Some remarks on the instability flag

Ramanan, S.; Ramanathan, A.

doi:10.2748/tmj/1178228852

Cited by 150 publications

(177 citation statements)

References 11 publications

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“…Given a homomorphism G → H and a principal G-bundle P , the space P × G H, where G acts as left translations of H, has a natural structure of a principal Hbundle. This construction of a principal H-bundle from the principal G-bundle P is called the extension of the structure group of P to H. From [12] we know that if P is a semistable (respectively, polystable) G-bundle, then P × G H is a semistable (respectively, polystable) H-bundle.…”

Section: Definition 31 ([12]mentioning

confidence: 99%

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Principal bundles with parabolic structure

Balaji

Biswas

Nagaraj

2001

Electron. Res. Announc. Amer. Math. Soc.

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Section: Definition 31 ([12]mentioning

confidence: 99%

“…If H is reductive and P is a semistable G-bundle, then the extension P × G H is also semistable ( [12]). This indicates how we may define semistability in the context of parabolic G-bundles.…”

Section: Definition 31 ([12]mentioning

confidence: 99%

Principal bundles with parabolic structure

Balaji

Biswas

Nagaraj

2001

Electron. Res. Announc. Amer. Math. Soc.

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“…We use this normalization convention also for the degree of torsion-free coherent sheaves on X, and for the degree of torsion-free coherent sheaves on X × G/P with respect to the Kähler form p * 1 ω + p * 2 ω ε . We now recall the definition of a semistable principal H-bundle (see [24,23,5]). Let Z 0 (H) be the connected component of the center of H containing the identity element.…”

Section: Dimensional Reduction For Equivariant Principal Bundlesmentioning

confidence: 99%

A Tannakian approach to dimensional reduction of principal bundles

Álvarez-Cónsul

Biswas

Garcı́a-Prada

2017

Journal of Geometry and Physics

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Abstract. Let P be a parabolic subgroup of a connected simply connected complex semisimple Lie group G. Given a compact Kähler manifold X, the dimensional reduction of G-equivariant holomorphic vector bundles over X × G/P was carried out by the first and third authors [2]. This raises the question of dimensional reduction of holomorphic principal bundles over X × G/P . The method used for equivariant vector bundles does not generalize to principal bundles. In this paper, we adapt to equivariant principal bundles the Tannakian approach of Nori, to describe the dimensional reduction of Gequivariant principal bundles over X × G/P , and to establish a Hitchin-Kobayashi type correspondence. In order to be able to apply the Tannakian theory, we need to assume that X is a complex projective manifold. IntroductionDimensional reduction is a very powerful construction in the context of gauge theory, both in physics and geometry. Many important gauge-theoretic equations appear as symmetric solutions of fundamental equations for connections, like the instanton equations on Riemannian 4-manifolds. Examples of these are the Bogomol'nyi equations for magnetic monopoles in 3 dimensions and the vortex equation in 2 dimensions, as well as many other important integrable systems and soliton equations. In the context of model building, some physicists have applied for a long time the method of 'coset-space dimensional reduction' in the construction of gauge unified theories (see, e.g., [15,7,20,26]).In [16,17,11,1] the dimensional reduction techniques were brought into the context of holomorphic vector bundles over Kähler manifolds, to study the dimensional reduction of stable SL(2, C)-equivariant bundles over X × P 1 and the corresponding Hermitian-YangMills equations, where X is a compact Kähler manifold and P 1 is the Riemann sphere. In [2] this construction was generalised to G-equivariant vector bundles on X ×G/P , where G is a connected simply connected complex semisimple Lie group and P ⊂ G is a parabolic subgroup. These gave rise to quiver bundles with relations, that is representations of a quiver with relations in the category of vector bundles, where the quiver with relations (Q, K) is determined by P . In particular when X is a point, one has a description of 2000 Mathematics Subject Classification. 53C07, 14D21, 16G20.

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“…We recall that ad(E H ) is the vector bundle over X associated to the principal H-bundle E H for the adjoint action of H on its Lie algebra h. To prove that the principal H-bundle E H is strongly semistable, it suffices to show that the vector bundle ad(E H ) is strongly semistable. To see this, let E P ⊂ (F r X ) * E H be a reduction of structure group to a maximal parabolic subgroup P ⊂ H that violates the semistability condition, or in other words, we have (3.4) degree(((F r X ) * ad(E H ))/ad(E P )) < 0 (see [9], [8]). Consider the subbundle…”

Section: Monodromy Of Principal Bundlesmentioning

confidence: 99%

“…See [9], [8] for the definition of a (strongly) semistable principal bundle with a reductive group as the structure group. We will show that the above definition coincides with the usual definition when G is reductive.…”

Section: Monodromy Of Principal Bundlesmentioning

confidence: 99%

Monodromy group for a strongly semistable principal bundle over a curve, II

Biswas¹,

Parameswaran²

2008

J. K-Theory

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Abstract. Let X be a geometrically irreducible smooth projective curve defined over a field k. Assume that X has a k-rational point; fix a k-rational point x ∈ X. From these data we construct an affine group scheme G X defined over the field k as well as a principal G X -bundle E GX over the curve X. The group scheme G X is given by a Q-graded neutral Tannakian category built out of all strongly semistable vector bundles over X. The principal bundle E GX is tautological. Let G be a linear algebraic group, defined over k, that does not admit any nontrivial character which is trivial on the connected component, containing the identity element, of the reduced center of G. Let E G be a strongly semistable principal G-bundle over X. We associate to E G a group scheme M defined over k, which we call the monodromy group scheme of E G , and a principal M -bundle E M over X, which we call the monodromy bundle of E G . The group scheme M is canonically a quotient of G X , and E M is the extension of structure group of E GX . The group scheme M is also canonically embedded in the fiber Ad(E G ) x over x of the adjoint bundle.

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Some remarks on the instability flag

Cited by 150 publications

References 11 publications

Principal bundles with parabolic structure

Principal bundles with parabolic structure

A Tannakian approach to dimensional reduction of principal bundles

Monodromy group for a strongly semistable principal bundle over a curve, II

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