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4th Iberoamerican Conference on Complex Geometry Location: Ouro Preto, BRAZIL Date: AUG 12-18, 2007., This is a survey of work done in the past two decades relating a basic equation of general relativity and quantum field theory with the theory of stable vector bundles and stable principal bundles, as well as providing moduli spaces for such objects, and compactifying them., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

We describe completely the so called jet like functors of a vector bundle E over an m-dimensional manifold M, i.e. bundles FE over M canonically depending on E such that F(E-1 x(M) E-2) = FE1 x(M) FE2 for any vector bundles E-l and E-2 over M. Then we study how a linear vector field on E can induce canonically a vector field on FE.

We describe completely the so called jet like functors of a vector bundle E over an m-dimensional manifold M, i.e. bundles FE over M canonically depending on E such that F(E-1 x(M) E-2) = FE1 x(M) FE2 for any vector bundles E-l and E-2 over M. Then we study how a linear vector field on E can induce canonically a vector field on FE.

We describe completely the so called jet like functors of a vector bundle E over an m-dimensional manifold M, i.e. bundles FE over M canonically depending on E such that F(E-1 x(M) E-2) = FE1 x(M) FE2 for any vector bundles E-l and E-2 over M. Then we study how a linear vector field on E can induce canonically a vector field on FE.

Let E be a stable vector bundle of rank r and slope 2g-1 on a smooth irreducible complex projective curve C of genus g > 2. In this paper we show a relation between theta divisor associated to E and the geometry of the tautological model of E. In particular, we prove that for r > g-1, if C is a Petri curve and E is general in its moduli space, its theta divisor defines an irreducible component of the variety parametrizing (g-2)-linear spaces which are g-secant the tautological model of E. Conversely, for a stable, (g-2)-very ample vector bundle E, the existence of an irreducible non special component of dimension g-1 of the above variety implies that E admits theta divisor

In this article we describe the life and work of Wolf Barth who died on 30th; December 2016. Wolf Barth's contributions to algebraic variety span a wide; range of subjects. His achievements range from what is now called the Barth-Lefschetz theorems to his fundamental contributions to the theory of algebraic surfaces and moduli of vector bundles, and include his later work on algebraic surfaces with many singularities, culminating in the famous Barth sextic.

Let C be a smooth complex projective curve of genus g ≥ 2 and let p ∈ C be a point. From Hecke correspondence, any stable bundle on C of rank r and determinant OC(p) defines a rational family of semistable vector bundles on C of rank r and trivial determinant. In this paper, we study linear systems of theta divisors associated to these families.

Tiny fluctuations of the Cosmic Microwave Background as well as various observable quantities obtained by spin raising and spin lowering of the effective gravitational lensing potential of distant galaxies and galaxy clusters are described mathematically as isotropic random sections of homogeneous spin and tensor bundles. We consider the three existing approaches to rigourous construction of the above objects, emphasising an approach based on the theory of induced group representations. Both orthogonal and unitary representations are treated in a unified manner. Several examples from astrophysics are included.

We develop a method, initially due to Salamon, for computing the space of “invariant” forms on an associated bundle X = P ×G V , with a suitable notion of invariance. We determine sufficient conditions for this space to be d-closed. We apply our method to the construction of Calabi–Yau metrics on TCP^1 and TCP^2

We develop a method, initially due to Salamon, for computing the space of “invariant” forms on an associated bundle X = P ×G V , with a suitable notion of invariance. We determine sufficient conditions for this space to be d-closed. We apply our method to the construction of Calabi–Yau metrics on TCP^1 and TCP^2

By Schwarzenberger's property, a complex vector bundle of dimension t over the complex projective space CP^n is extendible to CP^ for any k ≥ 0 if and only if it is stably equivalent to a Whitney sum of t complex line bundles. In this paper, we show some conditions for a negative multiple of a complex line bundle over CP^n to be extendible to CP^ or CP^, and its application to unextendibility of a normal bundle of CP^n.

By Schwarzenberger's property, a complex vector bundle of dimension t over the complex projective space CP^n is extendible to CP^ for any k ≥ 0 if and only if it is stably equivalent to a Whitney sum of t complex line bundles. In this paper, we show some conditions for a negative multiple of a complex line bundle over CP^n to be extendible to CP^ or CP^, and its application to unextendibility of a normal bundle of CP^n.

