The Hitchin--Kobayashi Correspondence for Quiver Bundles over Generalized K\"ahler Manifolds

TL;DR

This paper proves a correspondence between stability and Hermitian--Einstein metrics for quiver bundles over generalized K"ahler manifolds, extending classical results to a broader geometric setting.

Contribution

It establishes the Hitchin--Kobayashi correspondence for $I_ ext{pm}$-holomorphic quiver bundles on generalized K"ahler manifolds with Gauduchon metrics, generalizing known results.

Findings

Proves the equivalence of polystability and existence of Hermitian--Einstein metrics for these bundles.

Extends the Hitchin--Kobayashi correspondence to a new class of complex manifolds.

Provides a framework for analyzing quiver bundles in generalized K"ahler geometry.

Abstract

In this paper, we establish the Hitchin--Kobayashi correspondence for the $I_\pm$-holomorphic quiver bundle $\mathcal{E}=(E,\phi)$ over a compact generalized K\"{a}hler manifold $(X, I_+,I_-,g, b)$ such that $g$ is Gauduchon with respect to both $I_+$ and $I_-$, namely $\mathcal{E}$ is $(\alpha,\sigma,\tau)$-polystable if and only if $\mathcal{E}$ admits an $(\alpha,\sigma,\tau)$-Hermitian--Einstein metric.

Equations

$$J_{1,2}=\frac{1}{2}\left(\begin{array}[]{cc}I_{+}\pm I_{-}&-(\omega^{-1}_{+}\mp\omega_{-}^{-1})\\ \omega_{+}\mp\omega_{-}&-(I_{+}^{*}\pm I_{-}^{*})\\ \end{array}\right)$$

$$({\nabla }_M\pm \frac{1}{4}M┘\mathbb{H})\cdot {\xi }_{\pm }=$$

$$(df\pm \frac{1}{2}\mathbb{H})\cdot {\xi }_{\pm }=$$

$$R_{\mu \nu }^{(g)}-\frac{1}{4}{\mathbb{H}}_{\mu \alpha \beta }{\mathbb{H}}_{\nu \gamma \delta }{g}^{\alpha \delta }{g}^{\beta \delta }+2{\nabla }_{\mu }{\nabla }_{\nu }f$$

$$g^{\mu \alpha }{\nabla }_{\mu }(e^{-2f}{\mathbb{H}}_{\alpha \beta \gamma })$$

$$R^{(g)}+4g^{\mu \nu }{\nabla }_{\mu }{\nabla }_{\nu }f-4g^{\mu \nu }{\nabla }_{\mu }{\nabla }_{\nu }f-\frac{1}{12}{g}^{\mu \alpha }{g}^{\nu \beta }{g}^{\lambda \gamma }{\mathbb{H}}_{\mu \nu \lambda }{\mathbb{H}}_{\alpha \beta \gamma }$$

$${\mathrm{deg}}_{\pm }(E):=\frac{\sqrt{-1}}{2\pi }{\int }_X\mathrm{Tr}({\mathbb{F}}_H^{\pm })\wedge {\omega }_{\pm }^{n-1},$$

$$\begin{array}{ccc}{E}_{t(a)}& \stackrel{{\varphi }_a}{\to }& E_{h(a)}\\ f_{t(a)}\downarrow & & f_{h(a)}\downarrow \\ F_{t(a)}& \stackrel{{\phi }_a}{\to }& F_{h(a)}.\end{array}$$

$$\{\begin{array}{r}{\varphi }_a\circ {\bar{\partial }}_{+t(a)}={\bar{\partial }}_{+h(a)}\circ {\varphi }_a,\\ {\varphi }_a\circ {\bar{\partial }}_{-t(a)}={\bar{\partial }}_{-h(a)}\circ {\varphi }_a.\end{array}$$

$$\begin{array}{ccc}(E_{t(a)},{\bar{\partial }}_{+t(a)},{\bar{\partial }}_{-t(a)})& \stackrel{{\varphi }_a}{\to }& (E_{h(a)},{\bar{\partial }}_{+h(a)},{\bar{\partial }}_{-h(a)})\\ f_{t(a)}\downarrow & & f_{h(a)}\downarrow \\ (F_{t(a)},{\bar{\partial }}_{+t(a)}^{\prime },{\bar{\partial }}_{-t(a)}^{\prime })& \stackrel{{\phi }_a}{\to }& (F_{h(a)},{\overline{\partial }}_{+t(a)}^{\prime },{\bar{\partial }}_{-t(a)}^{\prime }).\end{array}$$

$${\mathrm{deg}}_{\alpha ,\sigma ,\tau }(\mathcal{F}):=$$

$${\mu }_{\alpha ,\sigma ,\tau }(\mathcal{F}):=$$

$$\sqrt{-1}({\alpha }_i{\sigma }_i{\mathbb{F}}_{H_i}^{+}\wedge {\omega }_{+}^{n-1}+(1-{\alpha }_i){\sigma }_i{\mathbb{F}}_{H_i}^{-}\wedge {\omega }_{-}^{n-1})$$

$$+(n-1)!\left(\sum _{a\in h^{-1}(i)}{\varphi }_a\circ {\varphi }_a^{\ast H}-\sum _{a\in t^{-1}(i)}{\varphi }_a^{\ast H}\circ {\varphi }_a\right){\mathrm{Vol}}_g$$

$$=$$

$$C_1(E_i)=$$

$$C_2(E_i)=$$

$$C_2^{\prime }(E_i)=$$

$$\sum _{i\in Q_0}{\sigma }_i(C_2(E_i)-C_2^{\prime }(E_i))+2\lambda \sum _{i\in Q_0}({\tau }_i+{\mu }_{(\alpha ,\sigma ,\tau )}(\mathcal{E}){\sigma }_i)C_1(E_i)\ge 0.$$

$$0=$$

$$\displaystyle\ \ \ \ \ \ \ \ \ \ \ -\lambda(\tau_{i}+\mu_{\alpha,\sigma,\tau}(\mathcal{E})\sigma_{i})\mathrm{Id}_{E_{i}}\Big{|}\Big{|}^{2}_{L^{2}}$$

$$=$$

$$\displaystyle\ \ \ \ \ \ \ \ \ +\frac{1}{\sigma_{i}}\bigg{|}\bigg{|}\sum_{a\in h^{-1}(i)}\phi_{a}\circ\phi_{a}^{*H}-\sum_{a\in t^{-1}(i)}\phi_{a}^{*H}\circ\phi_{a}-\lambda(\tau_{i}+\mu_{\alpha,\sigma,\tau}(\mathcal{E})\sigma_{i})\mathrm{Id}_{E_{i}}\bigg{|}\bigg{|}^{2}_{L^{2}}\Bigg{)}$$

$$\displaystyle-2\lambda\sum_{i\in Q_{0}}{(\tau_{i}+\mu_{(\alpha,\sigma,\tau)}(\mathcal{E})\sigma_{i})}\int_{X}{\rm Tr}\bigg{(}\sqrt{-1}\Big{(}\alpha\Lambda_{+}\mathbb{F}^{+}_{H_{i}}+(1-\alpha)\Lambda_{-}\mathbb{F}^{-}_{H_{i}}\Big{)}\bigg{)}d{\rm Vol}_{g}$$

$$+2\sum _{a\in Q_1}\mathrm{Re}⟨⟨{\varphi }_a,[\sqrt{-1}(\alpha {\Lambda }_{+}{\mathbb{F}}_H^{+}+(1-\alpha ){\Lambda }_{-}{\mathbb{F}}_H^{-}),\varphi {]}_a⟩{⟩}_{L^2},$$

$$||{\Lambda }_{\pm }{\mathbb{F}}_{H_i}^{\pm }|{|}_{L^2}^2=$$

$$⟨⟨{\varphi }_a,[\sqrt{-1}{\Lambda }_{\pm }{\mathbb{F}}_H^{\pm },\varphi {]}_a⟩{⟩}_{L^2}=$$

$$+⟨⟨{\varphi }_a,\frac{\sqrt{-1}}{(n-1)!}{\star }_g({\bar{\partial }}_{\pm }({\omega }_{\pm }^{n-1})\wedge {\partial }_H^{\pm }{\varphi }_a)⟩{⟩}_{L^2}$$

$$=$$

$${\sigma }_i{\sigma }_j(m_j-m_i)<{\sigma }_i{\tau }_j-{\sigma }_j{\tau }_i.$$

$${\sigma }_i{\tau }_j-{\sigma }_j{\tau }_i>0,$$

$$g=\frac{1}{4\pi |z{|}^2}(d{z}_{1}d{\bar{z}}_1+d{z}_{2}d{\bar{z}}_2),$$

$${\omega }_{+}=\frac{\sqrt{-1}}{4\pi |z{|}^2}(d{z}_1\wedge d{\bar{z}}_1+d{z}_2\wedge d{\bar{z}}_2).$$

$$\Omega =\frac{1}{|z{|}^4}({\bar{z}}_{1}d{z}_1+z_{2}d{\bar{z}}_2)\wedge ({\bar{z}}_{1}d{z}_2-z_{2}d{\bar{z}}_1)$$

$${\omega }_{-}=\frac{\sqrt{-1}}{4\pi |z{|}^4}(({\bar{z}}_{1}d{z}_1+z_{2}d{\bar{z}}_2)\wedge (z_{1}d{\bar{z}}_1+{\bar{z}}_{2}d{z}_2)+({\bar{z}}_{1}d{z}_2-z_{2}d{\bar{z}}_1)\wedge (z_{1}d{\bar{z}}_2-{\bar{z}}_{2}d{z}_1)).$$

$${\mathrm{deg}}_{+}{L}_{+}(m)$$

$${\mathrm{deg}}_{+}{L}_{-}(m)$$

$$2m{\sigma }_i{\sigma }_j({\alpha }_j-{\alpha }_i)<{\sigma }_i{\tau }_j-{\sigma }_j{\tau }_i.$$

Full text

1. 1 Introduction
2. 2 Setups
3. 3 The Hitchin–Kobayashi Correspondence

1. Introduction

The Hitchin–Kobayashi correspondence exhibits a deep relation between the algebraic notion of stability and the existence of special metrics on holomorphic vector bundles. There are several generalizations for this correspondence along different directions. For example, one replaces base manifolds with Hermitian manifolds with Gauduchon metric [22] or non-compact Kähler manifolds satisfying some analytic conditions [26]; one generalizes Yang–Mills system to other gauge theoretic systems, such as introducing Higgs fields or vortex fields via dimensional reduction [14, 27, 4], introducing singularities for Hermitian–Einstein connection and parabolic structure on vector bundle [24, 25]; introducing frame structure via vacuum expectation value of the scalar fields in $N=2$ vector multiplet [8]; one changes the stability condition, typically relaxes to semistability and approximate Hermitian–Einstein metric [6, 7, 21]; one considers an analog of such correspondence in positive characteristic or mixed characteristic [9, 20].

