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Let X be a compact Riemann surface of genus g ≥ 2 and M (G 2) be the moduli space of polystable principal G 2 -bundles over X. The Harder-Narasimhan types of the bundles induced a stratification of the moduli space M (G 2) called Shatz stratification. In this paper, a description of the Shatz strata of the unstable locus of M (G 2) corresponding to certain family of Harder-Narasimhan types (specifically, those of the form (λ , μ , 0 , − μ , − λ) with μ < λ ≤ 0) was given. For this purpose, a family of vector bundles was constructed in which a 3-form and a 2-form were defined so that it was proved that they were strictly polystable principal G 2 -bundles. From this, it was proved that, when the genus of X was g ≥ 12 , these Shatz strata were the disjoint union of a family of G 2 -Hecke curves in M (G 2) that will be constructed along the paper. Therefore, the presented results provided an advance in the knowledge of the geometry of M (G 2) through the study of its Shatz strata and presented a methodological innovation, by using Hecke curves for this study. [ABSTRACT FROM AUTHOR]

Let G be a semisimple complex algebraic group with a simple Lie algebra , and let ℳ G 0 denote the moduli stack of topologically trivial stable G -bundles on a smooth projective curve C. Fix a theta characteristic κ on C which is even in case dim is odd. We show that there is a nonempty Zariski open substack κ of ℳ G 0 such that H i (C , ad (E G) ⊗ κ) = 0 , i = 1 , 2 , for all E G ∈ κ . It is shown that any such E G has a canonical connection. It is also shown that the tangent bundle T U κ has a natural splitting, where U κ is the restriction of κ to the semi-stable locus. We also produce an isomorphism between two naturally occurring Ω M G r s 1 -torsors on the moduli space of regularly stable M G r s . [ABSTRACT FROM AUTHOR]

Let X be a compact Riemann surface of genus g ≥ 2 and let G be the group Spin(8, ℂ) or PSO(8, ℂ). Let τ be a choice of a non-trivial outer automorphism of G of order 3, called triality. Suppose that X is equipped with a holomorphic order 3 automorphism α. Then α acts on the moduli space M(G) of principal G-bundles over X by taking the pull-back, and τ also acts on M(G), by modifying the action of G on the principal bundles through any automorphism of G of order 3 representing τ. The combination of these two actions gives an automorphism of order 3 of the moduli space M(G). In this work, the notion of α-trialitarian G-bundle is introduced to describe the fixed points of the automorphism of M(G) mentioned above. In particular, an equivalent notion of stability and polystability for such bundles is proved, and an explicit construction of them is provided. [ABSTRACT FROM AUTHOR]

Let X be a compact algebraic curve of genus g ≥ 2. The nontrivial outer automorphism σ of the complex Lie group E6 acts on the moduli space M(E6) of principal E6-bundles over X, and this action defines an automorphism fσ of M(E6). The group H¹ (X, Z(E6)) of principal Z(E6)-bundles over X also acts on M(E6) by tensor product, Z(E6) being the center of E6, so each choice of an element L ∈ H¹ (X, Z(E6)) defines an automorphism fL of M(E6). In this paper two theorems describing the simple fixed points of the automorphism fL of M(E6) and the composition fL o fσ are proved. [ABSTRACT FROM AUTHOR]

LetXbe a compact connected Riemann surface of genusg≥2 equippedwith a holomorphic involutionσX, and letGbe a semisimple complex Liegroup which admits an outer involutionσ. A principal (G,σX,σ)-bundleoverXis a pair (E,ρ), whereEis a principalG-bundle overXandρ:E→σ∗X(σ(E)) is an isomorphism such that (σ∗Xρ)◦ρ:E→Eis anautomorphism of E which acts as the product by an element of the centerofG. In this paper, principal (G,σX,σ)-bundles overXare introduced andthe study is particularized to the case of G= SO(2n,C). It is shown thatthe stability and polystability conditions for a principal (SO(2n,C),σX,σ)-bundle coincide with those of the corresponding principal SO(2n,C)-bundle. Finally, the explicit form that a principal (SO(2n,C),σX,σ)-bundle takes isprovided, and the stability of these principal (SO(2n,C),σX,σ)-bundles isdiscussed. [ABSTRACT FROM AUTHOR]

