Proof. We begin to start with the following:

We remember that \(E\) is \(\alpha^{n-1}\)-slope stable. Let us define Kähler classes \(\alpha_{\varepsilon}:=\alpha+\varepsilon\omega_0\) for small \(\varepsilon>0\). The we snow that \(E\) is \(\alpha_{\varepsilon}^{n-1}\)-slope stable for small \(\varepsilon>0\). Let \(0\neq\mathcal{F}_0\subsetneq E\) be a torsion free subsheaf with maximal \(\alpha^{n-1}\)-slope in Lemma 18 \((2)\). Then there is a large \(\varepsilon_0>0\) which is independent of subsheaves of \(E\) such that for any \(0<\varepsilon<\varepsilon_0\) and \(0\neq \mathcal{F}\subsetneq E\), the following inequality holds: \[\begin{align} &\mu_{\alpha_{\varepsilon}}(E)-\mu_{\alpha_{\varepsilon}}(\mathcal{F})\\ &=\mu_{\alpha}(E)-\mu_{\alpha}(\mathcal{F})+\Sigma_{j=1}^{n-1}C_j\varepsilon^j\left(\frac{\int_Xc_1(E)\wedge\alpha^{n-1-j}\wedge\omega_0^j}{{\rm{rk}}E}-\frac{\int_Xc_1(\mathcal{F})\wedge\alpha^{n-1-j}\wedge\omega_0^j}{{\rm{rk}}\mathcal{F}} \right)\\ &\ge \mu_{\alpha}(E)-\mu_{\alpha}(\mathcal{F}_0)-\varepsilon C\\ &>0. \end{align}\] Here the inequality in the third line follows from Lemma 18 since \(\alpha^{n-1-j}\wedge\omega_0^j\) is represented by a positive current \(\langle T_{\min}^{n-1-j}\rangle\wedge\omega_0^j\), where \(T_{\min}\) is a closed positive \((1,1)\)-current with minimal singularities in \(\alpha\). And the last inequality comes from the \(\alpha^{n-1}\)-slope stability of \(E\). ◻

Proof. Since the result follows from the proof of [20] Theorem 5.1, we only sketch the proof. Let \(T\) be a closed positive \((1,1)\)-current in \(\alpha\) with minimal singularities which satisfies the complex Monge-Ampère equation \[\langle T^n\rangle=e^f\omega_0^n\] where \(f\in C^{\infty}(X)\). By [20] (refer to Theorem 13), such \(T\) exists and it is smooth Kähler on \(X\setminus D\). We denote by \(T=\theta+dd^c\varphi\). Let us define a smooth form by \(\theta_i:=\theta+(1/i)\omega_0\) in a Kähler class \(\alpha+(1/i)\omega_0\) and consider a smooth Kähler metric \(\omega_i=\theta_i+dd^c\varphi_i\) defined by \[\omega_i^n=c_i(e^f+\frac{1}{i})\omega_0^n.\] Such \(\omega_i\) exists by [26]. We also consider \(\varphi_{D'}\) defined by \([D']=\theta-\omega_0+dd^c\varphi_{D'}\). Then \(\varphi_{D'}\ge \log|s_D|^{2k}-C\) for some \(k>0\). Here we remark that \(f\) is in particular upper bounded. Then, by the proof of [20] Theorem 5.1, we have \[\sqrt{-1}\Lambda_{\omega_0}dd^c\varphi_i=\Delta_{\omega_0}\varphi_i\le C_1\exp(C(C_2-\varphi_{D'}))\le C|s_D|^{-2k}\] for any \(i>0\). Since \(\theta_i\le \theta_1\) is smooth, we obtain \(\sqrt{-1}\Lambda_{\omega_0}\omega_i\le C|s_D|^{-2k}\). Let us consider \(\omega_0=\Sigma_{k=1}^n\sqrt{-1}dz_k\wedge d\overline{z_k}\) and \(\omega_i=\Sigma_{k=1}^n\sqrt{-1}g_{i,k}dz_k\wedge d\overline{z_k}\) at a point \(p\). Then we have \(g_{i,k}\le C|s_D|^{-2k}\). Since \(\omega_i\) satisfies the Monge-Ampère equation, we have \(g_{i,k}\ge C|s_D|^{2m}\) for some \(m>0\). Hence we have \(\omega_i\ge C|s_D|^{2m}\omega_0\) for all \(i\).