In present paper, our considerations focus on generalized Kähler manifold as the base manifold and quiver bundle as the gauge theoretic system. Generalized Kähler manifold was first discovered by Gates, Hull, and Roček as the target space of $N=(2,2)$ sigma model [10], and then reformulated under the context of Hitchin’s generalized complex geometry [15, 11] by Gualtieri [12]. There are abundant candidates for generalized Kähler manifold, for example, all degenerate del Pezzo surfaces and all Hirzebruch surfaces admits non-trivial generalized Kähler structures [16]. On the other hand, quiver bundle coming from quiver gauge theory consists of a set of vector bundles and a set of morphisms between these bundles [1, 2].

We will define the notion of holomorphic quiver bundles over a generalized Kähler manifold, and introduce suitable stability and good metric for them. We should be faced with some new features in our setting: such stability depends on several real parameters reflecting the generalized Kähler structure on base manifold and quiver structure on gauge theoretic system, and such metric satisfies a series of mutually coupled equations. Then we prove certain set-theoretic Hitchin–Kobayashi correspondence, namely we have the following main theorem which generalizes the results in [1, 18].

Theorem 1.1** (= Theorem 3.15).**

Let $Q=(Q_{0},Q_{1})$ be a quiver, and $\mathcal{E}=(E,\phi)$ be an $I_{\pm}$-holomorphic $Q$-bundle over an $n$-dimensional compact generalized Kähler manifold $(X,I_{+},I_{-},g,b)$ such that $g$ is Gauduchon with respect to both $I_{+}$ and $I_{-}$, then $\mathcal{E}$ is $(\alpha,\sigma,\tau)$-polystable if and only if $\mathcal{E}$ admits an $(\alpha,\sigma,\tau)$-Hermitian–Einstein metric.

More related questions are proposed. Since a type of interesting generalized Kähler manifolds, so-called generalized Calibi–Yau manifolds appear in compactification of Type II string theory, must be non-compact, we need generalize such correspondence to the non-compact case. The parameters in the definition of stability form a parameter space of stability conditions which is partitioned into chambers, studying the wall-crossing on this space is also an interesting topic, maybe the Hitchin–Kobayashi correspondence can play some role.

2. Setups

In this paper, a generalized Kähler manifold refers to the geometric object defined by the following two equivalent approaches.

Definition 2.1**.**

([11])A manifold $X$ is called a generalized Kähler manifold if it carries two generalized complex structures $J_{1},J_{2}\in\mathrm{End}(TX\oplus T^{*}X)$ satisfying

• •

$J_{1}J_{2}=J_{2}J_{1}$,

• •

the symmetric pairing $G(A,B)=\langle J_{1}(A),J_{2}(B)\rangle$ is positive-definite for any non-zero $A,B\in TX\oplus T^{*}X$, where $\langle\cdot,\cdot\rangle$ denotes the natural inner product on $TX\oplus T^{*}X$.

Definition 2.2**.**

([12]) A manifold $X$ is called a generalized Kähler manifold if it carries the data $(I_{+},I_{-},g,b)$, where

• •

$I_{\pm}$ are two complex structures on $X$,

• •

$g$ is a Riemannian metric on $X$,

• •

$b$ is a two-form on $X$,

• •

$I_{\pm}$ are parallel with respect to the connections $\nabla^{\pm}=\nabla\pm\frac{1}{2}g^{-1}\mathbb{H}$, respectively, where $\nabla$ is the Levi-Civita connection of $g$ and $\mathbb{H}=db$.

The generalized Calabi–Yau manifold is an important kind of generalized Kähler manifold.

Definition 2.3**.**

([13])A generalized Calabi–Yau manifold is a generalized Kähler manifold $(X,J_{1},J_{2})$ such that both nowhere vanishing pure spinors $\psi_{1},\psi_{2}$ corresponding to $J_{1},J_{2}$, respectively satisfy the following conditions

• •

$d\psi_{1}=d\psi_{2}=0$,

• •

$(\psi_{1},\bar{\psi}_{1})=(\psi_{2},\bar{\psi}_{2})$,

where $(\bullet,\bullet)$ is the Mukai pairing.

Remark 2.4**.**

More generally, one defines the twisted generalized Kähler manifold as the manifold $X$ with 4-tuple $(I_{+},I_{-},g,\mathbb{H})$, where $I_{\pm},g$ is the same as above, and $\mathbb{H}$ is a closed 3-form such that $I_{\pm}$ are parallel with respect to the connections $\nabla^{\pm}=\nabla\pm\frac{1}{2}g^{-1}\mathbb{H}$, respectively. Similarly, one can also introduce the twisted generalized Calabi–Yau manifold by replacing the first condition on pure spinors by $d_{\mathbb{H}}\psi_{1}=d_{\mathbb{H}}\psi_{2}=0$ for $d_{\mathbb{H}}=d+\mathbb{H}\wedge$. When $\mathbb{H}$ is an exact 3-form, they reduce to the generalized Kähler manifold and generalized Calabi–Yau manifold defined as above.

Definition 2.5**.**

([18]) Let $(X,I_{+},I_{-},g,b)$ be a generalized Kähler manifold, and $E$ be a complex vector bundle over $X$. $E$ is called $I_{\pm}$-holomorphic if there are two operators $\bar{\partial}_{\pm}:C^{\infty}(E)\rightarrow C^{\infty}(E\otimes T^{0,1}_{I_{\pm}}X)$ such that they define a holomorphic structure on $E$ with respect to $I_{\pm}$ respectively.

Given an $I_{\pm}$-holomorphic vector bundle $(E,\bar{\partial}_{+},\bar{\partial}_{-})$, denote by $\iota$ the natural isomorphisms between $\bar{L}_{\pm}$ and $T^{0,1}_{I_{\pm}}X$, one defines $\bar{D}_{\pm,s}(v):=\bar{\partial}_{\pm,\iota(s)}(v)$ for $s\in C^{\infty}(\bar{L}_{\pm})$ and $v\in C^{\infty}(E)$, where $L_{+}=L_{1}\cap L_{2}$, $L_{-}=L_{1}\cap\bar{L}_{2}$ with $L_{1}$, $L_{2}$ be $\sqrt{-1}$-eigensubbundles of $(TX\oplus T^{*}X)\otimes\mathbb{C}$ with respect to the generalized complex structure $J_{1,2}$ determined by

[TABLE]

for Kähler forms $\omega_{\pm}=g(I_{\pm}\cdot,\cdot)$. Then $\bar{D}=\bar{D}_{+}+\bar{D}_{-}:C^{\infty}(E)\rightarrow C^{\infty}(E\otimes\bar{L}_{1})$ defines a generalized holomorphic bundle with respect to $J_{1}$ if and only if $\bar{\partial}_{+}\bar{\partial}_{-}+\bar{\partial}_{-}\bar{\partial}_{+}=0$ [18].

Moreover, we make the following assumptions on the $n$-dimensional generalized Kähler manifold $(X,I_{+},I_{-},g,b)$ in this paper:

• •

$g$ is Gauduchon, i.e., $dd^{c}_{\pm}\omega_{\pm}^{n-1}=0$ and $d\mathrm{Vol}_{g}=\frac{1}{n!}\omega_{\pm}^{n}$, where $d^{c}_{\pm}=I_{\pm}\circ d\circ I_{\pm}$;

• •

$X$ is compact.

The first assumption is not too restrictive. It can be satisfied for generalized Kähler 4-manifolds automatically, and for real compact Lie groups. On the second assumption, we have the following no-go type theorem.

Proposition 2.6**.**

1. (1)

A compact twisted generalized Kähler surface has even first Betti number if $\mathbb{H}$ is exact, and has odd first Betti number if $\mathbb{H}$ is not exact. 2. (2)

A compact twisted generalized Calabi–Yau manifold must be a usual Calabi–Yau manifold.

Proof.

