On abelian principal bundles

Let T be a complex torus and E _T a holomorphic principal T-bundle over a connected complex manifold M. We prove that the total space of E _T admits a K?hler structure if and only if M admits a K?hler structure and E _T admits a flat holomorphic connection whose monodromy preserves a K?hler form on T. If E _T admits a K?hler structure, then is isomorphic to . Received: 2 September 2005     

On extensions of principal bundles

Communicated by T. J. Willmore     

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.     

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.     

On vector bundles of finite order

We define Finsler metrics of finite order on a holomorphic vector bundle by imposing estimates on the holomorphic bisectional curvature. We generalize the vanishing theorem of Griffiths and Cornalba regarding Hermitian bundles of finite order to the Finsler context. We develop a value distribution theory for holomorphic maps from the projectivization of a vector bundle to projective space. We show that the projectivization of a Finsler bundle of finite order can be immersed into a projective space of sufficiently large dimension via a map of finite order.     

Monodromy for principal bundles

Given a strongly semistable principal bundle E_G over a curve, in Biswas et al. (2006) [4], a group-scheme for it was constructed, which was named as the monodromy group-scheme. Here we extend the construction of the monodromy group-scheme to principal bundles over higher dimensional varieties.     

On the homotopy type of principal classical group bundles over spheres

The total spaces of principalSU(n−1) bundles overS ²ⁿ⁻¹ are classified. The classification ofSp(n−1) bundles overS ⁴ⁿ−1, is studied as well. As an intermediate step the homotopy equivalences ofSU andSp are classified.     

Extended affine principal fibre bundles

Summary In I, extended affine geometry obtained by extending the affine group parameters to functions of coordinates is established and is then realized in the differentiable manifolds. In II, a connections is introduced into the extended affine principal fibre bundle of the extended structure group. To Enrico Bompiani on his scientific Jubilee     

Restriction theorems for principal bundles

Let E be a semistable (or stable) principal bundle over a smooth complex projective variety X, and let DX be a complete intersection. We study the (semi)stability of the restriction E| _D . Some of the results known for vector bundles, such as Grauert–Mülich, Flenner and Mehta–Ramanathan theorems, are generalized to principal bundles. Mathematics Subject Classification (2000): 14F05, 32L05The authors are members of VBAC (Vector Bundles on Algebraic Curves), which is partially supported by EAGER (EC FP5 Contract no. HPRN-CT-2000-00099) and by EDGE (EC FP5 Contract no. HPRN-CT-2000-00101). T.G. was supported by a postdoctoral fellowship of Ministerio de Educación y Cultura (Spain), and wants to thank the Tata Institute of Fundamental Research, where this work was done while he was a postdoctoral student.     

Note on the stability of principal bundles

We compare various notions of stability for principal bundles, and show that over a compact Riemann surface of genus greater than 2, there exist principal -bundles that are Ad-stable.

     

On a characterization of finite vector bundles as vector bundles admitting a flat connection with finite monodromy group

We prove that a holomorphic vector bundle over a compact connected Kähler manifold admits a flat connection, with a finite group as its monodromy, if and only if there are two distinct polynomials and , with nonnegative integral coefficients, such that the vector bundle is isomorphic to . An analogous result is proved for vector bundles over connected smooth quasi-projective varieties, of arbitrary dimension, admitting a flat connection with finite monodromy group.

When the base space is a connected projective variety, or a connected smooth quasi-projective curve, the above characterization of vector bundles admitting a flat connection with finite monodromy group was established by M. V. Nori.

     

On semistable principal bundles over a complex projective manifold,II

Let (X, ω) be a compact connected Kähler manifold of complex dimension d and \({E_G\,\longrightarrow\,X}\) a holomorphic principal G–bundle, where G is a connected reductive linear algebraic group defined over \({\mathbb{C}}\). Let Z(G) denote the center of G. We prove that the following three statements are equivalent:

1. (1)
   There is a parabolic subgroup \({P\,\subset\,G}\) and a holomorphic reduction of structure group \({E_P\,\subset\,E_G}\) to P, such that the corresponding L(P)/Z(G)–bundle

   $E_{L(P)/Z(G)}\,:=\,E_P(L(P)/Z(G))\,\longrightarrow\,X$

   admits a unitary flat connection, where L(P) is the Levi quotient of P.
    
2. (2)
   The adjoint vector bundle ad(E _G ) is numerically flat.
    
3. (3)
   The principal G–bundle E _G is pseudostable, and

   $\int\limits_X c_2({\rm ad}(E_G))\omega^{d-2}\,=\,0.$
    

If X is a complex projective manifold, and ω represents a rational cohomology class, then the third statement is equivalent to the statement that E _G is semistable with c ₂(ad(E _G )) = 0.

     

Mukai-Sakai bound for equivariant principal bundles

Mukai and Sakai proved that given a vector bundleE of rankn on a connected smooth projective curve of genusg and anyr∈[1,n], there is subbundleS of rankr such that deg Hom(S, E/S)≤r(n−r)g. We prove a generalization of this bound for equivariant principal bundles. Our proof even simplifies the one given by Holla and Narasimhan for usual principal bundles.     

Restriction theorems for Higgs principal bundles

We prove analogues of Grauert–Mülich and Flenner?s restriction theorems for semistable principal Higgs bundle over any smooth complex projective variety.     

Foliate partial holomorphic structures on principal bundles

Thefoliate partial holomorphic (f.p.h.) pseudogroup is the pseudogroup of the local diffeomorphisms of ^m which preserve a distribution of the form span , wherey ^a are real coordinates,z are complex coordinates, and ^m may have also some other real coordinates. The f.p.h. structures on manifolds are described geometrically by the Nirenberg-Frobenius theorem [N], and occur in many interesting situations [R], [FW], [DK1], [V3], etc. The present paper discusses f.p.h. structures on principal bundles, and associates with such structures adapted connections, and forms with values in an associated bundle of Lie algebras.     

EQUIVARIANT SELF EQUIVALENCES OF PRINCIPAL FIBRE BUNDLES

Let E be a compact Lie group, G a closed subgroup of E, and H a closed normal sub-group of G. For principal fibre bundle (E,p, E,/G;G) tmd (E/H,p‘,E/G;G/H), the relation between auta(E) (resp. autce (E)) and autG/H(E/H) (resp. autGe/H(E/H)) is investigated by using bundle map theory and transformation group theory. It will enable us to compute the group JG(E) (resp. SG(E)) while the group J G/u(E/H) is known.     

Root stacks,principal bundles and connections

We investigate principal bundles over a root stack. In case of dimension one, we generalize the criterion of Weil and Atiyah for a principal bundle to have an algebraic connection.