Reduction Of Structure Group Research Articles

Toric principal bundles, Tits buildings and reduction of structure group

A toric principal G -bundle is a principal G -bundle over a toric variety together with a torus action commuting with the G -action. Recently, extending the Klyachko classification of toric vector bundles, Kaveh–Manon give a classification of toric principal G -bundles using piecewise linear maps to the (extended) Tits building of G . In this paper, we use the Kaveh–Manon classification to give a description of the (equivariant) automorphism group of a toric principal bundle as well as a simple criterion for (equivariant) reduction of structure group, recovering results of Dasgupta et al. Finally, motivated by the equivariant splitting problem for toric principal bundles, we introduce the notion of Helly’s number of a building and pose the problem of giving sharp upper bounds for Helly’s numbers of Tits buildings of semisimple algebraic groups G .

Automorphisms of the moduli space of principal $G$-bundles induced by outer automorphisms of $G$

In this work we study finite-order automorphisms of the moduli space of principal $G$-bundles coming from outer automorphisms of the structure group when $G$ is a simple complex Lie group. We do this by describing the subvarieties of fixed points for the action of that automorphisms on the moduli space of principal $G$-bundles. In particular, we prove that these fixed points are reductions of structure group to the subgroup of fixed points of the outer automorphism. Moreover, we study the way in which these fixed points fall into the stable or nonstable locus of the moduli.

ON THE ALGEBRAIC HOLONOMY OF STABLE PRINCIPAL BUNDLES

Let EG be a stable principal G–bundle over a compact connected Kähler manifold, where G is a connected reductive linear algebraic group defined over ℂ. Let H ⊂ G be a complex reductive subgroup which is not necessarily connected, and let EH ⊂ EG be a holomorphic reduction of structure group to H. We prove that EH is preserved by the Einstein–Hermitian connection on EG. Using this we show that if EH is a minimal reductive reduction (which means that there is no complex reductive proper subgroup of H to which EH admits a holomorphic reduction of structure group), then EH is unique in the following sense: For any other minimal reduction of structure group (H′, EH′) of EG to some reductive subgroup H′, there is some element g ∈ G such that H′ = g-1Hg and EH′ = EHg. As an application, we show the following: Let M be a simply connected, irreducible smooth complex projective variety of dimension n such that the Picard number of M is one. If the canonical line bundle KM is ample, then the algebraic holonomy of the holomorphic tangent bundle T1, 0M is GL (n, ℂ). If [Formula: see text] is ample, the rank of the Picard group of M is one, the biholomorphic automorphism group of M is finite, and M admits a Kähler–Einstein metric, then the algebraic holonomy of T1, 0M is GL (n, ℂ). These answer some questions posed in V. Balaji and J. Kollár, Publ. Res. Inst. Math. Sci.44 (2008) 183–211.

PRINCIPAL BUNDLES ON RATIONALLY CONNECTED FIBRATIONS OVER ABELIAN VARIETIES

Let f : M → A be a smooth surjective algebraic morphism, where M is a connected complex projective manifold and A a complex abelian variety, such that all the fibers of f are rationally connected. We show that an algebraic principal G-bundle EG over M admits a flat holomorphic connection if EG admits a holomorphic connection; here G is any connected reductive linear algebraic group defined over ℂ. We also show that EG admits a holomorphic connection if and only if any of the following three statements holds. (1) The principal G-bundle EG is semistable, c2( ad (EG)) = 0, and all the line bundles associated to EG for the characters of G have vanishing rational first Chern class. (2) There is an algebraic principal G-bundle E'G on A such that f*E'G = EG, and all the translations of E'G by elements of A are isomorphic to E'G itself. (3) There is a finite étale Galois cover [Formula: see text] and a reduction of structure group [Formula: see text] to a Borel subgroup B ⊂ G such that all the line bundles associated to ÊB for the characters of B have vanishing rational first Chern class. In particular, the above three statements are equivalent.

Principal bundles over a smooth real projective curve of genus zero

Abstract Let H 0 denote the kernel of the endomorphism, defined by , of the real algebraic group given by the Weil restriction of . Let X be a nondegenerate anisotropic conic in . The principal -bundle over the complexification defined by the ample generator of Pic(), gives a principal H 0-bundle over X through a reduction of structure group. Given any principal G-bundle EG over X, where G is any connected reductive linear algebraic group defined over ℝ, we prove that there is a homomorphism such that EG is isomorphic to the principal G-bundle obtained by extending the structure group of using . The tautological line bundle over the real projective line and the principal -bundle together give a principal -bundle F on , Given any principal G-bundle EG over , where G is any connected reductive linear algebraic group defined over ℝ, we prove that there is a homomorphism such that EG is isomorphic to the principal G-bundle obtained by extending the structure group of F using .

Reduction of structure group of principal bundles over aprojective manifold with Picard number one

Let X be a complex projective manifold with Pic(X) ≅ Z. Let G be a connected reductive algebraic group. A principal G-bundle will becalled split if it admits a reductionof structure group to a maximal torus of G. This corresponds tothe usual definition of a split vector bundle as a direct sum ofline bundles. Let E be a principal G-bundle on X, and let ρ G→ G* be an injective homomorphism.We prove that if the associated principal G*-bundle E(G*) is split, then E admits a reduction to a Borel subgroupof G. Furthermore, if we assume that X is Fano or has trivialcanonical bundle, then E is also split. We apply this to obtain generalizations for principal bundles of two results about vector bundles on projective spaces. Namely, a principal G-bundle on CP is split if its restriction to a plane CP CPis split. Also, a principal G-bundle E on CP is trivial if there is a point p such that the restriction of E to every line through p is trivial.

Hitchin's self-duality equations on complete Riemannian manifolds

Let M be a complete Riemannian manifold. Let G be a real semisimple noncompact algebraic group, and let K be its maximal compact subgroup. Let P be a principal G-bundle over M. Fix a reduction of the structure group G of P to K, and consider the space d of smooth G-connections on P. d is an affine space modeled on the vector space ~21(ad(P)) of 1-forms on M with values in the adjoint bundle associated to P. For any connection D E d , we have a decomposition D = D + + 0, where D + is a connection preserving the K-structure and 0 is a 1-form on M with values in the orthogonal complement in ad(P) of ad(PK). Here Px is the K-bundle defined by the reduction of structure group. I f D is a flat connection, we have