Semi-stable quiver bundles over Gauduchon manifolds

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Introduction
The classical Hitchin-Kobayashi correspondence establishes a deep equivalent relation between the stability and the existence of the canonical metric (or connection) on holomorphic vector bundles. The study of Hitchin-Kobayashi correspondence has a huge story line, which can be trace back to the 1980s [5,6,11,18,19]. In the new century, this correspondence still attracted lots of researchers' attention (see [2-4, 8, 14, 17, 20-22] and references therein). And a lot of important and interesting applications of the correspondence come out. Takuro Mochizuki awarded the 2022 Breakthrough Prize in Mathematics, due to his excellent work in holonomic D-modules. Among these excellent work is the complete proof of a stimulating conjecture of Masaki Kashiwara about an extension of the Hard Lefschetz Theorem and other nice properties from pure sheaves to semisimple D-modules [15]. It is amazing that the Hitchin-Kobayashi correspondence on the filtered flat bundle plays a key role in the proof of Kashiwara's conjecture. In some sense, this reveals that the Hitchin-Kobayashi correspondence plays an important role in the development of modern mathematics.
In an earlier paper,Álvarez-Cónsul and García-Prada [1] established a Hitchin-Kobayashi correspondence on quiver bunldes over the compact Kähler manifold. Recently, their result has been generalized by Hu-Huang [7] to a more general base manifold (generalized Kähler manifold). To be specific, they proved that the stability and the existence of Hermitian Yang-Mills metric on the quiver bundle are equivalent. The stability condition they considered is given by a strict inequality. When the inequality is not strict, such inequality condition is nothing but semistability. And we will consider such semi-stability case in the quiver bundle. In fact we can prove the following theorem. Theorem 1.1. Let Q = (Q 0 , Q 1 ) be a quiver, and R = (E, φ) be a holomorphic Qbundle over a compact Gauduchon manifold (X, ω). Assume σ and τ are collections of positive real numbers σ v and then it admits an approximate (σ, τ )-Hermitian Yang-Mills structure, i.e. the metrics satisfying the inequality (2.1). Remark 1.1. By the result of Nie-Zhang [16], every semi-stable holomorphic vector bundle E v over compact Gauduchon manifold X must admit a Hermitian metric with negative mean curvature √ −1Λ ω F H0,v if the slope of E v is negative. In a recently paper, Li-Zhang-Zhang [10] gave a brief characterization of mean curvature negativity of holomorphic vector bundles over compact Gauduchon manifold.
At first, we can not useÁlvarez-Cónsul and García-Prada's techniques [1] to our setting directly. Since their proof is rather rely on the Donaldson's functional on Kähler manifold, but this functional is not well-defined on the Gauduchon manifold. Secondly, we can not use Hu-Huang's results [7] to our setting neither. This is because that they arrive at an inequality (not strictly) to get a contradiction with the strict inequality condition, and this is of course not valid to the semi-stable case. The proof of the main theorem will use the Uhlenbeck-Yau's continuity method [19]. It is also worth to mention that, the perturbed equation considered in this paper is also different to Hu-Huang [7]. In [7], the perturbed term is independent of the vertex numbers σ v . We observe that, once we add the vertex numbers σ v in the perturbed term, we can complete the proof of Theorem 1.1 by adapting with Simpson [18] and Nie-Zhang's [16] arguments.
An interesting aspect of this work is that the argument on the weakly L 2 1subbundles is different from the previous quiver bundle case [1,7]. In [7], they used Lübke-Teleman's argument [13] to run this step. To our best knowledge, we can not use this to our semi-stable setting. Hence, let us look back to the reference [1]. In [1], they construct a quantity χ [1, Page 22] by the eigenvalues λ j of u ∞ = ⊕ v u ∞,v , where u ∞,v is endomorphism on E v . In some sense, it is more natural to use eigenvalues λ j,v of E v to construct the quantity χ. Once we began by doing this to start the argument, another difficulty came out. The eigenvalues λ j,v , the real numbers σ v , the rank of E v and other quantities are intimately entangled, and these can not be seperated to run the next step. To fix this, we define the maximum of λ j,v and the minimum of , then we are lucky to construct a new and useful quantity χ, which may be of independent interest.

Preliminaries
In this section we introduce the basic setup and notation that will be used throughout the paper. More detailed information on quiver bundles can be found in [1,7].

Gauduchon manifold
Let X be an n−dimensional compact Hermitian manifold, and g be a Hermitian metric with associated Kähler form ω. g is called Gauduchon if ω satisfies ∂∂ω n−1 = 0. Throughout the paper we assume (X, ω) is a Gauduchon manifold.

Quiver bundle
A quiver is a pair Q = (Q 0 , Q 1 ) together with two maps h, t : Q 0 → Q 1 . Elements of Q 0 (resp. Q 1 ) are called vertices (resp. arrows) of the quiver. For each a ∈ Q 1 , ha (resp. ta) is called the head (resp. tail) of the arrow a.
A holomorphic Q-bundle over (X, ω) is a pair R = (E, φ), where E is a collection of holomorphic vector bundle E v over (X, ω), for each v ∈ Q 0 , and φ is a collection of morphisms φ a : E ta → E ha , for each a ∈ Q 0 , such that E v = 0 for all but finitely many v ∈ Q 0 , and φ a = 0 for all but finitely many a ∈ Q 1 .

(σ, τ )-Hermitian Yang-Mills structure
where Λ ω is the contraction with ω, F Hv is the curvature of the Chern connection D Hv with respect to the metric Alvarez-Cónsul-García-Prada [1] and Hu-Huang [7] proved a holomorphic Qbundle R = (E, φ) is said to be admitting a (σ, τ )-Hermitian Yang-Mills structure if and only if R = (E, φ) is poly-stable.

Stability and semi-stability
Given a holomorphic vector bundle E v on X, by Chern-Weil theory [23], its degree is given by where F Hv is the curvature of the Chern connection D Hv with respect to the metric H v on E v . The (σ, τ )-degree and (σ, τ )-slope of holomorphic Q-bundle R = (E, φ) are given by

Proof of Theorem 1.1
Fixing a proper background Hermitian metric H 0 on R = (E, φ), denote by H ε,v = H 0,v h ε . For each v ∈ Q 0 , we consider the following perturbed equation Following the techniques in [7,13], it is not hard to show that (3.1) is solvable for all ε ∈ (0, 1]. We omit this step here, since it is standard and tedious. Using the assumption of (σ, τ )-semi-stability, we can show that This implies that max X |Φ(H ε,v )| Hε,v converges to zero as ε → 0.
By an appropriate conformal change, we can assume that H 0 satisfies By Moser's iteration method, it is not hard to prove the following lemma, which is similar to [13].
where C 1 only depends on g and H 0 .
Before giving the detailed proof, let's recall some notations.
where we also assume rank(E v ) = r. We denote ϕ(η) and Ψ(η)(A) by [18, p. 880] Now, we are ready to prove the following identity.
where s ε,v = log h ε,v and in which the first inequality used [1, Lemma 3.5] Then it follows that max If the claim does not hold, then there exist δ > 0 and a subsequence ε it follows that v∈Q0 tr(σ v u εi,v ) = 0 and u εi,v L 2 = 1. Then combining (3.4) with Step 1 We will show that u εi,v L 2 1 are uniformly bounded. Since u εi,v L 2 = 1, we only need to prove du εi,v L 2 are uniformly bounded.