On Harder-Narasimhan reductions for Higgs principal bundles

The existence and uniqueness of H-N reduction for the Higgs principal bundles over nonsingular projective variety is shown. We also extend the notion of H-N reduction for (Γ,G)-bundles and ramifiedG-bundles over a smooth curve.     

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.     

Generalized Fredholm Properties for Invariant Pseudodifferential Operators

We define classes of pseudodifferential operators on G-bundles with compact base and give a generalized L ² Fredholm theory for invariant operators in these classes in terms of von Neumann’s G-dimension. We combine this formalism with a generalized Paley–Wiener theorem, valid for bundles with unimodular structure groups, to provide solvability criteria for invariant operators. This formalism also gives a basis for a G-index for these operators. We also define and describe a transversal dimension and its corresponding Fredholm theory in terms of anisotropic Sobolev estimates, valid also for similar bundles with nonunimodular structure group.     

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.     

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.     

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.     

The periodicity in stable equivariant surgery

The classical surgery theory (see [5] and [23]) computes the structure set S_m (M, rel ?) of manifolds homotopy equivalent to M relative to the boundary. Siebenmann showed that in topological category, the structure set is 4-periodic: S_m(M, rel ?) ? S_m+4(M × D⁴, rel ?) up to a copy of ?; see [12]. Cappell and Weinberger gave a geometric interpretation of this periodicity in [8]. By using Weinberger's stratified surgery theory (see [24]), we extend this to an equivariant periodicity result for topological manifolds with homotopically stratified actions by compact Lie groups, with D⁴ replaced by the unit ball of certain group representations. In particular, if G is an odd order group acting on a topological manifold M, then the equivariant stable structure sets satisfy S (M, rel ?) ? S(M × D(?⁴ ? ?G), rel ?) up to copies of ?. © 1993 John Wiley & Sons, Inc.     

Best Constants in Sobolev Inequalities on Riemannian Manifolds in the Presence of Symmetries

Let (M,g) be a smooth compact Riemannian manifold, and G a subgroup of the isometry group of (M,g). We compute the value of the best constant in Sobolev inequalities when the functions are G-invariant. Applications to non-linear PDEs of critical or upper critical Sobolev exponent are also presented.     

Orthogonality of natural sheaves on moduli stacks of SL(2)-bundles with connections on \Bbb P^1 minus 4 points

A special kind of SL(2)-bundles with connections on \Bbb P¹\{x₁,...,x₄}\Bbb P^1\setminus\{x_1,\dots,x_4\} is considered. We construct an equivalence between the derived category of quasicoherent sheaves on the moduli stack of such bundles and the derived category of modules over a TDO ring on some (non-separated) curve.     

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).     

Real Analytic Fibre Bundles: Some Results on Classification and Homotopies

The classification problem for holomorphic fibre bundles over Stein spaces was solved by H. GRAUERT. Along the same lines, the real coherent analytic case was considered by A. TOGNOLI and V. ANCONA. In this paper we propose a different approach, based on classifying spaces, in order to study the previous problem for real analytic fibre bundles over C -analytic subspaces of R ^m. So, let X be a C -analytic subspace of R ^m and G a compact Lie group. The main result is a characterization of the real analytic G-principal fibre bundles over X for which the analytic and topological equivalence coincide. Moreover, we prove that these bundles can be classified also by means of homotopy classes of analytic maps of X into classifying spaces. Among the others results, are worth recording: a relative approximation theorem of continuous cross sections by analytic ones, a theorem about the equivalence between analytical and topological homotopy between cross sections and a covering homotopy theorem.     

Generalized Solutions of Nonlocal Elliptic Problems

An elliptic equation of order 2m with general nonlocal boundary-value conditions, in a plane bounded domain G with piecewise smooth boundary, is considered. Generalized solutions belonging to the Sobolev space W ₂ ^m (G) are studied. The Fredholm property of the unbounded operator (corresponding to the elliptic equation) acting on L ₂(G), and defined for functions from the space W ₂ ^m (G) that satisfy homogeneous nonlocal conditions, is established.__________Translated from Matematicheskie Zametki, vol. 77, no. 5, 2005, pp. 665–682.Original Russian Text Copyright ©2005 by P. L. Gurevich.     

Symmetric products of surfaces and the cycle index

We study some of the combinatorial structures related to the signature ofG-symmetric products of (open) surfacesSP _G ^m (M)=M ^m/G whereG ⊂S _m.The attention is focused on the question, what information about a surfaceM can be recovered from a symmetric productSP ⁿ(M). The problem is motivated in part by the study of locally Euclidean topological commutative (m+k,m)-groups, [16]. Emphasizing a combinatorial point of view we express the signature Sign(SP _G ^m (M))in terms of the cycle index ofG, a polynomial which originally appeared in Pólya enumeration theory of graphs, trees, chemical structures etc. The computations are used to show that there exist punctured Riemann surfacesM _g,k,M _g′,k′such that the manifoldsSP ^m(M _g,k)andSP ^m(M)_g′,k′)are often not homeomorphic, although they always have the same homotopy type provided 2 _g +k=2 _g′ +k′ andk,k′≥1. Supported by the Serbian Ministry for Science and Technology, Grant No. 1643.     

Groups,group actions and fields definable in first‐order topological structures

Given a group (G, ·), G?M^m, definable in a first‐order structure $\mathcal {M}=(M,\ldots )Given a group (G, ·), G?M^m, definable in a first‐order structure $\mathcal {M}=(M,\ldots )$ equipped with a dimension function and a topology satisfying certain natural conditions, we find a large open definable subset V?G and define a new topology τ on G with which (G, ·) becomes a topological group. Moreover, τ restricted to V coincides with the topology of V inherited from M^m. Likewise we topologize transitive group actions and fields definable in $\mathcal {M}$. These results require a series of preparatory facts concerning dimension functions, some of which might be of independent interest.     

Morse theory for the space of Higgs <Emphasis Type="Italic">G</Emphasis>–bundles

Fix a C ^∞ principal G–bundle E⁰_G{E^0_G} on a compact connected Riemann surface X, where G is a connected complex reductive linear algebraic group. We consider the gradient flow of the Yang–Mills–Higgs functional on the cotangent bundle of the space of all smooth connections on E⁰_G{E^0_G}. We prove that this flow preserves the subset of Higgs G–bundles, and, furthermore, the flow emanating from any point of this subset has a limit. Given a Higgs G–bundle, we identify the limit point of the integral curve passing through it. These generalize the results of the second named author on Higgs vector bundles.     

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.     

Classification of Morse-Smale diffeomorphisms with one-dimensional set of unstable separatrices

Let M ⁿ be a closed orientable manifold of dimension n > 3. We study the class G ₁(M ⁿ ) of orientation-preserving Morse-Smale diffeomorphisms of M ⁿ such that the set of unstable separatrices of any f ∈ G ₁(M ⁿ ) is one-dimensional and does not contain heteroclinic intersections. We prove that the Peixoto graph (equipped with an automorphism) is a complete topological invariant for diffeomorphisms of class G ₁(M ⁿ ), and construct a standard representative for any class of topologically conjugate diffeomorphisms.     

Orbifold construction for topological field theories

An equivariant topological field theory is defined on a cobordism category of manifolds with principal fiber bundles for a fixed (finite) structure group. We provide a geometric construction which for any given morphism $G?H$ of finite groups assigns in a functorial way to a G-equivariant topological field theory an H-equivariant topological field theory, the pushforward theory. When H is the trivial group, this yields an orbifold construction for G-equivariant topological field theories which unifies and generalizes several known algebraic notions of orbifoldization.