2012/09/26 by Indranil Biswas, Oscar Garcı́a-Prada, Biswas, Indranil +4
Mathematics · #14P99 #32Q15 #53C07 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #math.AG #math.DG #msc:14P99 #msc:32Q15 #msc:53C07
paper · pdf · doi:10.48550/arxiv.1209.5814
arxiv created 2012/09/26 · openalex publication_date 2012/09/26 · arxiv updated 2012/09/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X be a compact connected Kähler manifold equipped with an anti-holomorphic involution which is compatible with the Kähler structure. Let G be a connected complex reductive affine algebraic group equipped with a real form σG. We define pseudo-real principal G--bundles on X; these are generalizations of real algebraic principal G--bundles over a real algebraic variety. Next we define stable, semistable and polystable pseudo-real principal G--bundles. Their relationships with the usual stable, semistable and polystable principal G--bundles are investigated. We then prove that the following Donaldson--Uhlenbeck--Yau type correspondence holds: a pseudo-real principal G--bundle admits a compatible Einstein-Hermitian connection if and only if it is polystable. A bijection between the following two sets is established: 1) The isomorphism classes of polystable pseudo-real principal G--bundles such that all the rational characteristic classes of the underlying topological principal G--bundle vanish. 2) The equivalence classes of twisted representations of the extended fundamental group of X in a σG--invariant maximal compact subgroup of G. (The twisted representations are defined using the central element in the definition of a pseudo-real principal G--bundle.) All these results are also generalized to the pseudo-real Higgs G--bundle.