(Sketch) The first result has been proved by the authors of [3]. We only prove the second claim. The structure of generalized Calabi–Yau reduces the structure group $O(2n,2n)$ of $TX\oplus T^{*}X$ to $SU_{+}(n)\times SU_{-}(n)$, then there are two globally defined $SU_{\pm}(n)$-invariant spinors $\xi_{\pm}$. The constraints on pure spinors can be rewritten in terms of $\xi_{\pm}$ [29]

[TABLE]

for $\forall M\in C^{\infty}(TX)$, exact three-form $\mathbb{H}=db$ and smooth function $f=\log\frac{1}{(\psi_{1},\bar{\psi}_{1})}$, where $\nabla$ denotes the spin connection with respect to $g$, and $\cdot$ stands for the Clifford multiplication. We only need to show if $X$ is compact then $\mathbb{H}$ vanishes. Indeed, the following equations are derived from the above conditions [17]

[TABLE]

After taking trace we get $g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}e^{-2f}-\frac{1}{6}e^{-2f}g^{\mu\alpha}g^{\nu\beta}g^{\lambda\gamma}\mathbb{H}_{\mu\nu\lambda}\mathbb{H}_{\alpha\beta\gamma}=0$, then integrating over $X$ implies the vanishing of $\mathbb{H}$ if $X$ is compact. ∎

Now let $(E,\bar{\partial}_{+},\bar{\partial}_{-})$ be an $I_{\pm}$-holomorphic bundle over a generalized Kähler manifold $X$, fix a Hermitian metric $H$ on $E$, then there is a unique Chern-connection compatible with the complex structures $I_{\pm}$ respectively, given by $D_{H}^{\pm}:=\partial_{H}^{\pm}+\overline{\partial}_{\pm}$, whose curvature form is denoted by $\mathbb{F}_{H}^{\pm}$. Then we define the degrees associated to the two Chern connections as follows:

[TABLE]

which are independent of the choice of Hermitian metric $H$ on $E$, since for any two Hermitian metrics $H$ and $H^{\prime}$ on $E$, we have ${\rm Tr}(\mathbb{F}^{\pm}_{H})={\rm Tr}(\mathbb{F}_{H^{\prime}}^{\pm})+\partial_{\pm}\overline{\partial}_{\pm}(\log\det((H^{\prime})^{-1}H))$.

Definition 2.7**.**

1. (1)

A quiver $Q=(Q_{0},Q_{1},h,t:Q_{1}\to Q_{0})$ is a $4$-tuple, where

• •

$Q_{0}$ and $Q_{1}$ are finite sets of vertices and arrows, respectively,

• •

$h,t:Q_{1}\to Q_{0}$ map each arrow $a\in Q_{1}$ to its head $h(a)$ and tail $t(a)$, respectively. 2. (2)

A $Q$-sheaf on a complex manifold $X$ is a pair $\mathcal{E}=(E,\phi)$, where $E=\{E_{i}\}_{i\in Q_{0}}$ is a collection of sheaves of $\mathcal{O}_{X}$-modules and $\phi=\{\phi_{a}\}_{a\in Q_{1}}$ a collection of morphisms $\phi_{a}:E_{t(a)}\to E_{h(a)}$. In particular, if each $E_{i}$ is locally free, $\mathcal{E}$ is called a $Q$-bundle. A $Q$-subsheaf of $\mathcal{E}=(E,\phi)$ is a $Q$-sheaf $\mathcal{E}^{\prime}=(E^{\prime},\phi^{\prime})$ such that $E^{\prime}_{i}$ is a subsheaf of $E_{i}$ for each vertex $i$ and $\phi^{\prime}_{a}=\phi_{a}|_{E_{h(a)}}$ for each arrow $a$. 3. (3)

A morphism $f:\mathcal{E}\to\mathcal{F}$ between two $Q$-sheaves $\mathcal{E}=(E,\phi)$ and $\mathcal{F}=(F,\varphi)$ is a collection of morphisms $f_{i}:E_{i}\to F_{i}$ such that for each arrow $a\in Q_{1}$, the following diagram commutes:

[TABLE] 4. (4)

A Hermitian metric on a $Q$-bundle $\mathcal{E}=(E,\phi)$ is a collection $H=\{H_{i}\}_{i\in Q_{0}}$ of Hermitian metrics $H_{i}$ on $E_{i}$. For each arrow $a\in Q_{1}$, by virtue of the Hermitian metrics at tail and head, the morphism $\phi_{a}$ has a smooth adjoint $\phi_{a}^{*H}:E_{h(a)}\to E_{t(a)}$ with respect to the Hermitian metrics at tail and head, that is, $H_{h(a)}(\phi_{a}(u),v)=H_{t(a)}(u,\phi_{a}^{*H}(v))$ for any sections $u,v$ of $E_{h(a)},E_{t(a)}$. 5. (5)

A $Q$-bundle $\mathcal{E}=(E,\phi)$ on a generalized Kähler manifold $(X,I_{+},I_{-},g,b)$ is called $I_{\pm}$-holomorphic if

• •

each $E_{i}$, $i\in Q_{0}$, is an $I_{\pm}$-holomorphic bundle, i.e., $E_{i}$ carries two holomorphic structures $\bar{\partial}_{+i},\bar{\partial}_{-i}$ with respect to $I_{\pm}$, respectively,

• •

each $\phi_{a}$, $a\in Q_{1}$, is $I_{\pm}$-holomorphic, namely

[TABLE] 6. (6)

A morphism $f:\mathcal{E}\to\mathcal{F}$ between two $I_{\pm}$-holomorphic $Q$-bundles $\mathcal{E}=(E,\phi)$ and $\mathcal{F}=(F,\varphi)$ is a collection of $I_{\pm}$-holomorphic morphisms $f_{i}:(E_{i},\bar{\partial}_{i+},\bar{\partial}_{i-})\to(F_{i},\bar{\partial}_{+i}^{\prime},\bar{\partial}_{-i}^{\prime})$, such that for each arrow $a\in Q_{1}$, the following diagram commutes:

[TABLE] 7. (7)

An $I_{\pm}$-holomorphic $Q$-bundle $\mathcal{E}=(E,\phi)$ is said to be simple if any endomorphism $f:\mathcal{E}\rightarrow\mathcal{E}$ must have the form $f=\{c\mathrm{Id}_{E_{i}}\}_{i\in Q_{0}}$ for a constant $c\in\mathbb{C}$.

Definition 2.8**.**

([1, 18]) Let $\mathcal{E}=(E,\phi)$ be an $I_{\pm}$-holomorphic $Q$-bundle.

1. (1)

A coherent $Q$-subsheaf $\mathcal{F}$ of $\mathcal{E}$ is a 4-tuple $\mathcal{F}=(\mathcal{F}_{+},\mathcal{F}_{-},\mathbb{S}_{+},\mathbb{S}_{-})$, where

• •

$\mathcal{F}_{\pm}=(F_{\pm},\varphi)$ are $Q$-subsheaves of the $Q$-sheaves $\mathcal{E}_{\pm}$, where $\mathcal{E}_{\pm}=(E_{\pm}=\{E_{\pm i}=(E_{i},\bar{\partial}_{\pm i})\}_{i\in Q_{0}},\phi=\{\phi_{a}\}_{a\in Q_{1}})$, respectively,

• •

$\mathbb{S}_{\pm}=\{S_{\pm i}\}_{i\in Q_{0}}$ are collections of analytic subsets of $(X,I_{\pm})$,respectively, such that for each $i\in Q_{0}$

• –

$S_{i}=S_{+i}\cup S_{-i}$ has codimension at least 2,

• –

$F_{\pm i}|_{X\backslash S_{\pm i}}$ are locally free and $F_{+i}|_{X\backslash S_{i}}=F_{-i}|_{X\backslash S_{i}}:=F_{i}$ as smooth vector bundles. 2. (2)

For any coherent subsheaf $\mathcal{F}$ of $\mathcal{E}$, we define $(\alpha,\sigma,\tau)$-degree and $(\alpha,\sigma,\tau)$-slope as follows:

[TABLE]

where $\alpha_{i}\in(0,1),\sigma_{i}\in\mathbb{R}^{+},\tau_{i}\in\mathbb{R}$, and ${\rm rk}(F_{i})={\rm rk}(F_{+i})={\rm rk}(F_{-i})$ denotes the rank of the corresponding sheaves. $\mathcal{E}$ is called $(\alpha,\sigma,\tau)$-stable (respectively, $(\alpha,\sigma,\tau)$-semistable) if for any proper coherent $Q$-subsheaf $\mathcal{F}$, we have $\mu_{\alpha,\sigma,\tau}(\mathcal{F})<\mu_{\alpha,\sigma,\tau}(\mathcal{E})$ (respectively, $\mu_{\alpha,\sigma,\tau}(\mathcal{F})\leq\mu_{\alpha,\sigma,\tau}(\mathcal{E}))$, and $\mathcal{E}$ is called polystable if it is the the direct sum of $(\alpha,\sigma,\tau)$-stable $Q$-subsheaves of the same slope with $\mathcal{E}$.

Due to the classical extension theorem [5], we have the following extension theorem for the coherent $Q$-subsheaves.

Proposition 2.9**.**

For each $i\in Q_{0}$, there are unique holomorphic bundles $\hat{F}_{\pm i}$ over $(X,I_{\pm})$ extending the bundles $F_{\pm i}|_{X\backslash S_{\pm i}}$, respectively, hence there is a unique $I_{\pm}$-bundle $(\hat{F}_{i},\hat{\bar{\partial}}_{+},\hat{\bar{\partial}}_{-})$ over $(X,I_{\pm})$ extending the $I_{\pm}$-bundle $(F_{i},\bar{\partial}_{+},\bar{\partial}_{-})$ over $X\backslash S_{i}$.

The following facts are analogs of classical cases [19].

Proposition 2.10**.**

Let $f:\mathcal{E}\to\mathcal{F}$ be a morphism between two $I_{\pm}$-holomorphic $Q$-bundles $\mathcal{E}=(E,\phi)$ and $\mathcal{F}=(F,\varphi)$.

1. (1)

If $\mathcal{E}$ and $\mathcal{F}$ are $(\alpha,\sigma,\tau)$-semistable, then $\mu_{\alpha,\sigma,\tau}(\mathcal{E})\leq\mu_{\alpha,\sigma,\tau}(\mathcal{F})$. 2. (2)

If $\mathcal{E}$ and $\mathcal{F}$ are stable of the same $(\alpha,\sigma,\tau)$-slope, then $f$ is an isomorphism. 3. (3)

If $\mathcal{E}$ is $(\alpha,\sigma,\tau)$-stable, then it is simple.

Definition 2.11**.**

A Hermitian metric $H$ on an $I_{\pm}$-holomorphic $Q$-bundle $\mathcal{E}=(E,\phi)$ is called an $(\alpha,\sigma,\tau)$-Hermitian–Einstein metric if for each vertex $i\in Q_{0}$ it satisfies the following equations

[TABLE]

with constants $\lambda=\frac{2\pi}{(n-1)!\int_{X}d{\rm Vol}_{g}}$ and $\gamma$.