In this article, we establish the Grothendieck–Serre conjecture over valuation rings: for a reductive group scheme $G$ over a valuation ring $V$ with fraction field $K$ , a $G$ -torsor over $V$ is trivial if it is trivial over $K$. This result is predicted by the original Grothendieck–Serre conjecture and the resolution of singularities. The novelty of our proof lies in overcoming subtleties brought by general nondiscrete valuation rings. By using flasque resolutions and inducting with local cohomology, we prove a non-Noetherian counterpart of Colliot-Thélène–Sansuc's case of tori. Then, taking advantage of techniques in algebraization, we obtain the passage to the Henselian rank-one case. Finally, we induct on Levi subgroups and use the integrality of rational points of anisotropic groups to reduce to the semisimple anisotropic case, in which we appeal to properties of parahoric subgroups in Bruhat–Tits theory to conclude. In the last section, by using extension properties of reflexive sheaves on formal power series over valuation rings and patching of torsors, we prove a variant of Nisnevich's purity conjecture. [ABSTRACT FROM AUTHOR]

Let X be a compact Riemann surface of genus g ≥ 2 and let n ≥ 3 be an integer number. The group of automorphisms of the moduli space of vector bundles over X with rank n and trivial determinant is isomorphic to H 1 (X , Z / (n)) ⋊ (Out (SL (n , C)) × Aut (X)) . Several papers have studied the subvarieties of fixed points for the action of the unique outer involution of SL (n , C) on this moduli space. In this paper, explicit descriptions of the fixed points for the actions of the elements of H 1 (X , Z / (n)) , H 1 (X , Z / (n)) ⋊ Out (SL (n , C)) , and Out (SL (n , C)) × Aut (X) on the moduli space of rank n and trivial determinant vector bundles over X are provided. For the description of the fixed points for the action of the elements of Out (SL (n , C)) × Aut (X) , the notion of Galois bundle is introduced. Specifically, Galois bundles over X admitting a nontrivial automorphism which commutes with the Galois structure are constructed associated with an involution σ X of X. Finally, it is discussed how the description of fixed points for the action of the elements of is covered by the descriptions above. [ABSTRACT FROM AUTHOR]

We study fiber bundles where the fibers are not a group G but a free G-space with disjoint orbits. The fibers are then not torsors but disjoint unions of these; hence, we like to call them semi-torsors. Bundles of semi-torsors naturally generalize principal bundles, and we call these semi-principal bundles. These bundles admit parallel transport in the same way that principal bundles do. The main difference is that lifts may end up in another group orbit, meaning that the change cannot be described by group translations alone. The study of such effects is facilitated by defining the notion of a basis of a G-set, in analogy with a basis of a vector space. The basis elements serve as reference points for the orbits so that parallel transport amounts to reordering the basis elements and scaling them with the appropriate group elements. These two symmetries of the bases are described by a wreath product group. The notion of basis also leads to a frame bundle, which is principal and so allows for a conventional treatment. In fact, the frame bundle functor is found to be a retraction from the semi-principal bundles to the principal bundles. The theory presented provides a mathematical framework for a unified description of geometric phases and exceptional points in adiabatic quantum mechanics.

We describe the structure of diffeological bundle of non formal classical pseudo-differential operators over formal ones, and its structure group. For this, we give results on diffeological principal bundles with (a priori) no local trivialization including an Ambrose-Singer theorem, use the smoothing connections alrealy exhibited by the author in previous works, and finish with open questions.

Let X be a hyperelliptic algebraic curve and let M (E 6) be the moduli space of polystable principal E 6 -bundles over X. Suppose, in addition, that the outer involution σ of E 6 acts as the hyperelliptic involution of X. Then, an automorphism of M (E 6) is defined which acts by E ↦ σ ∗ (E ∗) , where E is a principal E 6 -bundle over X seen as a vector bundle through the fundamental irreducible 27-dimensional representation of E 6 . In this paper, Galois E 6 -bundles over X are defined and related to the fixed points of the above automorphism of M (E 6) . If P E 6 is the centerless group with Lie algebra e 6 , then Galois P E 6 -bundles over X are also defined and related to Galois E 6 -bundles. Finally, a specific expression for a certain family of Galois E 6 -bundles over X is given and some implications of the study in terms of representations of the fundamental group π 1 (X) of the base curve are drawn. [ABSTRACT FROM AUTHOR]

We describe the structure of diffeological bundle of non formal classical pseudo-differential operators over formal ones, and its structure group. For this, we give results on diffeological principal bundles with (a priori) no local trivialization including an Ambrose-Singer theorem, use the smoothing connections alrealy exhibited by the author in previous works, and finish with open questions. [ABSTRACT FROM AUTHOR]