In [20], they showed that \(\varphi_i\to \varphi\) in \(L^1\)-topology. Let us fix an Euclidean ball \(B\) in a coordinate neighborhood in \({X}\setminus D\). Then, [20] showed that there exists a constant \(C_B>0\) such that the uniform estimate \(\|\Delta_{\omega_0}\varphi_i\|_{L^{\infty}(B)}\le C_B\) holds for any \(i\). It is also shown by GPTW24? that \(\|\varphi_i\|_{L^{\infty}(B)}\le C_B\). Then, by the standard elliptic estimate (e.g. [27] Theorem 9.11), we obtain the uniform \(L^{p}_2\)-estimate on \(B\), that is, we obtain \(\|\varphi_i\|_{L^p_2(B)}\le C_B\) for any \(i\) and \(p>0\). Then, the proof of [28] leads to the uniform \(C^{2,\alpha}(B)\) estimate \(\|\varphi_i\|_{C^{2,\alpha}(B)}\le C_B\). Then, by the classical Schauder estimates (e.g. [27] Problem 6.1), we obtain the uniform \(C^{k,\alpha}(B)\) estimate, that is, we have \(\|\varphi_i\|_{C^{k,\alpha}(B)}\le C_B\) for any \(k\). Therefore \(\varphi_i\) converges to \(\varphi\) locally smoothly on \({X}\setminus D\). ◻

Proof. The first statement (1) is a consequence of Lemma 17. We show (2), (3) and (4). We firstly prove that \(\nabla_i\) is a \(h_0\)-connection. It is a consequence of the direct computation as follows: For any local section \(u\) and \(v\) of \(E\), \[\begin{align} d\big(h_0(u,v)\big) &=d\big(h_i(\Psi_i^{-1} u,\Psi_i^{-1} v)\big)\\ &=h_i\big(\nabla_{h_i}(\Psi_i^{-1} u), \Psi_i^{-1} v\big)+h_0\big(\Psi_i^{-1} u, \nabla_{h_i}(\Psi_i^{-1} v)\big)\\ &=h_0\big((\Psi_i\circ\nabla_{h_i}\circ\Psi_i^{-1})u, v\big)+ h_0\big(u, (\Psi_i\circ\nabla_{h_i}\circ\Psi_i^{-1})v \big)\\ &=h_0(\nabla_i u, v) + h_0(u,\nabla_i v). \end{align}\] Next we see that \(\Psi_i:(E,\bar{\partial}^E)\to (F,\bar{\partial}^{\nabla_i})\) is holomorphic. Since \(\bar{\partial}^{\nabla_{h_i}}=\bar{\partial}^E\), the following computation works by the definition of \(\nabla_i\): \[(\bar{\partial}^{\nabla_i}\otimes\bar{\partial}^{E^*})\Psi_i =\bar{\partial}^{\nabla_i}\circ\Psi_i-\Psi_i\circ\bar{\partial}^E =\Psi_i\circ\bar{\partial}^{\nabla_{h_i}}-\Psi_i\circ\bar{\partial}^E=0.\] We recall that for any endomorphism \(A\) of \(E\), we have at each point \[A^{*_{h_i}}=h_i^{-1}\overline{A}^{t}h_i=\Psi_i^{-2}(h_0^{-1}\overline{A}^{t}h_0)\Psi_i^2=\Psi_i^{-2}A^{*_{h_0}}\Psi_i^2.\] Hence \[|A|_{h_i}^2={\rm{Tr}}(A\cdot A^{*_{h_i}})={\rm{Tr}}((\Psi_i A \Psi_i^{-1})(\Psi_i^{-1} A^{*_{h_0}} \Psi_i)).\] Then, since \(F_{\nabla_i}=\Psi_i\circ F_{h_i}\circ \Psi_i^{-1}\), we have that \[|F_{\nabla_i}|_{h_0}^2={\rm{Tr}}((\Psi_i\circ \sqrt{-1}F_{h_i}\circ\Psi_i^{-1})(\Psi_i^{-1}\circ\sqrt{-1}F_{h_i}^{*_{h_0}}\circ\Psi_i))=|F_{h_i}|_{h_i}^2.