Remark 2.12**.**

Taking trace and the sum over all vertices and then doing integral over $X$ on both sides, we see that $\gamma$ is exactly the slope $\mu_{(\alpha,\sigma,\tau)}(\mathcal{E})$.

We employ the following notations:

• •

$S(E_{i},H_{i})$ is the space of smooth $H_{i}$-Hermitian endomorphisms of $E_{i}$, $S^{+}(E_{i},H_{i})\subset S(E_{i},H_{i})$ is the open subset of positive-definite ones;

• •

$S(\mathcal{E},H)=\prod\limits_{i\in Q_{0}}S(E_{i},H_{i})$, $S^{+}(\mathcal{E},H)=\prod\limits_{i\in Q_{0}}S^{+}(E_{i},H_{i})$. The metric $H$ induces a metric on $S(\mathcal{E},H)$, also denoted by $H$, namely $\langle f,g\rangle_{H}=\sum_{i\in Q_{0}}\langle f_{i},g_{i}\rangle_{H_{i}}$ for $f=(f_{i})_{i\in Q_{0}},g=(g_{i})_{i\in Q_{0}}\in S(\mathcal{E},H)$.

• •

$L^{p}_{k}(S)$ denotes the corresponding Sobolev space.

• •

The pointwise or global norms and inner products $|\bullet|,\langle\bullet,\bullet\rangle,||\bullet||,\langle\langle\bullet,\bullet\rangle\rangle_{L^{2}}$ are defined with respect to the metrics $H_{i}$ or induced metric induced metrics on $E_{h(a)}\otimes(E_{t(a)})^{*}$ from the metrics $H_{h(a)}$ and $H_{t(a)}$ unambiguously depending on the contexts.

Proposition 2.13**.**

Let $H$ be an $(\alpha,\sigma,\tau)$-Hermitian–Einstein metric on an $I_{\pm}$-holomorphic $Q$-bundle $\mathcal{E}=(E,\phi)$ over $X$, then we define

[TABLE]

where $\Lambda_{\pm}$ is the adjoint of the operator of the wedge by $\omega_{\pm}$ with respect to the metric $g$. When $\alpha_{i}=\alpha_{j}=\alpha$ for $\forall i,j\in Q_{0}$, the following inequality holds

[TABLE]

Proof.

By assumption we have

[TABLE]

where $[A,\phi]_{a}=A_{h(a)}\circ\phi_{a}-\phi_{a}\circ A_{t(a)}$ for $A\in{\rm End}(\mathcal{E})$. Then we find the desired inequality by virtue of the following identities

[TABLE]

where $\star_{g}$ denotes the Hodge star with respect to $g$, the connections acting on $\phi_{a}$ are the induced connections on $E_{h(a)}\otimes(E_{t(a)})^{*}$, and the $I_{\pm}$-holomorphicity of $\phi_{a}$’s plays a crucial roal in the second identity. ∎

We end this section with some examples.

Example 2.14**.**

1. (1)

We first consider $X=\mathbb{P}^{1}$ with the standard Kähler structure $(I,\omega)$, it can be retreated as a generalized Kähler manifold by taking $I=I_{+}=I_{-},\omega=\omega_{+}=\omega_{-}$. Let $Q=(Q_{0},Q_{1},h,t)$ be a quiver with $Q_{0}=\{i,j\}$, $Q_{1}=\{a\}$ and $t(a)=i,h(a)=j$, then we consider the $I_{\pm}$-holomorphic $Q$-bundle $\mathcal{E}=(E,\phi)$ over $\mathbb{P}^{1}$ given by $E_{i}=\mathcal{O}(m_{i}),E_{j}=\mathcal{O}(m_{j})$ for $m_{j}\geq m_{i}$, and $0\neq\phi_{a}\in H^{0}(\mathbb{P}^{1},\mathcal{O}(m_{j}-m_{i}))$. Obviously, $\deg_{\pm}\mathcal{O}(m)=m$, hence for the stability parameters $\alpha_{i},\alpha_{j}$; $\sigma_{i},\sigma_{j}$ and $\tau_{i},\tau_{j}$, $\mathcal{E}$ is $(\alpha,\sigma,\tau)$-stable if and only if the following inequality holds

[TABLE]

In particular, the parameters $\sigma,\tau$ are subject to the condition

[TABLE]

which gives the constraints on these parameters as follows:

• •

if $\tau_{i}=0$, then $\tau_{j}>0$;

• •

if $\tau_{j}=0$, then $\tau_{i}<0$;

• •

if $\tau_{i},\tau_{j}\neq 0$, then $\frac{\sigma_{i}}{\sigma_{j}}>\frac{\tau_{i}}{\tau_{j}}$. 2. (2)

Now we consider the example of Hopf surfaces, which can be found in [12] (Example 1.21) and [18] (Section 4 for details). Let $X$ be a standard Hopf surface, namely $X=\mathbb{C}^{2}\backslash\{(0,0)\}/(2(z_{1},z_{2})\sim(z_{1},z_{2}))$, then $X$ is diffeomorphic to $S^{3}\times S^{1}$. Denote by $I_{+}$ the induced complex structure from $\mathbb{C}^{2}$, the Hermitian metric is given by

[TABLE]

for $|z|^{2}=z_{1}\bar{z}_{1}+z_{2}\bar{z}_{2}$, and the associated 2-form $\omega_{+}=gI_{+}$ is

[TABLE]

One can specify another complex structure $I_{-}$ by providing a generator

[TABLE]

for $\Omega^{2,0}((X,I_{+}))$. It is easy to check that $(g,I_{-})$ is also Hermitian, and the associated 2-form is given by

[TABLE]

Then $(I_{+},I_{-},g,\mathbb{H})$ defines a twisted generalized Kähler structure on $X$, where $\mathbb{H}=d^{c}_{+}\omega_{+}=-d^{c}_{-}\omega_{-}$ [12, 18]. Actually, the torsion of twisted generalized Kähler structures on $X$ cannot be exact [12]. There is a natural projection $\mathrm{pr}:X\rightarrow\mathbb{P}^{1}$ onto $\mathbb{P}^{1}$ via $(z_{1},z_{2})\mapsto[z_{1}:z_{2}]$, and this projection is holomorphic with respect to $I_{+}$. We set $\mathcal{O}_{+}(m):=\mathrm{pr}^{*}\mathcal{O}_{\mathbb{P}^{1}}(m)$ for all $m\in\mathbb{Z}$, where $\mathcal{O}_{\mathbb{P}^{1}}(m)$ denotes the holomorphic line bundle on $\mathbb{P}^{1}$ of degree $m$. Consider the inverse map $\varrho:X\rightarrow X$, $(z_{1},z_{2})\mapsto(z_{1},z_{2})^{-1}:=\frac{1}{|z|^{2}}(\bar{z}_{1},-z_{2})$, which is a biholomorphic map from $(X,I_{-})$ to $(X,I_{+})$, and we introduce $\mathcal{O}_{-}(m):=\varrho^{*}\mathcal{O}_{+}(m)$ for all $m\in\mathbb{Z}$. For simplicity, we denote $\mathcal{O}_{\pm}(0)$ by $\mathcal{O}_{\pm}$. By Proposition 4.5 of [18], $\mathcal{O}_{+}(m)$ can be made into an $I_{\pm}$-holomorphic line bundle $L_{+}(m):=(\mathbb{O},\bar{\partial}_{m,+},\bar{\partial}_{m,-})$ on $(X,g,I_{+},I_{-},\mathbb{H})$ such that $(\mathbb{O},\bar{\partial}_{m,+})\simeq\mathcal{O}_{+}(m)$ and $(\mathbb{O},\bar{\partial}_{m,-})\simeq\mathcal{O}_{-}(-m)$, where $\mathbb{O}$ denotes the topologically trivial line bundle $X\times\mathbb{C}$ on $X$. Similarly, the $I_{\pm}$-holomorphic line bundle associated to $\mathcal{O}_{-}(m)$ is denoted by $L_{-}(m):=(\mathbb{O},\bar{\partial}^{\prime}_{m,+},\bar{\partial}^{\prime}_{m,-})$ with isomorphisms $(\mathbb{O},\bar{\partial}^{\prime}_{m,+})\simeq\mathcal{O}_{+}(-m)$ and $(\mathbb{O},\bar{\partial}^{\prime}_{m,-})\simeq\mathcal{O}_{-}(m)$. Moreover, one can show that [18]

[TABLE]

Next we take the quiver $Q$ be the same as in (1), and an $I_{\pm}$-holomorphic $Q$-bundle $\mathcal{E}=(E,\phi)$ which is given by $E_{i}=L_{+}(m_{i}),E_{j}=L_{+}(m_{j})$ and $\phi_{a}$, where $\phi_{a}$ must vanish if $m_{i}\neq m_{j}$. Assume $m_{i}=m_{j}=m$ and $\phi_{a}$ is non-zero, then $\mathcal{E}$ is $(\alpha,\sigma,\tau)$-stable if and only if

[TABLE]

Finally, as the Example 4.11 in [18], let $V$ be a fixed smooth complex vector bundle of rank 2, we choose $I_{\pm}$-holomorphic structures $\bar{\partial}_{\pm}^{V}$ on $V$ as follows:

• •

$\bar{\partial}_{+}^{V}$ is $I_{+}$-holomorphic structure such that $V_{+}:=(V,\bar{\partial}_{+}^{V})$ is not isomorphic to a sum of two line bundles and is given by the non-trivial extension

[TABLE]

for $m_{+}\in\mathbb{Z}^{>0}$,

• •

$\bar{\partial}_{-}^{V}$ is $I_{-}$-holomorphic structure such that $V_{-}:=(V,\bar{\partial}_{-}^{V})$ is given by the non-trivial extension

[TABLE]

for $m_{-}\in\mathbb{Z}^{\geq 2}$.