By Schwarzenberger's property, a complex vector bundle of dimension t over the complex projective space CP^n is extendible to CP^ for any k ≥ 0 if and only if it is stably equivalent to a Whitney sum of t complex line bundles. In this paper, we show some conditions for a negative multiple of a complex line bundle over CP^n to be extendible to CP^ or CP^, and its application to unextendibility of a normal bundle of CP^n.

type:text, We study the stable unextendibility of vector bundles over the quaternionic projective space HP^n by making use of combinatorial properties of the Stiefel-Whitney classes and the Pontrjagin classes. First, we show that the tangent bundle of HP^n is not stably extendible to HP^ for n ≥ 2, and also induce such a result for the normal bundle associated to an immersion of HP^n into R^<4n+k>. Secondly, we show a sufficient condition for a quaternionic r-dimensional vector bundle over HP^n not to be stably extendible to HP^ for r ≤ n and l > 0, which is also a necessary condition when r = n and l = 1.

Let X be a smooth variety defined over an algebraically closed field of arbitrary characteristic and H be a very ample divisor on X. We show that for a semistable X-bundle E of rank two, there exists an integer m depending only on Delta(E).Hdim(X)-2 and Hdim(X) such that the restriction of E to a general divisor in the linear system |mH| is again semistable. As corollaries we obtain boundedness results, and weak versions of Bogomolov's theorem and Kodaira's vanishing theorem for surfaces in arbitrary characteristic.

Let X be a smooth variety defined over an algebraically closed field of arbitrary characteristic and H be a very ample divisor on X. We show that for a semistable X-bundle E of rank two, there exists an integer m depending only on Delta(E).Hdim(X)-2 and Hdim(X) such that the restriction of E to a general divisor in the linear system |mH| is again semistable. As corollaries we obtain boundedness results, and weak versions of Bogomolov's theorem and Kodaira's vanishing theorem for surfaces in arbitrary characteristic.

We develop the theory of invariant random fields in vector bundles. The spectral decomposition of an invariant random field in a homogeneous vector bundle generated by an induced representation of a compact connected Lie group $G$ is obtained. We discuss an application to the theory of cosmic microwave background, where $G=SO(3)$. A theorem about equivalence of two different groups of assumptions in cosmological theories is proved.

We develop the theory of invariant random fields in vector bundles. The spectral decomposition of an invariant random field in a homogeneous vector bundle generated by an induced representation of a compact connected Lie group $G$ is obtained. We discuss an application to the theory of cosmic microwave background, where $G=SO(3)$. A theorem about equivalence of two different groups of assumptions in cosmological theories is proved.

We develop the theory of invariant random fields in vector bundles. The spectral decomposition of an invariant random field in a homogeneous vector bundle generated by an induced representation of a compact connected Lie group $G$ is obtained. We discuss an application to the theory of cosmic microwave background, where $G=SO(3)$. A theorem about equivalence of two different groups of assumptions in cosmological theories is proved.

We develop the theory of invariant random fields in vector bundles. The spectral decomposition of an invariant random field in a homogeneous vector bundle generated by an induced representation of a compact connected Lie group $G$ is obtained. We discuss an application to the theory of cosmic microwave background, where $G=SO(3)$. A theorem about equivalence of two different groups of assumptions in cosmological theories is proved.

We develop the theory of invariant random fields in vector bundles. The spectral decomposition of an invariant random field in a homogeneous vector bundle generated by an induced representation of a compact connected Lie group $G$ is obtained. We discuss an application to the theory of cosmic microwave background, where $G=SO(3)$. A theorem about equivalence of two different groups of assumptions in cosmological theories is proved.

We study the stable unextendibility of vector bundles over the quaternionic projective space HP^n by making use of combinatorial properties of the Stiefel-Whitney classes and the Pontrjagin classes. First, we show that the tangent bundle of HP^n is not stably extendible to HP^ for n ≥ 2, and also induce such a result for the normal bundle associated to an immersion of HP^n into R^<4n+k>. Secondly, we show a sufficient condition for a quaternionic r-dimensional vector bundle over HP^n not to be stably extendible to HP^ for r ≤ n and l > 0, which is also a necessary condition when r = n and l = 1.

We develop the theory of invariant random fields in vector bundles. The spectral decomposition of an invariant random field in a homogeneous vector bundle generated by an induced representation of a compact connected Lie group $G$ is obtained. We discuss an application to the theory of cosmic microwave background, where $G=SO(3)$. A theorem about equivalence of two different groups of assumptions in cosmological theories is proved.