\] Hence, since \(h_i\) is \(\omega_i\)-HYM, we obtain that \(\nabla_i\) is an \(\omega_i\)-HYM connection of the complex hermitian vector bundle \((E, h_0)\). The uniform \(L^2\)-bound in (4) follows from the following standard inequality (refer to [24]): \[\begin{align} (2rc_2(E)-(r-1)c_1(E)^2)\cdot\{\omega_i\}^{n-2} &=c_n\int_X|F_{\nabla_{h_i}}|^2_{h_0}\omega_i^n-c_n\int_X|\Lambda_{\omega_i}F_{\nabla_{i}}|_{h_0}^2\omega_i^n\\ &\ge c_n\int_X|F_{\nabla_{h_i}}|^2_{h_0}\omega_i^n-C, \end{align}\] where the last inequality follows from the \(\omega_i\)-HYM equation \(\Lambda_{\omega_i}F_{\nabla_i}=\lambda_i{\rm{id}}\) and \[\lambda_i=\frac{1}{\omega_i^n}\frac{c_1(E)\cdot\{\omega_i\}^{n-1}}{{\rm{rk}}E}.\] ◻

Proof. (1) is a direct consequence of \(\det \Psi_i=1\). We prove (2). Let \(h_D\) be a smooth hermitian metric on \(\mathcal{O}(D)\). We denote by \(\nabla_D:=\nabla_{i}\otimes\nabla_{h_0}^*\otimes\nabla_{h_D}\) a \(h_0^*\otimes h_0\otimes h_D\)-connection on \({\rm{Hom}}(E,E)\otimes \mathcal{O}(D)\). If we use the identity \(\Box^{\partial^{\nabla_D}}_{\omega_i}=\Box^{\bar{\partial}^{\nabla_D}}_{\omega_i}-\sqrt{-1}[\Lambda_{\omega_i},F_{\nabla_D}]\), and \(T\ge |s_D|^{2m}\omega_0\), the following calculation works: \[\begin{align} \label{estimate32on32difference} &\int_X\langle \nabla_D(s_D^{m+1}{\Psi_i}),\nabla_D(s_D^{m+1}{\Psi_i})\rangle_{h_0^*\otimes h_0\otimes h_D,\omega_i}\omega_i^n\notag\\ &=\int_X\langle\Delta^{\nabla_D}_{\omega_i}(s_D^{m+1}{\Psi_i}),s_D^{m+1}{\Psi_i}\rangle_{h_0^*\otimes h_0\otimes h_D}\omega_i^n \notag\\ &=-\int_X\langle\sqrt{-1}\Lambda_{\omega_i}F_{\nabla_D}\cdot (s_D^{m+1}{\Psi_i}), s_D^{m+1}{\Psi_i}\rangle_{h_0^*\otimes h_0\otimes h_D}\omega_i^n \notag\\ &=\int_X\langle-\sqrt{-1}\Lambda_{\omega_i}F_{\nabla_{h_0}\otimes\nabla_{h_D}}\circ(|s_D|^{2m}s_D{\Psi_i}),s_D{\Psi}\rangle_{h_0^*\otimes h_0\otimes h_D}\omega_i^n\notag\\ &+\int_X\langle(|s_D|^{2m}s_D{\Psi_i})\circ(\sqrt{-1}\Lambda_{\omega_i}F_{\nabla_i}),s_D{\Psi_i}\rangle_{h_0^*\otimes h_0\otimes h_D}\omega_i^n\notag\\ &\le C\int_X\langle|s_D|^{2m}\sqrt{-1}\Lambda_{\omega_i}\omega_0\cdot(s_D{\Psi_i}),s_D{\Psi_i}\rangle_{h_0^*\otimes h_0\otimes h_D}\omega_i^n\notag\\ &+|\lambda_i|\int_X|s_D{\Psi_i}|_{h_0^*\otimes h_0\otimes h_D}^2\omega_i^n\notag\\ &\le C. \end{align}\tag{2}\] Since \(\sup_{X\setminus D}|s_D\Psi_i|\le C\) by the assumption, we obtain that \[\int_X|s_D^{m+1}(\nabla_i\otimes\nabla_{h_0}^*)\Psi_i|^2\omega_i^n\le C.\] Since \(\nabla_i=\Psi_i\circ\nabla_{h_i}\circ\Psi_i^{-1}\) by Lemma 30, the result follows by the follwing computation: \[\begin{align} (\nabla_i\otimes\nabla_{h_0}^*)\Psi_i &=\nabla_i\circ\Psi_i-\Psi_i\circ\nabla_{h_0}\\ &=\Psi_i\circ(\nabla_{h_i}-\nabla_{h_0})\\ &=\Psi_i\circ(\Psi_i^{-2}\partial^{\nabla_{h_0}}(\Psi_i^2))\\ &=2\partial^{\nabla_{h_0}}\Psi_i. \end{align}\] The statement for \(\Psi_i^{-1}\) is proved by applying the same argument to the holomorphic morphism \(\Psi_i^{-1}:(E,\bar{\partial}^{\nabla_i})\to (E,\bar{\partial}^E)\). ◻

