Let (X, ω) be a compact connected Kähler manifold of complex dimension d and EG 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 EP ⊂ EG to P, such that the corresponding L(P)/Z(G)-bundle EL(P)/Z(G):= EP(L(P)/Z(G)) → X admits a unitary flat connection, where L(P) is the Levi quotient of P. (2) The adjoint vector bundle ad(EG) is numerically flat. (3) The principal G-bundle EG is pseudostable, and If X is a complex projective manifold, and ω represents a rational cohomology class, then the third statement is equivalent to the statement that EG is semistable with c2(ad(EG)) = 0.

On semistable principal bundles on complex projective manifolds, II / Biswas, I; Bruzzo, Ugo. - In: GEOMETRIAE DEDICATA. - ISSN 0046-5755. - 146:1(2010), pp. 27-41. [10.1007/s10711-009-9424-8]

On semistable principal bundles on complex projective manifolds, II

Bruzzo, Ugo
2010-01-01

Abstract

Let (X, ω) be a compact connected Kähler manifold of complex dimension d and EG 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 EP ⊂ EG to P, such that the corresponding L(P)/Z(G)-bundle EL(P)/Z(G):= EP(L(P)/Z(G)) → X admits a unitary flat connection, where L(P) is the Levi quotient of P. (2) The adjoint vector bundle ad(EG) is numerically flat. (3) The principal G-bundle EG is pseudostable, and If X is a complex projective manifold, and ω represents a rational cohomology class, then the third statement is equivalent to the statement that EG is semistable with c2(ad(EG)) = 0.
2010
146
1
27
41
Biswas, I; Bruzzo, Ugo
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/13938
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