[Paper Review] Pseudo-real principal Higgs bundles on compact Kaehler manifolds

Indranil Biswas, Oscar Garcı́a-Prada+1 more|arXiv (Cornell University)|Sep 26, 2012

Algebraic Geometry and Number Theory12 references3 citations

TL;DR

This paper introduces pseudo-real principal Higgs bundles on compact Kähler manifolds equipped with an anti-holomorphic involution, generalizing real algebraic bundles to complex geometry. It establishes a Donaldson–Uhlenbeck–Yau correspondence: a pseudo-real principal G-bundle admits an Einstein–Hermitian connection if and only if it is polystable, and proves a bijection between polystable pseudo-real bundles (with vanishing rational characteristic classes) and twisted representations of the extended fundamental group in a σG-invariant maximal compact subgroup of G.

ABSTRACT

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 $σ_G$. 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 $σ_G$--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.

Motivation & Objective

• To generalize the notion of real algebraic principal bundles to complex geometry by introducing pseudo-real principal G-bundles on compact Kähler manifolds with anti-holomorphic involutions.
• To define and study stability conditions—stable, semistable, and polystable—for pseudo-real principal G-bundles, relating them to standard stability notions in complex geometry.
• To establish a Donaldson–Uhlenbeck–Yau type correspondence: a pseudo-real principal G-bundle admits a compatible Einstein–Hermitian connection if and only if it is polystable.
• To construct a natural bijection between isomorphism classes of polystable pseudo-real principal G-bundles (with vanishing rational characteristic classes) and equivalence classes of twisted representations of the extended fundamental group in a σG-invariant maximal compact subgroup of G.
• To extend all results to the Higgs bundle setting, proving analogous correspondence and representation-theoretic bijection for pseudo-real Higgs G-bundles.

Proposed method

• Define pseudo-real principal G-bundles using a central element c in the center of a maximal compact subgroup KG ⊂ G, compatible with a real form σG of G.
• Introduce twisted representations of the extended fundamental group Γ(X, x₀) into the semi-direct product K̃G = KG ⋊ ℤ/2ℤ, using the involution σG and the central element c.
• Construct a flat connection D = ∇ + θ + θ* on the underlying principal G-bundle from a completely reducible twisted representation, ensuring vanishing rational characteristic classes.
• Use harmonic reduction theory (Corlette’s theorem) to show that a harmonic reduction EKG ⊂ EG can be chosen to be invariant under the pseudo-real structure ρ.
• Construct the pseudo-real Higgs G-bundle (EG, ρ, θ) from a flat connection D with completely reducible monodromy, ensuring compatibility with the involution and the Higgs field.
• Prove that the resulting Higgs bundle is polystable and that the correspondence between twisted representations and polystable pseudo-real Higgs G-bundles is bijective.

Experimental results

Research questions

• RQ1 Does a pseudo-real principal G-bundle on a compact Kähler manifold admit a compatible Einstein–Hermitian connection if and only if it is polystable?
• RQ2 Is there a natural bijection between isomorphism classes of polystable pseudo-real principal G-bundles (with vanishing rational characteristic classes) and equivalence classes of twisted representations of the extended fundamental group in a σG-invariant maximal compact subgroup of G?
• RQ3 Can the Donaldson–Uhlenbeck–Yau correspondence be extended to the setting of pseudo-real Higgs G-bundles?
• RQ4 How does the presence of a Higgs field affect the stability and reduction properties of pseudo-real principal G-bundles?
• RQ5 Under what conditions does a flat connection on a principal G-bundle lift to a pseudo-real structure compatible with a harmonic reduction?

Key findings

• A pseudo-real principal G-bundle admits a compatible Einstein–Hermitian connection if and only if it is polystable, establishing a Donaldson–Uhlenbeck–Yau type correspondence.
• There is a natural bijective correspondence between isomorphism classes of polystable pseudo-real principal G-bundles with all rational characteristic classes of positive degree vanishing and equivalence classes of twisted representations of the extended fundamental group in a σG-invariant maximal compact subgroup of G.
• The correspondence is constructed via the monodromy of a flat connection D = ∇ + θ + θ* on the underlying principal G-bundle, where D arises from a completely reducible twisted representation.
• For pseudo-real Higgs G-bundles, the same correspondence holds: polystable Higgs bundles with vanishing rational characteristic classes correspond bijectively to twisted representations with completely reducible monodromy.
• A harmonic reduction EKG ⊂ EG can be chosen to be invariant under the pseudo-real structure ρ, ensuring compatibility with the Einstein–Hermitian condition.
• The Higgs field θ and its adjoint θ* combine with the Chern connection ∇ to yield a flat connection D, whose monodromy is completely reducible and gives rise to the twisted representation.