Vector bundles on p-adic curves and parallel transport

We define functorial isomorphisms of parallel transport along étale paths for a class of vector bundles on a p-adic curve. All bundles of degree zero whose reduction is strongly semistable belong to this class. In particular, they give rise to representations of the algebraic fundamental group of the curve. This may be viewed as a partial analogue of the classical Narasimhan-Seshadri theory of vector bundles on compact Riemann surfaces.     

Entire curves,integral sets and principal bundles

We compare the behaviour of entire curves and integral sets,in particular in relation to locally trivial fiber bundles,algebraic groups and finite ramified covers over semi-abelian varieties.     

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     

Lifting nonregular paths in fiber bundles

Siberian Mathematical Journal -     

Integrable almost complex structures in principal bundles and holomorphic curves

We consider almost complex structures that arise naturally in a particular class of principal fibre bundles, where the choice of a connection can be used to determine equivariant isomorphisms between the vertical and horizontal tangent bundles of the total space. For instance, such data always exist on the frame bundle of a 3-manifold, but also in many other situations. We study the integrability condition to a complex structure, obtaining a system of gauge invariant coupled first order partial differential equations. This yields to a few correspondences between complex-geometric properties on the total space and metric properties on the base.     

Moduli for principal bundles over algebraic curves: I

We classify principal bundles on a compact Riemann surface. A moduli space for semistable principal bundles with a reductive structure group is constructed using Mumford's geometric invarian theory.     

Moduli for principal bundles over algebraic curves: II

We classify principal bundles on a compact Riemann surface. A moduli space for semistable principal bundles with a reductive structure group is constructed using Mumford’s geometric invariant theory. This is the second and concluding part of the thesis of late Professor A Ramanathan; the first part was published in the previous issue.     

Fundamental groups of moduli of principal bundles on curves

Geometriae Dedicata - Let X be a compact connected Riemann surface of genus g, with $$g\, \ge \, 2$$ , and let $${\text {G}}$$ be a connected semisimple affine algebraic group defined over...     

On extensions of principal bundles

Communicated by T. J. Willmore     

On the finite principal bundles

Let G be a connected linear algebraic group defined over \({\mathbb C}\). Fix a finite dimensional faithful G-module V ₀. A holomorphic principal G-bundle E _G over a compact connected Kähler manifold X is called finite if for each subquotient W of the G-module V ₀, the holomorphic vector bundle E _G (W) over X associated to E _G for W is finite. Given a holomorphic principal G-bundle E _G over X, we prove that the following four statements are equivalent: (1) The principal G-bundle E _G admits a flat holomorphic connection whose monodromy group is finite. (2) There is a finite étale Galois covering \({f: Y \longrightarrow X}\) such that the pullback f*E _G is a holomorphically trivializable principal G-bundle over Y. (3) For any finite dimensional complex G-module W, the holomorphic vector bundle E _G (W) = E ×  ^G W over X, associated to the principal G-bundle E _G for the G-module W, is finite. (4) The principal G-bundle E _G is finite.     

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 line bundles over real algebraic curves

Let X be a geometrically irreducible smooth projective curve defined over the real numbers. Let nX be the number of connected components of the locus of real points of X. Let x₁,…,x? be real points from ? distinct components, with ?<nX. We prove that the divisor x₁+?+x? is rigid. We also give a very simple proof of the Harnack's inequality.     

Algebraic and analytic parallel transport on p-adic curves

We relate two different partial p-adic analogues of the classical Riemann-Hilbert correspondence on curves. The first one comes from Deninger-Werner and Faltings and is of algebraic nature. The second one comes from André and Berkovich and is defined on Berkovich analytic spaces.     

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 ample vector bundles over curves in positive characteristic

Let E be an ample vector bundle over a smooth projective curve defined over an algebraically closed field of positive characteristic. We construct a family of curves in the total space of E, parametrized by an affine space, that surjects onto the total space of E and give a deformation of (nonreduced) zero section of E. To cite this article: I. Biswas, A.J. Parameswaran, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

On diophantine approximation along algebraic curves

Let be a quadratic form such that the associated algebraic curve contains a rational point. Here we show that there exists a domain such that for almost all , there exists an infinite sequence of nonzero integer triples satisfying the following two properties: (i) For each , is an excellent rational approximation to , in the sense that

and (ii) is a rational point on the curve . In addition, we give explicit values of for which both (i) and (ii) hold, and produce a similar result for a certain class of cubic curves.