We assume the images of $\mathcal{O}_{\pm}$ in $V_{\pm}$ coincide as smooth line subbundles of $V$. Then $\widetilde{L}:=(\mathbb{O},\bar{\partial}_{0,+},\bar{\partial}^{\prime}_{0,-})$ is the only $I_{\pm}$-holomorphic line subbundle of $(V,\bar{\partial}^{V}_{+},\bar{\partial}^{V}_{-})$ [18]. The $I_{\pm}$-holomorphic $Q$-bundle $\mathcal{E^{\prime}}=(E^{\prime},\phi^{\prime})$ is given by $E^{\prime}_{i}=\widetilde{L},E^{\prime}_{j}=(V,\bar{\partial}^{V}_{+},\bar{\partial}^{V}_{-})$ and $\phi^{\prime}_{a}$ is determined by the inclusions $\chi_{\pm}$. To find the constraints on stability parameters, note that $\mathcal{E}^{\prime}$ has 3 proper $Q$-subbundles:

• (i)

$\mathcal{F}=(F,\phi)$, where $F_{i}=\tilde{L}$, $F_{j}=\tilde{L}$ and $\phi_{a}$ is induced by $\phi_{a}^{\prime}$, which is identity;

• (ii)

$\mathcal{F}=(F,\phi)$, where $F_{i}=0$, $F_{j}=\tilde{L}$ and $\phi_{a}=0$;

• (iii)

$\mathcal{F}=(F,\phi)$, where $F_{i}=0$, $F_{j}=(V,\bar{\partial}^{V}_{+},\bar{\partial}^{V}_{-})$ and $\phi_{a}=0$,

therefore, $\mathcal{E^{\prime}}$ is $(\alpha,\sigma,\tau)$-stable if and only if the following inequalities are satisfied

[TABLE]

3. The Hitchin–Kobayashi Correspondence

Lemma 3.1**.**

If there exists an $(\alpha,\sigma,\tau)$-Hermitian–Einstein metric on an $I_{\pm}$-holomorphic $Q$-bundle $\mathcal{E}=(E,\phi)$ over an $n$-dimensional generalized Kähler manifold $(X,I_{+},I_{-},g,b)$, then $\mathcal{E}$ is $(\alpha,\sigma,\tau)$-polystable.

Proof.

Let $\mathcal{E}^{\prime}$ be a proper coherent $Q$-subsheaf of $\mathcal{E}$. At each vertex $i\in Q_{0}$, one defines the orthogonal projections $p_{\pm i}:E_{\pm i}\rightarrow E^{\prime}_{\pm i}$, which are defined outside $S_{\pm i}$, respectively, via the metric $H_{i}$, then we have

[TABLE]

where $\xi_{i}^{\pm}=\bar{\partial}_{\pm i}p_{\pm i}$ denote the second fundamental forms which are of class $L^{2}$. Hence, by assumption that $H$ is a Hermitian–Einstein metric on $\mathcal{E}$, the degree is calculated as

[TABLE]

where $S=\bigcup\limits_{i\in Q_{0}}S_{i}$, $\phi_{a}^{\bot}$ is the composition $(E_{t(a)}^{\prime})^{\bot}\xrightarrow{\phi_{a}}E_{h(a)}\xrightarrow{p_{h(a)}}E^{\prime}_{h(a)}$ for the orthogonal complement $(E_{t(a)}^{\prime})^{\bot}$ of $E_{t(a)}^{\prime}$ in $E_{t(a)}$ defined outside $S_{h(a)}\bigcup S_{t(a)}$, and $|\phi_{a}^{\bot}|^{2}_{H}$ is defined via the induced metric $H$ on $E_{h(a)}\otimes(E_{t(a)})^{*}$. It follows that $\mathcal{E}$ is semistable. Assume $\mathcal{E}=(E,\phi)$ is indecomposable, i.e. $\mathcal{E}$ cannot be written as a direct sum of two $Q$-bundles, then either $\xi_{i}\neq 0$ for some $i\in Q_{0}$ or $\phi_{a}^{\bot}\neq 0$ for some $a\in Q_{1}$, therefore $\mu_{\alpha,\sigma,\tau}(\mathcal{E}^{\prime})<\mu_{\alpha,\sigma,\tau}(\mathcal{E})$, thus $\mathcal{E}$ is stable. Finally, we find that $\mathcal{E}$ is polystable. ∎

Next we will use the continuity method to show the converse direction, thus to show that if an $I_{\pm}$-holomorphic $Q$-bundle $\mathcal{E}=(E,\phi)$ is $(\alpha,\sigma,\tau)$-stable, then there exist an $(\alpha,\sigma,\tau)$-Hermitian–Einstein metric $H$ on it. The approach of proof we employed here mainly follows from [18, 23].

We fix a Hermitian metric $H$ on an $I_{\pm}$-holomorphic $Q$-bundle $\mathcal{E}=(E,\phi)$. If $\tilde{H}=Hf=\{H_{i}f_{i}\}_{i\in Q_{0}}$ is an $(\alpha,\sigma,\tau)$-Hermitian–Einstein metric for $f\in S^{+}(H,\mathcal{E})$, then at each vertex $i$ we have

[TABLE]

where

[TABLE]

The perturbed equation is given by

[TABLE]

for $\varepsilon\in[0,1]$. Consider the set

[TABLE]

Proposition 3.2**.**

1. (1)

There exists a Hermitian metric $H=\{H_{i}\}_{i\in Q_{0}}$ on an $I_{\pm}$-holomorphic $Q$-bundle $\mathcal{E}=(E,\phi)$, such that the simultaneous equations $\{L^{1}_{{(\alpha,\sigma,\tau)i}}(f)=0\}_{i\in Q_{0}}$ has a solution $f^{(1)}\in S^{+}(H,\mathcal{E})$ with

[TABLE] 2. (2)

If $\sigma_{i}=\sigma_{j}$ for all $i,j\in Q_{0}$, then there exists a Hermitian metric $H=\{H_{i}\}_{i\in Q_{0}}$ on an $I_{\pm}$-holomorphic $Q$-bundle $\mathcal{E}=(E,\phi)$, such that

[TABLE]

and

[TABLE]

for any solution $f^{(\varepsilon)}$ of $\{L^{\varepsilon}_{{(\alpha,\sigma,\tau)i}}(f)=0\}_{i\in Q_{0}}$.

Proof.

(1) For any Hermitian metric $G=\{G_{i}\}$ on $\mathcal{E}$ one defines the operator

[TABLE]

where $P_{\pm}^{G_{i}}:=\sqrt{-1}\Lambda_{\pm}\bar{\partial}_{\pm}\partial^{\pm}_{G_{i}}$ for each vertex $i$, and in particular, it is denoted by $P_{i}$ when acting on functions. Since

[TABLE]

there exist a function $\chi_{i}$ such that

[TABLE]

Hence by taking $\tilde{H}_{i}=e^{\chi_{i}}G_{i}$, we obtain

[TABLE]

Let us define

[TABLE]

Since $K_{(\alpha,\sigma,\tau)}(\tilde{H}_{i})$ is $\tilde{H}_{i}$-Hermitian for each $i\in Q_{0}$, and $\phi_{a}\circ(\phi_{a})^{*\tilde{H}}$ is also $\tilde{H}_{i}$-Hermitian for any $a\in h^{-1}(i)$ or $a\in t^{-1}(a)$ according to the definition of $\phi_{a}^{*H}$, $f_{i}^{(1)}$ is an $H_{i}$-Hermitian positive-definite endomorphism. Obviously, it follows from the traceless-ness of $\sum_{i\in Q_{0}}K_{(\alpha,\sigma,\tau)}(\tilde{H}_{i})$ and $\sum_{i\in Q_{0}}(\sum_{a\in h^{-1}(i)}\phi_{a}\circ(\phi_{a})^{*\tilde{H}}-\sum_{a\in t^{-1}(i)}(\phi_{a})^{*\tilde{H}}\circ\phi_{a})$ that

[TABLE]

For $H$ and $f^{(1)}$, we have

[TABLE]

(2) The metric $H$ is chosen as above, then the following identities hold

[TABLE]

where we note that $P_{i}+P_{i}^{*H_{i}}=\sigma_{i}\Delta_{g}$ when acting on functions, here $\Delta_{g}$ is the Laplacian associated to the metric $g$. As a consequence, if $\sigma_{i}=\sigma_{j}=\sigma$ for all $i,j\in Q_{0}$, we get

[TABLE]

Moreover, let $\{\rho_{i}={\rm Tr}(\log f_{i}^{(\varepsilon)})\}_{i\in Q_{0}}$ for a solution $f^{(\varepsilon)}$ of $\{L^{\varepsilon}_{{(\alpha,\sigma,\tau)i}}(f)=0\}_{i\in Q_{0}}$, then we have

[TABLE]

which implies $\sum_{i\in Q_{0}}\rho_{i}=\log(\prod_{i\in Q_{0}}\det f_{i}^{(\varepsilon)})=0$, i.e. $\prod_{i\in Q_{0}}\det f_{i}^{(\varepsilon)}=1$. ∎

From now on we fix a background metric $H$ on $\mathcal{E}$ as in the above proposition. For $f\in S^{+}(\mathcal{E},H)$, one defines

[TABLE]

Obviously, $\hat{L}_{(\alpha,\sigma,\tau)i}(\varepsilon,f)\in S(E_{i},H_{i})$. Denote by $\mathbb{L}_{(\alpha,\sigma,\tau)i}^{\varepsilon,f}$ the linearization of $\hat{L}_{(\alpha,\sigma,\tau)i}(\varepsilon,f)$ , which is calculated as follows:

[TABLE]

where $\eta=(\eta_{i})\in S(\mathcal{E},H)$, and $[\phi^{*H},\eta\circ f^{-1}]_{a}=\phi_{a}^{*H}\circ\eta_{h(a)}\circ f^{-1}_{h(a)}-\eta_{t(a)}\circ f^{-1}_{t(a)}\circ\phi_{a}^{*H}$. It is clear that $\mathbb{L}_{(\alpha,\sigma,\tau)i}^{\varepsilon,f}$ extends to a second-order elliptic differential operator of index zero between Sobolev spaces $L^{p}_{k}(S(\mathcal{E},H))$ and $L^{p}_{k-2}(S(\mathcal{E},H))$.