Proof. It is a direct consequence of [3]. In fact, the \(\omega_i\)-HYM equation is locally written as \[\label{uniform32esti32diff32lemm32eq1} \sqrt{-1}\Lambda_{\omega_i}\bar{\partial}(\partial h_i\cdot h_i^{-1})=\lambda_i{\rm{id}}.\tag{3}\] By Lemma 32, we have \[\label{uniform32esti32diff32eq2} \sup_{X\setminus D}|s_D\Psi_i^{-1}|\le C.\tag{4}\] By (?? ), (3 ) and (4 ), [3] implies that for any relatively compact open subset \(K\subset X\setminus D\) and \(0<\alpha\rq{}<1\), there exists a constant \(C_K>0\) such that the following holds: \[\|\Psi_i\|_{C^{1,\alpha\rq{}}(K)}\le C_K.\] Here the \(C^{1,\alpha\rq{}}\)-norm is defined by \(\omega_0\). Since the natural inclusion \(C^{1,\alpha\rq{}}(\overline{K})\hookrightarrow C^{1,\alpha}(\overline{K})\) is compact for any \(0<\alpha<\alpha\rq{}<1\), we can find a subsequence of \(\{\Psi_i\}_i\) such that \(\Psi_i\to \Psi_{\infty, K}\) in \(C^{1,\alpha}(K)\). Then, if we consider a sequence of relatively compact open subsets by \(X\setminus D_{\varepsilon_k}\) for \(\varepsilon_k\searrow 0\) where \(D_{\varepsilon_k}\) is the \(\varepsilon_k\)-neighborhood of \(D\) with respect to \(\omega_0\), then the standard diagonal procedure gives a subsequence of \(\{\Psi_i\}_i\) such that \(\Psi_i\to \Psi_{\infty}\) in \(C^{1,\alpha}(K)\) for any relatively compact open subset \(K\subset X\setminus D\). Let us define a hermitian metric on \(E|_{X\setminus D}\) by \(h_{\infty}:=h_0\Psi_{\infty}^2\). Then it satisfies \[\label{uniform32esti32diff32eq3} \sqrt{-1}\Lambda_T\bar{\partial}(\partial h_{\infty}\cdot h_{\infty}^{-1})=\lambda{\rm{id}}\tag{5}\] where \(\lambda=\frac{1}{\alpha^n}\frac{c_1(E)\cdot \alpha^{n-1}}{{\rm{rk}}E}\). Since \(h_{\infty}^{\pm1}\) is in \(C^{1,\alpha}(K)\cup L^2_1(K)\) for any \(K\Subset X\setminus D\), the standard elliptic estimate concludes that \(\Psi_{\infty}\) is smooth on \(X\setminus D\). The remaining statements \((2)\) and \((3)\) are the direct consequences of Lemma 32, (?? ) and \(\det\Psi_i=1\). ◻

Proof. Since \(\det\Psi_i\equiv 1\), we have \(|\Psi_i|_{h_0}\ge 1\). Hence there exists a constant \(a>0\) such that \[\Omega_a=\{\log|s_D\Psi_i|^2>-a\}\] is not empty for all \(i\) and \(\Omega_a\) is a relatively compact open subset in \(X\setminus D\) since \(|\Psi_i|\ge 1\) on \(X\) and \(|s_D|=0\) on \(D\). Hence \(\omega_i\) smoothly converges to \(T\) on \(\Omega_a\) and \(T\) is smooth Kähler metric on \(\Omega_a\). Then, as [31], there exists a constant \(C>0\) independent of \(i\) such that \[\Delta_{\omega_i}\left(\log|s_D\Psi_i|^2+a-\int_X\log|s_D\Psi_i|^2\omega_i^n\right) \ge -2|\Lambda_{\omega_i}F_{h_0}|_{h_0}-2|\Lambda_{\omega_i}F_{h_D}|_{h_D}-C_1 \ge -C\] holds on \(\Omega_a\). In fact, let us consider the following composition of holomorphic morphisms: \[(E,\bar{\partial}^E)\xrightarrow{\Psi_i} (E,\bar{\partial}^{\nabla_i})\xrightarrow{s_D}(E\otimes\mathcal{O}(D), \bar{\partial}^{\nabla_i\otimes\nabla_{h_D}}).