Numerically flat principal bundles (original) (raw)

Holomorphic principal bundles over a compact Kähler manifold

Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 2000

Let ρ : G −→ H be a homomorphism between connected reductive algebraic groups over C such that the center of the Lie algebra g is sent to the center of h. If EG is a holomorphic principal G-bundle over a compact connected Kähler manifold M , and EG is semistable (resp. polystable), then the principal H-bundle EG ×G H is also semistable (resp. polystable). A G-bundle over M is polystable if and only if it admits an Einstein-Hermitian connection; this is an analog of a theorem of Uhlenbeck and Yau for G-bundles. Two different formulations of the G-bundle analog of the Harder-Narasimhan reduction have been established. The equivalence of the two formulations is a consequence of a group theoretic result. © 2000 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS Fibrés holomorphes principaux sur une variété kählérienne compacte Résumé. Soit ρ : G −→ H un homomorphisme entre deux groupes algébriques connexes et réductifs sur C tels que le centre de l'algèbre de Lie g de G s'envoie sur le centre de l'algèbre de Lie h de H. Si EG est un fibré principal holomorphe semi-stable (resp. polystable) sur une variété compacte kählérienne M , alors le fibré principal associé EG ×G H est aussi semi-stable (resp. polystable). Un G-fibré sur M est polystable si et seulement s'il admet une connexion d'Einstein-Hermite ; ceci est l'analogue du théorème d'Uhlenbeck et Yau pour les G-fibrés. Deux différentes formulations pour les G-fibrés du théorème de réduction de Harder-Narasimhan sont établies. L'équivalence des deux formulations est une conséquence de certains résultats sur les groupes algébriques.

Flat principal bundles over an abelian variety

Journal of Geometry and Physics, 2004

We prove that a principal G-bundle E G over a complex abelian variety A, where G is a complex reductive algebraic group, admits a flat holomorphic connection if and only if E G is isomorphic to all the translations of it by the group structure of A.

On semistable principal bundles over a complex projective manifold, II

Geometriae Dedicata, 2010

Let (X, ω) be a compact connected Kähler manifold of complex dimension d and E G −→ X a holomorphic principal G-bundle, where G is a connected reductive linear algebraic group defined over C. Let Z(G) denote the center of G. We prove that the following three statements are equivalent: (1) There is a parabolic subgroup P ⊂ G and a holomorphic reduction of structure group E P

Semistable and numerically effective principal (Higgs) bundles

Advances in Mathematics, 2011

We study Miyaoka-type semistability criteria for principal Higgs G-bundles E on complex projective manifolds of any dimension. We prove that E has the property of being semistable after pullback to any projective curve if and only if certain line bundles, obtained from some characters of the parabolic subgroups of G, are numerically effective. One also proves that these conditions are met for semistable principal Higgs bundles whose adjoint bundle has vanishing second Chern class. In a second part of the paper, we introduce notions of numerical effectiveness and numerical flatness for principal (Higgs) bundles, discussing their main properties. For (non-Higgs) principal bundles, we show that a numerically flat principal bundle admits a reduction to a Levi factor which has a flat Hermitian-Yang-Mills connection, and, as a consequence, that the cohomology ring of a numerically flat principal bundle with coefficients in R is trivial. To our knowledge this notion of numerical effectiveness is new even in the case of (non-Higgs) principal bundles.

On Semistable Principal Bundles over a Complex Projective Manifold

International Mathematics Research Notices, 2010

Let G be a simple linear algebraic group defined over the field of complex numbers. Fix a proper parabolic subgroup P of G, and also fix a nontrivial antidominant character χ of P. We prove that a holomorphic principal G-bundle E G over a connected complex projective manifold M is semistable satisfying the condition that the second Chern class c 2 (ad(E G)) ∈ H 4 (M, Q) vanishes if and only if the line bundle over E G /P defined by χ is numerically effective. Also, a principal G-bundle E G over M is semistable with c 2 (ad(E G)) = 0 if and only if for every pair of the form (Y , ψ), where ψ is a holomorphic map to M from a compact connected Riemann surface Y , and for every holomorphic reduction of structure group E P ⊂ ψ * E G to the subgroup P , the line bundle over Y associated to the principal P-bundle E P for χ is of nonnegative degree. Therefore, E G is semistable with c 2 (ad(E G)) = 0 if and only if for each pair (Y , ψ) of the above type the G-bundle ψ * E G over Y is semistable. Similar results remain valid for principal bundles over M with a reductive linear algebraic group as the structure group. These generalize an earlier work of Y. Miyaoka, [Mi], where he gave a characterization of semistable vector bundles over a smooth projective curve. Using these characterizations one can also produce similar criteria for the semistability of parabolic principal bundles over a compact Riemann surface.

Semistability of Principal Bundles on a Kähler Manifold with a Non-Connected Structure Group

Symmetry, Integrability and Geometry: Methods and Applications, 2014

We investigate principal G-bundles on a compact Kähler manifold, where G is a complex algebraic group such that the connected component of it containing the identity element is reductive. Defining (semi)stability of such bundles, it is shown that a principal G-bundle E G admits an Einstein-Hermitian connection if and only if E G is polystable. We give an equivalent formulation of the (semi)stability condition. A question is to compare this definition with that of [Gómez T.

On the Algebraic Holonomy of Stable Principal Bundles

International Journal of Mathematics, 2011

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 a...

A Criterion for Homogeneous Principal Bundles

International Journal of Mathematics, 2010

We consider principal bundles over G/P, where P is a parabolic subgroup of a semi-simple and simply connected linear algebraic group G defined over ℂ. We prove that a holomorphic principal H-bundle EH → G/P, where H is a complex reductive group, and is homogeneous if the adjoint vector bundle ad (EH) is homogeneous. Fix a faithful H-module V. We also show that EH is homogeneous if the vector bundle EH ×H V associated to it for the H-module V is homogeneous.

Hermitian–Einstein Connections on Principal Bundles Over Flat Affine Manifolds

International Journal of Mathematics, 2012

Let M be a compact connected special flat affine manifold without boundary equipped with a Gauduchon metric g and a covariant constant volume form. Let G be either a connected reductive complex linear algebraic group or the real locus of a split real form of a complex reductive group. We prove that a flat principal G-bundle EG over M admits a Hermitian–Einstein structure if and only if EG is polystable. A polystable flat principal G-bundle over M admits a unique Hermitian–Einstein connection. We also prove the existence and uniqueness of a Harder–Narasimhan filtration for flat vector bundles over M. We prove a Bogomolov type inequality for semistable vector bundles under the assumption that the Gauduchon metric g is astheno-Kähler.