Proposition 3.3**.**

Let $\varepsilon\in(0,1]$, $\lambda\in\mathbb{R}$, $f\in S^{+}(\mathcal{E},H)$, $\eta\in S(\mathcal{E},H)$, and we denote $\eta_{i}^{f_{i}}=f_{i}^{-\frac{1}{2}}\circ\eta_{i}\circ f_{i}^{-\frac{1}{2}}$, $\phi_{a}^{f}=f_{h(a)}^{\frac{1}{2}}\circ\phi_{a}\circ f_{t(a)}^{-\frac{1}{2}}$. If $f$ is a solution of $\{L^{\varepsilon}_{(\alpha,\sigma,\tau)i}(f)=0\}_{i\in Q_{0}}$, and the equality $\mathbb{L}_{(\alpha,\sigma,\tau)i}^{\varepsilon,f}(\eta)+\lambda f_{i}\circ\log f_{i}=0$ holds at each vertex, then we have the following inequality

[TABLE]

where $(\partial^{\pm}_{H_{i}})^{f_{i}}=\mathrm{Ad}_{f_{i}^{-\frac{1}{2}}}\circ\partial^{\pm}_{H_{i}}\circ\mathrm{Ad}_{f_{i}^{\frac{1}{2}}}$ and $\bar{\partial}_{\pm}^{f_{i}}=\mathrm{Ad}_{f_{i}^{\frac{1}{2}}}\circ\bar{\partial}_{\pm}\circ\mathrm{Ad}_{f_{i}^{-\frac{1}{2}}}$ with the adjoint action $\mathrm{Ad}_{\bullet}\circ\star=\bullet\circ\star\circ\bullet^{-1}$.

Proof.

Since $\mathbb{L}_{(\alpha,\sigma,\tau)i}^{\varepsilon,f}(\eta)=\eta_{i}\circ L^{\varepsilon}_{(\alpha,\sigma,\tau)i}(f)+f_{i}\circ\frac{d}{dt}|_{t=0}L_{(\alpha,\sigma,\tau)i}^{\varepsilon}(f+t\eta)$, we have

[TABLE]

Indeed, the left hand side has been calculated as

[TABLE]

where $P^{H_{i}}_{f_{i}}=\alpha_{i}\sigma_{i}P^{H_{i}}_{+f_{i}}+(1-\alpha_{i})\sigma_{i}P^{H_{i}}_{-f_{i}}$ for $P^{H_{i}}_{\pm f_{i}}=\sqrt{-1}\Lambda_{\pm}\bar{\partial}_{\pm}^{f_{i}}(\partial^{\pm}_{H_{i}})^{f_{i}}$ and $\Phi=\mathrm{Ad}_{f_{i}^{\frac{1}{2}}}\circ\frac{d}{dt}|_{t=0}\log(f_{i}+t\eta_{i})$. Taking inner product with $\eta_{i}^{f_{i}}$ and then taking sum over all vertices on both sides, we have

[TABLE]

The estimate \big{\langle}\Phi,\eta_{i}^{f_{i}}\big{\rangle}_{H_{i}}\geq\big{|}\eta_{i}^{f_{i}}\big{|}^{2}_{H_{i}} leads to the desired inequality. ∎

Proposition 3.4**.**

$J$* is a non-empty open subset of $(0,1]$.*

Proof.

This claim is an application of implicit function theorem for Banach spaces. We only need to show the operator $\mathbb{L}_{(\alpha,\sigma,\tau)}^{\varepsilon,f}:=\bigoplus_{i\in Q_{0}}\mathbb{L}_{(\alpha,\sigma,\tau)i}^{\varepsilon,f}:L^{p}_{k}(S(H,\mathcal{E}))\rightarrow L^{p}_{k-2}(S(H,\mathcal{E}))$ sending $\eta\in L^{p}_{k}(S(H,\mathcal{E}))$ to $(\mathbb{L}_{(\alpha,\sigma,\tau)i}^{\varepsilon,f}(\eta))$ is injective. Indeed, assume $\mathbb{L}_{(\alpha,\sigma,\tau)i}^{\varepsilon,f}(\eta)=0$, then since $P^{H_{i}}$ is a positive operator and $\varepsilon,\alpha_{i},1-\alpha_{i},\sigma_{i}>0$, we find that $|\eta_{i}^{f_{i}}|^{2}_{H_{i}}=0$ for $\forall i\in Q_{0}$ due to maximal principle. Therefore, it follows that $\eta=0$ from the above proposition (put $\lambda=0$). Moreover, one can show that any solution $f\in L_{k}^{p}(S^{+}(H,\mathcal{E}))$ is in fact smooth by some rather standard arguments. ∎

Let $\varepsilon_{0}\in(0,1]$ and suppose there exists a solution $f^{(\varepsilon)}\in S^{+}(\mathcal{E},H)$ of $\{L^{\varepsilon}_{(\alpha,\sigma,\tau)i}(f)=0\}_{i\in Q_{0}}$ for any $\varepsilon>\varepsilon_{0}>0$ with $\varepsilon\in(0,1]$. From the proof of Proposition 3.2, we may assume $\int_{X}\log(\prod_{i\in Q_{0}}\det f_{i}^{(\varepsilon)})d{\rm Vol}_{g}=0$. Let $\mu^{(\varepsilon)}=\frac{df^{(\varepsilon)}}{d\varepsilon}$, $\nu^{(\varepsilon)}=(f^{(\varepsilon)})^{-\frac{1}{2}}\circ\mu^{(\varepsilon)}\circ(f^{(\varepsilon)})^{-\frac{1}{2}}$, $\gamma^{(\varepsilon)}=\mathrm{Ad}_{(f^{(\varepsilon)})^{-\frac{1}{2}}}\circ\nu^{(\varepsilon)}$, and $m^{(\varepsilon)}=(m_{i}^{(\varepsilon)})$ for $m_{i}^{(\varepsilon)}=\max\limits_{X}\{|\log f_{i}^{(\varepsilon)}|\}$. Sometimes for convenience, we will drop the upper index $(\varepsilon)$ when there is no ambiguity.

Proposition 3.5**.**

Let $\mathcal{E}=(E,\phi)$ be a simple $I_{\pm}$-holomorphic $Q$-bundle. Then there exist positive constant $C(m),D(m)$ depending only on $m$ such that we have the following inequalities

[TABLE]

Proof.

(1) Firstly, we have the inequality

[TABLE]

Integrating both sides over $X$ leads to

[TABLE]

Define the operator $\Delta_{i}:L^{p}_{k}(S(\mathcal{E},H))\rightarrow L^{p}_{k-2}(S(E_{i},H_{i}))$ as

[TABLE]

for $\eta\in L^{p}_{k}(S(H,\mathcal{E}))$, and define

[TABLE]

as $\Delta(\eta)=(\Delta_{i}(\eta))$. Then we have

[TABLE]

Obviously, $\Delta$ is an elliptic self-adjoint positive operator, and the assumption that $\mathcal{E}$ is simple implies ${\rm Ker}\Delta=\mathbb{C}\mathrm{Id}_{\mathcal{E}}$. Therefore,

[TABLE]

where $\kappa$ is the smallest positive eigenvalue of $\Delta$ because we have the restriction

[TABLE]

The desired inequality is then obtained by $\sum_{i\in Q_{0}}|\gamma_{i}|^{2}_{H_{i}}\geq C_{1}(m)\sum_{i\in Q_{0}}|\nu_{i}|^{2}_{H_{i}}$.

(2) Since $\frac{d}{d\varepsilon}\hat{L}_{(\alpha,\sigma,\tau)i}(\varepsilon,f)=\mathbb{L}_{(\alpha,\sigma,\tau)i}^{\varepsilon,f}(\mu)+f_{i}\circ\log f_{i}=0$, applying Proposition 3.3, we have

[TABLE]

The inequality in (1) gives rise to

[TABLE]

hence $||\nu_{i}||_{L^{2}}\leq C_{3}(m)$ for any vertex $i\in Q_{0}$. On the other hand, again by Proposition 3.3, there exists a second-order elliptic operator $P$ such that

[TABLE]

which implies

[TABLE]

thus $\max\limits_{X}|\mu_{i}|_{H_{i}}\leq D(m)$ for any vertex $i\in Q_{0}$. ∎

Proposition 3.6**.**

For any $f\in S^{+}(\mathcal{E},H)$, we have the following inequalities

[TABLE]

Proof.

(1) At each point of $X$, we write

[TABLE]

where $r_{i}={\rm rk}(E_{i})$, $\{e_{A_{i}}\}_{A_{i}}$ forms a $H_{i}$-unitary frame of $E_{i}$ and $\{e^{A_{i}}\}_{A_{i}}$ stands for the dual frame, and $\theta_{A_{i}}$’s are real numbers. Then we calculate pointwisely

[TABLE]

(2) It is similar to Lemma 5.14 in [18].∎

Corollary 3.7**.**

Assume $f\in S^{+}(H,\mathcal{E})$ is a solution of $\{L_{(\alpha,\sigma,\tau)i}^{\varepsilon}(f)=0\}_{i\in Q_{0}}$. Then

• (1)

$m_{i}\leq\frac{1}{\varepsilon}m_{K}$, where $m_{K}:=|Q_{0}|\underset{i\in Q_{0}}{\sum}\underset{X}{\max}|K^{\phi}_{(\alpha,\sigma,\tau)}(H_{i})|_{H_{i}}$.