\] Then the following computation works on \(\Omega_a\Subset X\setminus D\): \[\begin{align} \Delta_{\omega_i}\log|s_D\Psi_i|^2_{h_0\otimes h_0^*\otimes h_D} &=2\sqrt{-1}\Lambda_{\omega_i}\partial\left(|s_D\Psi_i|^{-2}\langle s_D\Psi_i,\partial^{\nabla_i\otimes\nabla_{h_0}^*\otimes\nabla_{h_D}}(s_D\Psi_i)\rangle \right)\\ &=-2|s_D\Psi_i|^{-4}\left|\langle s_D\Psi_i,\partial^{\nabla_i\otimes\nabla_{h_0}^*\otimes\nabla_{h_D}}(s_D\Psi_i)\rangle \right|^2\\ &+2|s_D\Psi_i|^{-2}\left|\partial^{\nabla_i\otimes\nabla_{h_0}^*\otimes\nabla_{h_D}}(s_D\Psi_i)\right|^2\\ &-2|s_D\Psi_i|^{-2}\langle s_D\Psi_i, \sqrt{-1}\Lambda_{\omega_i}F_{\nabla_i\otimes\nabla_{h_0}^*\otimes\nabla_{h_D}}(s_D\Psi_i)\rangle\\ &\ge -2|\lambda_i|-2|\Lambda_{\omega_i}F_{h_0}|-2|\Lambda_{\omega_i}F_{h_D}|. \end{align}\] Then, by the mean value type inequality of Guo-Phong-Sturm (Lemma 31), we obtain that there is a positive constant \(C>0\) independent of \(i\) such that \[\begin{align} &\sup_X\left(\log|s_D\Psi_i|^2+a-\int_X\log|s_D\Psi_i|^2\omega_i^n\right)\notag\\ &\le C\left(1+\int_X\left|\log|s_D\Psi_i|^2+a-\int_X\log|s_D\Psi_i|^2\omega_i^n\right|\omega_i^n\right) \end{align}\] holds. Since the constant \(a>0\) is independent of \(i\), we can find a constant \(C>0\) independent of \(i\) such that the following holds: \[\label{C032to32L132eq1} \sup_X\left(\log|s_D|^2+\log|\Psi_i|^2\right)\le C\big(1+\int_X|\log\left|s_D\Psi_i|\right|\omega_i^n\big).\tag{6}\] By applying the same argument to \(\log|s_D\Psi^{-1}_i|^2\), we obtain \[\label{C032to32L132eq2} \sup_X\left(\log|s_D|^2+\log|\Psi_i^{-1}|^2\right)\le C\big(1+\int_X|\log\left|s_D\Psi_i^{-1}|\right|\omega_i^n\big),\tag{7}\] We denote by \(\Psi_i=e^{s_i}\) for some \(h_0\)-hermitian endomorphism \(s_i\) of \(E\). Choosing a suitable \(h_0\)-orthonormal basis \(e_1,\ldots, e_r\) around \(x\in X\) so that \(s_i\) becomes a diagonal matrix. We denote by \(s_i=\lambda_1e_1+\cdots+\lambda_re_r\). Then, for any \(i\), we have \[\label{C032to32L132eq3} \log|\Psi_i^{\pm1}|^2 =\log(e^{\pm2\lambda_1}+\cdots+e^{\pm2\lambda_r})\ge \pm2\lambda_i\tag{8}\] and \[\begin{align} \label{C032to32L132eq4} \log|\Psi_i^{\pm1}|^2 &\le \log(e^{2|\lambda_1|}+\cdots+e^{2|\lambda_r|})\notag\\ &\le \log(e^{2|\lambda_1|+\cdots+2|\lambda_r|}+C)\notag\\ &\le C(|\lambda_1|+\cdots+|\lambda_r|+1)\notag\\ &\le C(|s_i|_{h_0}+1) \end{align}\tag{9}\] Then, by (6 ), (7 ), (8 ) and (9 ), we obtain that \[\sup_X\left(\log|s_D|^2+|s_i|^2\right) \le \sup_X\log|s_D\Psi_i^{\pm1}|_{h_0\otimes h_D}^2 \le C(1+\int_X\left|\log|s_D|^2+|s_i|_{h_0}\right|\omega_i^n).\] ◻

Proof. The proof of this lemma is inspired by the argument of [32]. We will divide the proof into four-steps as [32].

