized to Lie groups of any type. In this article, we show that there is a similar correspondence between del Pezzo surfaces of degree 0 with an A _d -singularity containing Σ as an anti-canonical curve and Kac–Moody [(E)\tilde]_k{\widetilde{E}_{k}}-bundles over Σ with k = 8 − d. In the degenerate case where surfaces are rational elliptic surfaces, the corresponding [(E)\tilde]_k{\widetilde{E}_k}-bundles over Σ can be reduced to E _k -bundles.     

Semistable and numerically effective principal (Higgs) bundles

We study Miyaoka-type semistability criteria for principal Higgs G-bundles E on complex projective manifolds of any dimension. We prove that E has the property of being semistable after pullback to any projective curve if and only if certain line bundles, obtained from some characters of the parabolic subgroups of G, are numerically effective. One also proves that these conditions are met for semistable principal Higgs bundles whose adjoint bundle has vanishing second Chern class.In a second part of the paper, we introduce notions of numerical effectiveness and numerical flatness for principal (Higgs) bundles, discussing their main properties. For (non-Higgs) principal bundles, we show that a numerically flat principal bundle admits a reduction to a Levi factor which has a flat Hermitian–Yang–Mills connection, and, as a consequence, that the cohomology ring of a numerically flat principal bundle with coefficients in R is trivial. To our knowledge this notion of numerical effectiveness is new even in the case of (non-Higgs) principal bundles.     

Principal bundles whose restrictions to a curve are isomorphic

Let X be a normal projective variety defined over an algebraically closed field k. Let |O _X (1)| be a very ample invertible sheaf on X. Let G be an affine algebraic group defined over k. Let E _G and F _G be two principal G-bundles on X. Then there exists an integer n > > 0 (depending on E _G and F _G ) such that if the restrictions of E _G and F _G to a curve C ∈ |O _X (n)| are isomorphic, then they are isomorphic on all of X.