• (2)

There exist positive constants $C,C^{\prime}$ independent of $m$ such that for any vertex $i\in Q_{0}$ we have the inequality

[TABLE]

Proof.

(1) Since $L_{(\alpha,\sigma,\tau)i}^{\varepsilon}(f)=0$, we have

[TABLE]

where $K^{\phi}_{(\alpha,\sigma,\tau)}(H_{i})=K_{(\alpha,\sigma,\tau)}(H_{i})+\sum\limits_{a\in h^{-1}(i)}\phi_{a}\circ(\phi_{a})^{*H}-\sum\limits_{a\in t^{-1}(i)}(\phi_{a})^{*H}\circ\phi_{a}$. Therefore we arrive at

[TABLE]

which implies

[TABLE]

for $m_{K}:=|Q_{0}|\underset{i\in Q_{0}}{\sum}\underset{X}{\max}|K^{\phi}_{(\alpha,\sigma,\tau)}(H_{i})|_{H_{i}}$.

(2) We have seen that

[TABLE]

As in Proposition 3.5 (2), we get the desired inequalities. ∎

Proposition 3.8**.**

Let $\mathcal{E}=(E,\phi)$ be a simple $I_{\pm}$-holomorphic $Q$-bundle. Assume there is a smooth family solution $f^{(\varepsilon)}\in S^{+}(\mathcal{E},H)$ to $\{L^{\varepsilon}_{(\alpha,\sigma,\tau)i}(f)=0\}_{i\in Q_{0}}$ and that there is a uniform $m^{\prime}$ so that $m^{(\varepsilon)}<m^{\prime}$ for all $\varepsilon\in(\varepsilon_{0},1]$. Then there exits a constant $C(m^{\prime})$ independent of $\varepsilon$ such that $||f_{i}^{(\varepsilon)}||_{L_{2}^{p}}\leq C(m^{\prime})$ for each vertex $i\in Q_{0}$.

Proof.

By Kähler identities on Gauduchon manifold (Lemma 7.2.5 in [23]), we have

[TABLE]

Since $\Delta+\mathrm{Id}$ is self-adjoint and has strictly positive spectrum and by Proposition 3.5 (2), there is a positive constant $C$ such that

[TABLE]

On the other hand, it follows from $\mathbb{L}_{(\alpha,\sigma,\tau)i}^{\varepsilon,f}(\mu)+f_{i}\circ\log f_{i}=0$ and $L^{\varepsilon}_{(\alpha,\sigma,\tau)i}(f)=0$ that the variation $\mu_{i}$ satisfies the equation

[TABLE]

We need to estimate the $L^{p}$-norms of the terms on the right hand side (cf. Proposition 3.3.5 in [23]). Indeed, we have

• •

the first term is exactly $-\sqrt{-1}\mu_{i}\circ(K^{\phi}_{(\alpha,\sigma,\tau)}(H_{i})+\varepsilon\log f_{i})$, hence the norm is bounded by a constant $C_{1}(m^{\prime})$ thanks to Proposition 3.5 (2);

• •

the norm of second term is bounded by $C_{2}(m^{\prime})(||\mu_{i}||_{L_{1}^{2p}}||f_{i}||_{L_{1}^{2p}}+||f_{i}||^{2}_{L_{1}^{2p}})$ due to Hölder’s inequality;

• •

the norms of the third and the forth terms are obviously bounded by some constants $C_{3}(m^{\prime}),C_{4}(m^{\prime})$;

• •

the norm of the last term is also bounded by a constant $C_{5}(m^{\prime})$ since

[TABLE]

Consequently, we obtain

[TABLE]

hence,

[TABLE]

When $\sum\limits_{i\in Q_{0}}||\mu_{i}||_{L_{2}^{p}}\geq|Q_{0}|$, we have $\sum\limits_{i\in Q_{0}}||\mu_{i}||_{L_{1}^{2p}}\leq C_{8}(m^{\prime})\sum\limits_{i\in Q_{0}}||\mu_{i}||^{\frac{1}{2}}_{L_{2}^{p}}$ by an interpolation inequality of Aubin. We may assume $\sum\limits_{i\in Q_{0}}||f_{i}||_{L_{2}^{p}}\geq|Q_{0}|$, otherwise the conclusion has already holds truly, then similarly $\sum\limits_{i\in Q_{0}}||f_{i}||_{L_{1}^{2p}}\leq C_{9}(m)\sum\limits_{i\in Q_{0}}||f_{i}||^{\frac{1}{2}}_{L_{2}^{p}}$. Therefore we arrive at

[TABLE]

Let $\underline{\mu}=\max_{i\in Q_{0}}\{||\mu_{i}||_{L_{2}^{p}}\}$, $\underline{f}=\max_{i\in Q_{0}}\{||f_{i}||_{L_{2}^{p}}\}$, then

[TABLE]

which implies for each vertex $i\in Q_{0}$, we have

[TABLE]

Clearly, the above inequality is also satisfied when $\sum\limits_{i\in Q_{0}}||\mu_{i}||_{L_{2}^{p}}\leq|Q_{0}|$. Now take $i_{0}$ be the vertex such that $||f_{i_{0}}||_{L_{2}^{p}}=\underline{f}$, then

[TABLE]

Integration over $[\varepsilon,1]$ leads to the final inequality $||f_{i}||_{L_{2}^{p}}\leq\underline{f}\leq C(m^{\prime})$. ∎

Lemma 3.9**.**

$\mathcal{E}=(E,\phi)$* and $f^{(\varepsilon)}$ are as above. Then*

1. (1)

$J=(0,1]$. 2. (2)

If there is a constant $C$ such that $\max\limits_{i\in Q_{0}}\{||f_{i}^{(\varepsilon)}||_{L^{2}}\}_{i}\leq C$ for all $\varepsilon\in(0,1]$ then there exists a solution $f^{(0)}$ of the equations $\{L_{(\alpha,\sigma,\tau)i}(f)=0\}_{i\in Q_{0}}$.

Proof.

(1) Assume $J=(\varepsilon_{0},1]$ for $\varepsilon_{0}>0$. If one shows the solution $f^{(\varepsilon)}$ actually extends to $[\varepsilon_{0},1]$, contradicting with the openness of $J$, the claim follows. By Corollary 3.7 (1), $m_{i}^{(\varepsilon)}\leq\frac{1}{\varepsilon}m_{K}<\frac{1}{\varepsilon_{0}}m_{K}:=m^{\prime}$, then $||f^{(\varepsilon)}_{i}||_{L_{2}^{p}}\leq C(m^{\prime})$ as a result of Proposition 3.8. This uniform estimate guarantees the existence of the solution $f^{(\varepsilon_{0})}$. Choose $p>2n$. The uniform $L_{2}^{p}$ norm bound implies that there is a sequence $\varepsilon_{\mathfrak{k}}\rightarrow\varepsilon_{0}$, $f_{i}^{(\varepsilon_{\mathfrak{k}})}\rightarrow f_{i}^{(\varepsilon_{0})}$ for each vertex $i\in Q_{0}$ converges weakly in $L_{2}^{p}$-norm and strongly in $L_{1}^{p}$-norm. We need to show $L^{\varepsilon_{0}}_{(\alpha,\sigma,\tau)i}(f^{(\varepsilon_{0})})=0$. For any smooth section $\psi=(\psi_{i})\in C^{\infty}({\rm End}(\mathcal{E}))$, we compute

[TABLE]

Note that the maps $\log:L_{1}^{p}\rightarrow L^{2}$ and $\exp:L^{2}\rightarrow L^{2}$, hence $(\bullet)^{-1}=\exp(-\log\bullet):L_{1}^{p}\rightarrow L^{2}$ are continuous, we then conclude that the terms on the right hand converge to zero when $\varepsilon_{\mathfrak{k}}\rightarrow\varepsilon_{0}$.

(2) This claim follows from Corollary 3.7 (2), Proposition 3.8 and similar arguments as above. ∎

To complete the Hitchin–Kobayashi correspondence, we should prove when the boundedness of $||f_{i}^{(\varepsilon)}||_{L^{2}}$’s is not satisfied, i.e., there is a vertex $i\in Q_{0}$ such that $\limsup\limits_{\varepsilon\rightarrow 0}||f_{i}^{(\varepsilon)}||_{L^{2}}=\infty$, the $I_{\pm}$-holomorphic $Q$-bundle $\mathcal{E}$ is not $(\alpha,\sigma,\tau)$-stable. Firstly, we observe that under such assumption on the norms, $\sum\limits_{i\in Q_{0}}{\rm rk}(E_{i})>1$, otherwise, there is only one vertex $i$ with $f_{i}^{(\varepsilon)}=1$.

Proposition 3.10**.**

Let $f\in S^{+}(\mathcal{E},H)$ be the solution to $\{L^{\varepsilon}_{(\alpha,\sigma,\tau)i}(f)=0\}_{i\in Q_{0}}$. Then for any $0<\varsigma\leq 1$, we have the inequality

[TABLE]

Proof.

Firstly, we have the following equality

[TABLE]

According to Lemma 5.21 in [18], we have

[TABLE]

As the proof of Proposition 3.6, we calculate the third summand on the right side

[TABLE]

Combining the above two (in)equalities provides the desired result. ∎

Corollary 3.11**.**

With the same conditions as in Proposition 3.10, there is a constant $C$ independent of $\varepsilon$ such that for each vertex $i\in Q_{0}$, we have

[TABLE]

Proof.