1st Step: This is the only step that requires an argument different from that in [32]. Let \(\Phi(x,y):=\frac{\exp(x-y)-1}{x-y}\) for \(x,y\in\mathbb{R}\). Then, since \(h_i=h_0e^{s_i}\) defines an \(\omega_i\)-HYM metric on \(E\) and \(u_i=\delta_is_i\), we have (refer to [32]) \[\label{L132est32eq0} 0=\delta_i^{-1}\int_X\langle\Phi(u_i/\delta_i)(\partial^{\nabla_{h_0}}u_i),\partial^{\nabla_{h_0}}u_i\rangle_{h_0,\omega_i}\omega_i^n+\int_X{\rm{Tr}}(u_i\sqrt{-1}\Lambda_iF_{h_0})\omega_i^n.\tag{10}\] We then prove \(\int_X{\rm{Tr}}(u_i\sqrt{-1}\Lambda_{\omega_i}F_{h_0})\omega_i^n\) is uniformly bounded from above and below. Since \(h_0\) is a smooth hermitian metric on \(E\) and \(\omega_0\) is a smooth Kähler metric on \(X\), there is a constant \(C>0\) such that \[-C\omega_0\cdot {\rm{Id}}_E\le \sqrt{-1}F_{h_0}\le C\omega_0\cdot {\rm{Id}}_E.\] By Lemma 34 and the definition of \(u_i\), we have \[\label{L132est32eq1} |u_i|_{h_0}\le C-C\delta_i\log|s_D|_{h_D}^2.\tag{11}\] Then we have \[\label{L132est32eq14655} |u_i|_{h_0}|\sqrt{-1}\Lambda_{\omega_i}F_{h_0}|_{h_0}\omega_i^n\le C(C-\delta_i\log|s_D|^2)\omega_0\wedge\omega_i^{n-1}.\tag{12}\] By Claim 36 below, we obtain \[\label{L132est32eq1465} \left|\delta_i^{-1}\int_X\langle\Phi(u_i/\delta_i)(\partial^{\nabla_{h_0}}u_i),\partial^{\nabla_{h_0}}u_i\rangle_{h_0,\omega_i}\omega_i^n\right| \le \int_X|u_i|_{h_0}|\sqrt{-1}\Lambda_{\omega_i}F_{h_0}|_{h_0}\omega_i^n\le C.\tag{13}\]

Since the rest of the proof of (1) and (2) is the same with [32], we only describe the outline.

2nd Step: In this step, we show that \(u_i\) subsequencially weakly converges to \(u_{\infty}\) in \(L^2_{1,{\rm{loc}}}(X\setminus D, T)\)-topology. By (11 ), the eigenvalues of \(u_i\) on a relatively compact open subset \(K\Subset X\setminus D\) is contained in an interval \([\alpha, \beta]\) for any \(i\). We recall that \(\Phi(x,y)=\frac{{\rm{exc}}(x-y)-1}{x-y}\) defined in the 1st Step satisfies \(\Phi(x,y)\ge \frac{{\rm{exc}}(\alpha-\beta)-1}{\alpha-\beta}\) for any \(x, y\in [\alpha,\beta]\) (refer to [32]). Then we have \[\langle \Phi(u_i)\partial^{\nabla_{h_0}}u_i,\partial^{\nabla_{h_0}}u_i\rangle_{h_0,\omega_i} \ge C_K^{-1}|\partial^{\nabla_{h_0}}u_i|_{h_0,\omega_i}^2on K.\] By \(\delta_i^{-1}=\int_X\left|\log|s_D|^2+|s_i|\right|\omega_i^n\ge 1\) and (13 ), we obtain \[\label{L132est32eq2} \int_K|\partial^{\nabla_{h_0}}u_i|_{h_0,\omega_i}^2\omega_i^n \le C_K\delta_i^{-1}\int_X\langle\Phi(\delta_i^{-1}u_i)\partial^{\nabla_{h_0}}u_i,\partial^{\nabla_{h_0}}u_i\rangle_{h_0,\omega_i}\omega_i^n\le C_K.\tag{18}\] Let us consider a sequence of relatively compact open subsets in \(X\setminus D\) by \(X\setminus D_{\varepsilon_i}\) where \(D_{\varepsilon_i}\) is an \(\varepsilon_i\)-neighborhood of \(D\) with respect to \(\omega_0\) with \(\varepsilon_i\searrow 0\). We recall that \(\omega_i\) smoothly converges to \(T\) on each \(X\setminus D_{\varepsilon_i}\) and \(T\) is a smooth Kähler metric on \(X\setminus D\). Then by (11 ) and (18 ), the standard diagonal procedure gives a subsequence of \(\{u_i\}_i\), also denoted by \(\{u_i\}_i\), such that \[\label{L132est32eq2465} u_i\to u_{\infty}weakly in L^2_{1,{\rm{loc}}}(X\setminus D,T) and strongly in L^2_{{\rm{loc}}}(X\setminus D, T^n).