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Geometriae Dedicata

, Volume 146, Issue 1, pp 27–41 | Cite as

On semistable principal bundles over a complex projective manifold, II

  • Indranil Biswas
  • Ugo Bruzzo
Original Paper

Abstract

Let (Xω) be a compact connected Kähler manifold of complex dimension d and \({E_G\,\longrightarrow\,X}\) a holomorphic principal G–bundle, where G is a connected reductive linear algebraic group defined over \({\mathbb{C}}\). Let Z(G) denote the center of G. We prove that the following three statements are equivalent:
  1. (1)
    There is a parabolic subgroup \({P\,\subset\,G}\) and a holomorphic reduction of structure group \({E_P\,\subset\,E_G}\) to P, such that the corresponding L(P)/Z(G)–bundle
    $$E_{L(P)/Z(G)}\,:=\,E_P(L(P)/Z(G))\,\longrightarrow\,X$$
    admits a unitary flat connection, where L(P) is the Levi quotient of P.
     
  2. (2)

    The adjoint vector bundle ad(E G ) is numerically flat.

     
  3. (3)
    The principal G–bundle E G is pseudostable, and
    $$\int\limits_X c_2({\rm ad}(E_G))\omega^{d-2}\,=\,0.$$
     
If X is a complex projective manifold, and ω represents a rational cohomology class, then the third statement is equivalent to the statement that E G is semistable with c 2(ad(E G )) = 0.

Keywords

Principal bundle Pseudostability Numerical effectiveness 

Mathematics Subject Classification (2000)

32L05 14L10 14F05 

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Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.School of MathematicsTata Institute of Fundamental ResearchBombayIndia
  2. 2.Scuola Internazionale Superiore di Studi AvanzatiTriesteItaly
  3. 3.Istituto Nazionale di Fisica NucleareSezione di TriesteItaly

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