[Paper Review] Kobayashi-Hitchin correspondence for analytically stable bundles

Takuro Mochizuki|arXiv (Cornell University)|Dec 25, 2017

Geometry and complex manifolds43 references3 citations

TL;DR

This paper establishes the existence of Hermitian-Einstein metrics on holomorphic vector bundles satisfying an analytic stability condition over certain non-compact Kähler manifolds, extending the Kobayashi-Hitchin correspondence beyond compact settings. The key result proves that analytic stability implies the existence of a Hermitian-Einstein metric with controlled curvature decay, applicable to instantons and monopoles on quotients of R⁴ by closed subgroups, with explicit examples of doubly periodic monopoles linked to algebraic data.

ABSTRACT

We prove the existence of a Hermitian-Einstein metric on holomorphic vector bundles with a Hermitian metric satisfying the analytic stability condition, under some assumption for the underlying Kähler manifolds. We also study the curvature decay of the Hermitian-Einstein metrics. It is useful for the study of the classification of instantons and monopoles on the quotient of $4$-dimensional Euclidean space by some types of closed subgroups. We also explain examples of doubly periodic monopoles corresponding to some algebraic data.

Motivation & Objective

• To extend the Kobayashi-Hitchin correspondence to non-compact Kähler manifolds by establishing the existence of Hermitian-Einstein metrics for analytically stable holomorphic vector bundles.
• To analyze the curvature decay properties of Hermitian-Einstein metrics on such bundles, particularly in relation to instantons and monopoles on quotients of R⁴.
• To provide a differential-geometric framework for classifying instantons and monopoles via algebraic data, especially in doubly periodic settings.
• To generalize Simpson’s analytic stability approach to non-compact settings, proving existence of metrics satisfying boundedness and L² conditions on the connection.

Proposed method

• Introduces an analytic stability condition for holomorphic vector bundles equipped with a Hermitian metric on non-compact Kähler manifolds, generalizing the notion from compact and parabolic settings.
• Applies techniques from Simpson’s work on Higgs bundles and harmonic metrics, adapting them to non-compact base spaces with finite volume and specific Kähler metric assumptions.
• Uses the Chern connection associated to a Hermitian metric and defines the curvature operator ΛF(h), requiring its trace-free part to vanish for Hermitian-Einstein condition.
• Employs a pullback construction via a covering map p: C_z × C_w^* → R × C_w^*, relating the bundle structure on the base to a product bundle with monodromy action.
• Analyzes S¹-invariant holomorphic subbundles by studying the monodromy operator Φ induced by parallel transport, showing irreducibility under non-degenerate eigenvalue conditions.
• Establishes boundedness of the metric and L²-summability of the connection difference (h h₀⁻¹) via detailed analysis of differential operators and holomorphic subbundle invariance.

Experimental results

Research questions

• RQ1 Does the Kobayashi-Hitchin correspondence hold for holomorphic vector bundles on non-compact Kähler manifolds under analytic stability?
• RQ2 Can Hermitian-Einstein metrics be constructed with controlled curvature decay on such bundles, particularly for instantons and monopoles?
• RQ3 How do the curvature and connection behavior of Hermitian-Einstein metrics relate to algebraic data in doubly periodic monopole constructions?
• RQ4 What conditions on the Kähler manifold ensure the existence of Hermitian-Einstein metrics for analytically stable bundles?

Key findings

• The paper proves the existence of a Hermitian-Einstein metric h on a holomorphic vector bundle satisfying the analytic stability condition, under suitable assumptions on the Kähler manifold.
• The metric h is mutually bounded with the initial Hermitian metric h₀, and satisfies det(h) = det(h₀), ensuring volume preservation.
• The curvature F(h) of the Hermitian-Einstein metric is bounded, and the difference (∂ + θ)(h h₀⁻¹) is L², ensuring integrability of the connection variation.
• The trace-free part of ΛF(h) vanishes, confirming the Hermitian-Einstein condition ΛF(h) = (Tr ΛF(h)/rank E) id_E.
• The construction yields a non-trivial example of doubly periodic monopoles via algebraic data, linking differential geometry to algebraic structures.
• The proof relies on showing that any S¹-invariant holomorphic subbundle must be trivial or the full bundle, via monodromy analysis of the induced connection.