\tag{19}\] By (12 ) and Claim 36, the dominant convergence theorem gives \[\label{L132est32eq3} \lim_{i\to\infty}\sqrt{-1}\int_X{\rm{Tr}}(u_i\Lambda_{\omega_i}F_{h_0})\omega_i^n\; =\sqrt{-1}\int_{X\setminus D}{\rm{Tr}}(u_{\infty}\Lambda_TF_{h_0})T^n.\tag{20}\] 3rd Step: In this step, we show that \(u_{\infty}\), the weak limit of \(u_i\) constructed in 2nd step, is not identically 0. Since \(\delta_i=\left(\int_X(-\log|s_D|^2+|s_i|)\omega_i^n\right)^{-1}\) and \(u_i=\delta_is_i\), we have \[-\int_X\delta_i\log|s_D|^2\omega_i^n+\int_X|u_i|\omega_i^n=1.\] If we recall that \(\int_X(-\log|s_D|)\omega_i^n\le C\), then we obtain \[\label{L132est32eq4} 2\int_X(-\delta_i\log|s_D|^2)\omega_i^n+\int_X|u_i|\omega_i^n \ge1-\delta_iC.\tag{21}\] By (11 ) and again by \(\int_X(-\log|s_D|)\omega_i^n\le C\), if we take the limit as \(i\to \infty\), the dominant convergence theorem allows us to interchange the limit and the integral in the LHS of (21 ). Then we obtain \[\int_{X\setminus D}|u_{\infty}|T^n\ge 1.\] Hence, if \(u_{\infty}=0\), then it implies \(0\ge 1\) which is a contradiction. Thus \(u_{\infty}\ne 0\).

4th Step: In this step, we show the inequality in (2) in the statement. Let \(\Psi:\mathbb{R}\times\mathbb{R}\to \mathbb{R}_{>0}\) be a smooth positive function such that \(\Psi(a,b)<\frac{1}{b-a}\) when \(a<b\). We fix \(K\subset X\setminus D\) a relatively compact open subset. We recall that \(\delta_i^{-1}\Phi(\delta_i^{-1}a,\delta_i^{-1}b)\) tends to \(\frac{1}{b-a}\) when \(a<b\) and to \(\infty\) when \(b\le a\) (refer to [32]). We also recall that, by (11 ), the eigenvalues of \(u_i\) lies in an interval \([\alpha,\beta]\) for all \(i\). Hence, for sufficiently large \(i>0\), we have \[\delta_i^{-1}\Phi(\delta_i^{-1}a,\delta_i^{-1}b)\ge \Psi(a,b)for all a,b\in[\alpha,\beta].\] Together with (10 ), we obtain \[\label{L132est32eq5} 0\ge \sqrt{-1}\int_X{\rm{Tr}}(u_i\Lambda_{\omega_i}F_{h_0})\omega_i^n +\int_K\langle\Psi(u_i)\partial^{\nabla_{h_0}}u_i,\partial^{\nabla_{h_0}}u_i\rangle_{h_0,\omega_i}\omega_i^n.\tag{22}\] By (11 ) and (19 ), we can see that \(\Psi^{\frac{1}{2}}(u_i)\to \Psi^{\frac{1}{2}}(u_{\infty})\) in the operator norm topology on \({\rm{Hom}}(L^2(K, T^n), L^q(K, T^n))\) for any \(1\le q<2\) ([31], see also [32]). Thus, by (20 ) and (22 ), we obtain that for any \(\delta>0\), there exists \(i_0>0\) such that for any \(i>i_0\) the following holds: \[\label{L132est32eq6} 2\delta\ge \|\Psi^{\frac{1}{2}}(u_{\infty})\partial^{\nabla_{h_0}}\|_{L^q(K,T)}^2 +\sqrt{-1}\int_{X\setminus D}{\rm{Tr}}(u_{\infty}\Lambda_TF_{h_0})T^n\tag{23}\] Since \(\partial^{\nabla_{h_0}}u_i\to\partial^{\nabla_{h_0}}u_{\infty}\) weakly on \(K\) (refer to (19 )), we can again use [31] (see also [32]) to obtain that \(\Psi(u_{\infty})\partial^{\nabla_{h_0}}u_i \to \Psi(u_{\infty})\partial^{\nabla_{h_0}}u_{\infty}\) weakly in \(L^q(K, T)\). Then, by (23 ), we obtain \[\label{L132est32eq7} 2\delta\ge \|\Psi^{\frac{1}{2}}(u_{\infty})\partial^{\nabla_{h_0}}u_{\infty}\|_{L^q(K,T)}^2+\sqrt{-1}\int_{X\setminus D}{\rm{Tr}}(u_{\infty}\Lambda_TF_{h_0})T^n.