Frame bundle

About: Frame bundle is a research topic. Over the lifetime, 1600 publications have been published within this topic receiving 23049 citations.

A compactification of the moduli space of principal Higgs bundles over singular curves

Abstract: A principal Higgs bundle $(P,\phi)$ over a singular curve $X$ is a pair consisting of a principal bundle $P$ and a morphism $\phi:X\to\text{Ad}P \otimes \Omega^1_X$ We construct the moduli space of principal Higgs G-bundles over an irreducible singular curve $X$ using the theory of decorated vector bundles More precisely, given a faithful representation $\rho:G\to Sl(V)$ of $G$, we consider principal Higgs bundles as triples $(E,q,\phi)$ where $E$ is a vector bundle with $\rk{E}=\dim V$ over the normalization $\xtilde$ of $X$, $q$ is a parabolic structure on $E$ and $\phi:E\ab{}\to L$ is a morphism of bundles, being $L$ a line bundle and $E\ab{}\doteqdot (E^{\otimes a})^{\oplus b}$ a vector bundle depending on the Higgs field $\phi$ and on the principal bundle structure Moreover we show that this moduli space for suitable integers $a,b$ is related to the space of framed modules

Orbifold principal bundles on an elliptic fibration and parabolic principal bundles on a Riemann surface

Abstract: Let X be a compact Riemann surface and associated to each point p-i of a finite subset S of X is a positive integer m-i. Fix an elliptic curve C. To this data we associate a smooth elliptic surface Z fibered over X. The group C acts on Z with X as the quotient. It is shown that the space of all vector bundles over Z equipped with a lift of the action of C is in bijective correspondence with the space of all parabolic bundles over X with parabolic structure over S and the parabolic weights at any p-i being integral multiples of 1 / m-i. A vector bundle V over Z equipped with an action of C is semistable (respectively, polystable) if and only if the parabolic bundle on X corresponding to V is semistable (respectively, polystable). This bijective correspondence is extended to the context of principal bundles.

The natural operators lifting projectable vector fields to some fiber product preserving bundles

Abstract: Admissible b er product preserving bundle functors F onFMm are de- ned. For every admissible b er product preserving bundle functorF onFMm all natural operators B : T projjFMm;n ! TF lifting projectable vector elds to F are classied.

The natural affinors on some fiber product preserving gauge bundle functors of vector bundles

Abstract: Dedicated to Professor Ivan Kolář on the occasion of his 70th bithday with respect and gratitude Abstra t. We classify all natural affinors on vertical fiber product preserving gauge bundle functors F on vector bundles. We explain this result for some more known such F . We present some applications. We remark a similar classification of all natural affinors on the gauge bundle functor F ∗ dual to F as above. We study also a similar problem for some (not all) not vertical fiber product preserving gauge bundle functors on vector bundles.

Natural transformations of higher order cotangent bundle functors

Abstract: We determine all natural transformations of the rth order cotangent bundle functor T r∗ into T s∗ in the following cases: r = s, r < s, r > s. We deduce that all natural transformations of T r∗ into itself form an r-parameter family linearly generated by the pth power transformations with p = 1, . . . , r. Using general methods developed in [2]–[5], we deduce that all natural transformations of the rth order cotangent bundle functor T r∗ into itself form an r-parameter family generated by the pth power transformations A p with p = 1, . . . , r. Then we deduce that all natural transformations of T r∗ into T (r+q)∗ form an r-parameter family generated by the generalized pth power transformations A p with p = q + 1, . . . , q + r. Moreover, we deduce that all natural transformations of T r∗ into T (r−q)∗ form an (r − q)-parameter family generated by the generalized pth power transformations Ar,r−q p with p = 1, . . . , r − q. The author is grateful to Professor I. Kolář for suggesting the problem and for valuable remarks and useful discussions. 1. Let M be a smooth n-dimensional manifold. Let T r∗M = J(M,R)0 be the space of all r-jets j xf of smooth functions f : M → R with source at x ∈M and target at 0 ∈ R. The fibre bundle πM : T r∗M →M with source r-jet projection πM : j xf 7→ x has a canonical structure of a vector bundle with (1.1) j xf + j r xg = j r x(f + g) , k · j xf = j x(k · f) for x ∈M and k ∈ R [1]. 1991 Mathematics Subject Classification: Primary 58A20.