Taking $\varsigma=1$, the above proposition shows

[TABLE]

where the second inequality applies Corollary 3.7 (1), and the third one follows from the inequalities $C_{2}^{-1}({\rm Tr}f_{i})\leq|f_{i}|_{H_{i}}\leq C_{2}({\rm Tr}f_{i})$ since $f\in S^{+}(\mathcal{E},H)$. By the Lemma 3.2.2 in [23], we have

[TABLE]

as desired. ∎

For $\varepsilon>0$, $x\in X$ and $i\in Q_{0}$, we denote by $e_{i}(\varepsilon,x)$ the largest eigenvalue of $\log f_{i}^{(\varepsilon)}$ for the solution $f^{(\varepsilon)}$ of $\{L^{\varepsilon}_{(\alpha,\sigma,\tau)i}(f)=0\}_{i\in Q_{0}}$ and define $\rho_{i}^{(\varepsilon)}=\exp(-M_{i}^{(\varepsilon)})$ for $M_{i}^{(\varepsilon)}=\max\limits_{X}e_{i}(\varepsilon,x)$, $\mathfrak{F}_{i}^{(\varepsilon)}=\rho_{i}^{(\varepsilon)}f_{i}^{(\varepsilon)}$. Let $i_{0}\in Q_{0}$ be the vertex depending on $\varepsilon$ such that $||f_{i_{0}}^{(\varepsilon)}||_{L^{2}}=\max\limits_{i\in Q_{0}}\{||f_{i}^{(\varepsilon)}||_{L^{2}}\}$.

Proposition 3.12**.**

Assume $\limsup\limits_{\varepsilon\rightarrow 0}||f_{i_{0}}^{(\varepsilon)}||_{L^{2}}=\infty$, then there is a sequence $\varepsilon_{\mathfrak{k}}\rightarrow 0$ such that $\rho^{(\varepsilon_{\mathfrak{k}})}_{i_{0}}\rightarrow 0$ and $\mathfrak{F}_{i_{0}}^{(\varepsilon_{\mathfrak{k}})}$ converges weakly in $L_{1}^{2}$-norm to an $\mathfrak{F}_{\infty}\neq 0$.

Proof.

Firstly, by definition, we have $\rho_{i}^{(\varepsilon)}f_{i}^{(\varepsilon)}\leq\mathrm{Id}_{E_{i}}$ and $\max\limits_{X}(\rho_{i}^{(\varepsilon)}|f_{i}^{(\varepsilon)}|_{H_{i}})\geq 1$ for $\forall i\in Q_{0}$. Then we get

[TABLE]

where the second inequality follows from Corollary 3.11. Therefore, there is a constant $C_{0}$ independent of $\varepsilon$ such that

[TABLE]

In particular, the above inequality on the left hand side means that if $\mathfrak{F}_{i_{0}}^{(\varepsilon_{\mathfrak{k}})}$ converges to $\mathfrak{F}_{\infty}$ weakly in $L_{1}^{2}$-norm then we may assume it converges strongly in $L^{2}$-norm, hence $\mathfrak{F}_{\infty}\neq 0$. On the other hand, by Corollary 3.7 (1) and Proposition 3.10, we have

[TABLE]

where $D^{+}_{H_{i}}$ is the induced Chern connection on the endomorphism bundle. This means that $\rho^{(\varepsilon)}_{i_{0}}f_{i_{0}}^{(\varepsilon)}$ is $L_{1}^{2}$-bounded uniformly. By assumption that $L^{2}$-norm of $f_{i_{0}}^{(\varepsilon)}$ is unbounded, there is a sequence $\varepsilon_{\mathfrak{k}}\rightarrow 0$ such that $\rho^{(\varepsilon_{\mathfrak{k}})}_{i_{0}}\rightarrow 0$ and $\mathfrak{F}_{i_{0}}^{(\varepsilon_{\mathfrak{k}})}$ converges weakly in $L_{1}^{2}$-norm. ∎

We have shown that there is a sequence $\varepsilon_{\mathfrak{k}}\rightarrow 0$ such that $\mathfrak{F}^{(\varepsilon_{\mathfrak{k}})}_{i_{0}}$ converges weakly to a non-zero $L_{1}^{2}$-endomorphism $\mathfrak{F}^{\infty}$ of $\mathcal{E}$. Similarly, for $0<\varsigma\leq 1$, define $(\mathfrak{F}_{i}^{(\varepsilon)})^{\varsigma}=(\rho_{i}^{(\varepsilon)}f_{i}^{(\varepsilon)})^{\varsigma}$ and $(\mathfrak{F}^{(\varepsilon)})^{\varsigma}=\bigoplus\limits_{i\in Q_{0}}(\mathfrak{F}_{i}^{(\varepsilon)})^{\varsigma}$, then there exists a sequence $\varsigma_{\mathfrak{k}}\rightarrow 0$ and $\widetilde{\mathfrak{F}^{\infty}}\neq 0$ such that $(\mathfrak{F}^{\infty})^{\varsigma_{\mathfrak{k}}}\rightarrow\widetilde{\mathfrak{F}^{\infty}}$ weakly in $L_{1}^{2}$-norm. Then we introduce the endomorphism

[TABLE]

Proposition 3.13**.**

$\Theta$* satisfies the following identities in $L^{1}$-norm*

1. (1)

$\Theta^{2}=\Theta=\Theta^{*H}$, 2. (2)

$(\mathrm{Id}_{\mathcal{E}}-\Theta)\circ\bar{\partial}_{\pm}\Theta=0$, 3. (3)

$(\mathrm{Id}-\Theta)_{h(a)}\circ\phi_{a}\circ\Theta_{t(a)}=0$* for $\forall a\in Q_{1}$.*

Therefore, $\Theta$ defines a $Q$-coherent subsheaf $\mathcal{F}$ of $\mathcal{E}$ with the property that

[TABLE]

Proof.

(1) These facts are obvious.

(2) As the proof of Proposition 3.4.6 iii) in [23], the following calculations lead to the identities in (2):

[TABLE]

for $0\leq\varsigma_{1}\leq\frac{\varsigma_{2}}{2}\leq 1$, where the first inequality is due to Fatou lemma, the second one is because of the inequality $\mathrm{Id}_{E_{i}}-(\mathfrak{F}^{(\varepsilon_{\mathfrak{k}})}_{i})^{\varsigma_{1}}\leq\frac{2\varsigma_{1}}{2\varsigma_{1}+\varsigma_{2}}(\mathfrak{F}^{(\varepsilon_{\mathfrak{k}})}_{i})^{-\frac{\varsigma_{2}}{2}}$, the third one is as the result of Proposition 3.10, and the last one follows from Corollary 3.7 (1).

(3) Similarly as above, since

[TABLE]

we have

[TABLE]

which indicates the desired identities.

The existence of $\mathcal{F}$ is due to the classical result of Uhlenbeck and Yau [28]. Non-vanishing of $\mathfrak{F}^{\infty}$ implies $\sum\limits_{i\in Q_{0}}{\rm rk}(F_{i})<\sum\limits_{i\in Q_{0}}{\rm rk}(E_{i})$. On the other hand, $\int_{X}\log(\prod\limits_{i\in Q_{0}}\det f_{i}^{(\varepsilon)})d{\rm Vol}_{g}=0$ means that almost everywhere either there exists a vertex $i\in Q_{0}$ such that $\lim\limits_{\varepsilon\rightarrow 0}|f_{i}^{(\varepsilon)}|_{H_{i}}<\infty$ and $\det f_{i}^{(\varepsilon)}\rightarrow 0$, or $\lim\limits_{\varepsilon\rightarrow 0}f_{i}^{(\varepsilon)}$ has zero eigenvalue for some vertex $i$ with $|f_{i}^{(\varepsilon)}|_{H_{i}}\rightarrow\infty$, whenever we have $\sum\limits_{i\in Q_{0}}{\rm rk}(\mathfrak{F}_{i}^{(\varepsilon)})<\sum\limits_{i\in Q_{0}}{\rm rk}(E_{i})$, hence $\sum\limits_{i\in Q_{0}}{\rm rk}(F_{i})>0$. ∎

Lemma 3.14**.**

$\mu_{\alpha,\sigma,\tau}(\mathcal{F})\geq\mu_{\alpha,\sigma,\tau}(\mathcal{E})$.

Proof.

The slope of $\mathcal{F}$ is given by

[TABLE]

Since $\sum\limits_{i\in Q_{0}}\int_{X\backslash S}{\rm Tr}(K^{\phi}_{(\alpha,\sigma,\tau)}(H_{i}))d{\rm Vol}_{g}=0$, we have

[TABLE]

[TABLE]

where we have noted that

[TABLE]

since $\sum\limits_{i\in Q_{0}}\int_{X}{\rm Tr}(\log f_{i}^{(\varepsilon_{\mathfrak{k}})})d{\rm Vol}_{g}=0$ . On the other hand, we have

[TABLE]

where the second equality follows from the properties that $\Theta^{*H}=\Theta$ and $(\mathrm{Id}-\Theta)_{h(a)}\circ\phi_{a}\circ\Theta_{t(a)}=0$ for $\forall a\in Q_{1}$. Putting the above calculations together confirms the lemma. ∎

Now combining Lemmas 3.1, 3.9 and 3.14 gives us the main theorem:

Theorem 3.15**.**

Let $Q=(Q_{0},Q_{1})$ be a quiver, and $\mathcal{E}=(E,\phi)$ be an $I_{\pm}$-holomorphic $Q$-bundle over an $n$-dimensional compact generalized Kähler manifold $(X,I_{+},I_{-},g,b)$ such that $g$ is Gauduchon with respect to both $I_{+}$ and $I_{-}$, then $\mathcal{E}$ is $(\alpha,\sigma,\tau)$-polystable if and only if $\mathcal{E}$ admits an $(\alpha,\sigma,\tau)$-Hermitian–Einstein metric.

Acknowledgments

The author Z. Hu would like to thank Prof. Yang-Hui He and Prof. Kang Zuo for their useful discussions, and the author P. Huang would like to thank Prof. Jiayu Li and Prof. Xi Zhang for their kind help and encouragement. The author P. Huang was financial supported by China Scholarship Council (No. 201706340032). The authors would like to thank the referees for their careful reading.