[Paper Review] The limit of the Yang-Mills-Higgs flow on Higgs bundles

Jiayu Li, Xi Zhang|arXiv (Cornell University)|Oct 30, 2014

Geometry and complex manifolds21 references3 citations

TL;DR

This paper establishes that the Yang-Mills-Higgs flow on a Higgs bundle over a compact Kähler manifold converges at infinity to a limiting Higgs sheaf that is isomorphic to the double dual of the graded Higgs sheaf associated with the Harder-Narasimhan-Seshadri (HNS) filtration of the initial Higgs bundle. The convergence is shown via gauge-theoretic analysis, resolving singularities through blow-ups and using the uniqueness of reflexive extensions to identify the limit.

ABSTRACT

In this paper, we consider the gradient flow of the Yang-Mills-Higgs functional for Higgs pairs on a Hermitian vector bundle $(E, H_{0})$ over a compact Kähler manifold $(M, ω)$. We study the asymptotic behavior of the Yang-Mills-Higgs flow for Higgs pairs at infinity, and show that the limiting Higgs sheaf is isomorphic to the double dual of the graded Higgs sheaves associated to the Harder-Narasimhan-Seshadri filtration of the initial Higgs bundle.

Motivation & Objective

• To understand the asymptotic behavior of the Yang-Mills-Higgs flow on Higgs bundles over compact Kähler manifolds.
• To establish a correspondence between the Yang-Mills-Higgs flow and the Harder-Narasimhan-Seshadri (HNS) filtration of the initial Higgs bundle.
• To identify the limiting Higgs sheaf at infinity, particularly in the presence of singularities and bubbling phenomena.
• To prove that the limit is isomorphic to the double dual of the graded Higgs sheaf from the HNS filtration.

Proposed method

• Analyzing the Yang-Mills-Higgs flow as a gradient flow of the Yang-Mills-Higgs functional on Higgs pairs $(A, heta)$.
• Using gauge transformations and local convergence arguments to analyze the limit on the complement of the singular set $Σ_{an}$.
• Applying the Gauss-Codazzi equation to relate curvature and connection behavior on subbundles during the flow.
• Constructing a sequence of gauge transformations $u_j$ to normalize the flow and extract a limit on the quotient bundle $Q$.
• Resolving the singular set $Σ_{al}$ via finitely many blow-ups with smooth centers, preserving the HNS filtration structure.
• Using the uniqueness of reflexive extension (from [30]) to extend the limit Higgs sheaf to the whole manifold and identify it with $Gr^{HNS}(E)^{**}$.

Experimental results

Research questions

• RQ1 What is the limiting Higgs sheaf of the Yang-Mills-Higgs flow on a Higgs bundle over a compact Kähler manifold?
• RQ2 How does the flow behave at infinity when the initial Higgs bundle is not stable?
• RQ3 Can the limit be described in terms of the Harder-Narasimhan-Seshadri filtration of the initial Higgs structure?
• RQ4 Is the limiting Higgs sheaf isomorphic to the double dual of the graded Higgs sheaf from the HNS filtration?

Key findings

• The Yang-Mills-Higgs flow converges in $C^{∞}_{loc}$ away from the singular set $Σ_{an}$ of Hausdorff codimension 4.
• The limiting Higgs sheaf on $M \setminus (Σ_{al} \cup Σ_{an})$ is isomorphic to the graded Higgs sheaf $Gr^{HNS}(E, \overline{\partial}_{A_0}, \phi_0)$.
• After resolving the singular set $Σ_{al}$ via finitely many blow-ups, the limit extends to a reflexive Higgs sheaf on the entire manifold $M$.
• The limit Higgs sheaf is isomorphic to the double dual of the graded Higgs sheaf from the HNS filtration: $(E_\infty, \overline{\partial}_{A_\infty}, \phi_\infty) \simeq Gr^{HNS}(E, \overline{\partial}_{A_0}, \phi_0)^{**}$.