Let Γ be a finite group acting on a Lie group G. We consider a class of group extensions 1 → G → G ^ → Γ → 1 defined by this action and a 2-cocycle of Γ with values in the centre of G. We establish and study a correspondence between G ^ -bundles on a manifold and twisted Γ -equivariant bundles with structure group G on a suitable Galois Γ -covering of the manifold. We also describe this correspondence in terms of non-abelian cohomology. Our results apply, in particular, to the case of a compact or reductive complex Lie group G ^ , since such a group is always isomorphic to an extension as above, where G is the connected component of the identity and Γ is the group of connected components of G ^ . [ABSTRACT FROM AUTHOR]

We investigate the complex geometry of total spaces of reductive principal bundles over compact base spaces and establish a close relation between the Kähler property of the base, momentum maps for the action of a maximal compact subgroup on the total space, and the Kähler property of special equivariant compactifications. We provide many examples illustrating that the main result is optimal. [ABSTRACT FROM AUTHOR]

Criteria for a diffeomorphism of a smooth manifold M to be lifted to a linear automorphism of a given real vector bundle p : V → M, are stated. Examples are included and the metric and complex vector-bundle cases are also considered. [ABSTRACT FROM AUTHOR]

We survey recent work, published since 2015, on equivariant Oka theory. The main results described in the survey are as follows. Homotopy principles for equivariant isomorphisms of Stein manifolds on which a reductive complex Lie group G acts. Applications to the linearisation problem. A parametric Oka principle for sections of a bundle E of homogeneous spaces for a group bundle G , all over a reduced Stein space X with compatible actions of a reductive complex group on E, G , and X. Application to the classification of generalised principal bundles with a group action. Finally, an equivariant version of Gromov's Oka principle based on a notion of a G-manifold being G-Oka. [ABSTRACT FROM AUTHOR]

We introduce a class of G-invariant connexions on a homogeneous principal bundle Q over a Hermitian symmetric space M = G / K . The parameter space carries the structure of normal variety and has a canonical anti-holomorphic involution. The fixed points of the anti-holomorphic involution are precisely the integrable invariant complex structures on Q. This normal variety is closely related to quiver varieties and, more generally, to varieties of commuting matrix tuples modulo simultaneous conjugation. [ABSTRACT FROM AUTHOR]

We show that any extension of an Abelian group corresponds to a solution of the parametric Yang–Baxter equation. This statement is a generalization of the well-known construction of a braided set in terms of group structure to the case of group extensions. We also show that this construction in the case of a semidirect product is a specialization of a more general construction using principal bundles and that the case of vector bundles considered earlier is an infinitesimal version of the case of a solution coming from the principal bundle structure. [ABSTRACT FROM AUTHOR]

We investigate the category of regular coverings of a given smooth principal bundle. A covering map of one principal bundle to another principal bundle is understood as a morphism in the category of principal bundles consisting of covering maps of their structural groups, total spaces and bases. The main results are: the construction of an invariant of a regular covering, which is a subsequence of the homotopy sequence of the base bundle; the existence theorem of a covering with a given invariant; criteria for morphisms and isomorphisms; a theorem on the automorphism group of a given covering. [ABSTRACT FROM AUTHOR]

Let X be a connected smooth complex projective variety of dimension n ≥ 1 . Let D be a simple normal crossing divisor on X. Let G be a connected complex Lie group, and E G a holomorphic principal G-bundle on X. In this article, we give criterion for existence of a logarithmic connection on E G singular along D. [ABSTRACT FROM AUTHOR]

For complex connected, reductive, affine, algebraic groups G , we give a Lie-theoretic characterization of the semistability of principal G -co-Higgs bundles on the complex projective line ℙ1 in terms of the simple roots of a Borel subgroup of G. We describe a stratification of the moduli space in terms of the Harder–Narasimhan type of the underlying bundle. [ABSTRACT FROM AUTHOR]

We present a version of higher Hochschild homology for spaces equipped with principal bundles for a structure group G. As coefficients, we allow E∞-algebras with G-action. For this homology theory, we establish an equivariant version of excision and prove that it extends to an equivariant topological field theory with values in the (∞,1)-category of cospans of E∞-algebras. [ABSTRACT FROM AUTHOR]

We investigate connections, and more generally logarithmic connections, on holomorphic principal bundles over a compact connected Riemann surface.