\tag{24}\] Here we recall that \(1\le q<2\), \(K\Subset X\setminus D\) and \(\delta>0\) are chosen arbitrary. Thus (24 ) applies when \(q=2\), \(K=X\setminus D\) and \(\delta=0\), that is, we obtain \[0\ge \int_{X\setminus D}\langle\Psi(u_{\infty})\partial^{\nabla_{h_0}}u_{\infty},\partial^{\nabla_{h_0}}u_{\infty}\rangle_{h_0,T}+\sqrt{-1}\int_{X\setminus D}{\rm{Tr}}(u_{\infty}\Lambda_TF_{h_0})T^n.\]

(3) By \((1)\) and \((2)\), using Proposition 38 below, we can prove \((3)\) by the same argument with [31] (refer to the comment below [7]). We only sketch the proof. By [31], we see that the eigenvalues \(\lambda_1,\ldots,\lambda_r\) of \(u_{\infty}\) are all constants and \(\lambda_i\) are not all equal. Let \(\{\gamma\}\) be the set of intervals between the eigenvalues of \(u_{\infty}\) and for each \(\gamma\), we choose \(p_{\gamma}:\mathbb{R}\to \mathbb{R}\) with \(p_{\gamma}(\lambda_i)=1\) for \(\lambda_i<\gamma\) and \(p_{\gamma}(\lambda_i)=0\) for \(\lambda_i>\gamma\). Define \(\pi_{\gamma}:=p_{\gamma}(u_{\infty})\). Then, by [31], we can see that \(\pi_{\gamma}\) defines a \(T\)-weak holomorphic subbundle (see Definition 37 below). By [31], there exists \(\gamma\) such that \[\mu_{\alpha}(E)\le \frac{1}{{\rm{Tr}}(\pi_{\gamma})}\int_{X\setminus W}\left({\rm{Tr}}(\pi_{\gamma}\sqrt{-1}F_{h_0})-\sqrt{-1}{\rm{Tr}}(\bar{\partial}\pi_{\gamma}\wedge(\bar{\partial}\pi_{\gamma})^{*_{h_0}})\right)\wedge T^{n-1}\] where \(W\subset X\) is an analytic subset in \(X\). By Proposition 38 below, there exists a torsion free subsheaf \(\mathcal{F}\subset E\) on \(X\) such that \[\frac{1}{{\rm{Tr}}(\pi_{\gamma})}\int_{X\setminus W}\left({\rm{Tr}}(\pi_{\gamma}\sqrt{-1}F_{h_0})-\sqrt{-1}{\rm{Tr}}(\bar{\partial}\pi_{\gamma}\wedge(\bar{\partial}\pi_{\gamma})^{*_{h_0}})\right)\wedge T^{n-1}=\mu_{\alpha}(\mathcal{F}).\] It contradicts to the assumption that \(E\) is \(\alpha^{n-1}\)-slope stable. Hence we obtain a uniform estimate \(C>0\) such that the following holds: \[\int_X\left(-\log|s_D|^2+|s_i|\right)\omega_i^n\le C.\] ◻

Proof. Let \(\pi\) be a \(T\)-weak holomorphic subbundle of \((E,h_0)\) and \({\rm im}(\pi)\) be the torsion free coherent sheaf of \(E|_{X\setminus D}\) defined by the image of \(\pi\). Let us consider the following short exact sequence of torsion free sheaves on \(X\setminus D\): \[\label{weak32proj32eq1} 0\to \mathcal{F}\xrightarrow{\iota}E|_{X\setminus D}\xrightarrow{p}\mathcal{G}\to 0.\tag{25}\] Then \(p\) defines an \(\mathcal{O}_X\)-linear morphism \(\widetilde{p}:=i_*p:E\to i_*\mathcal{G}\) where \(i:X\setminus D\hookrightarrow X\) is a natural inclusion. Then \(\widetilde{\mathcal{G}}:={\rm im}(\widetilde{p})/{\rm{Tor}}\) is a torsion free coherent sheaf on \(X\). In fact, the dual of the surjection \(\widetilde{p}: E\to {\rm im}(\widetilde{p})/{\rm{Tor}}\) gives an injection of \(\mathcal{O}_X\)-module \(\widetilde{p}^*:({\rm im}(\widetilde{p})/{\rm{Tor}})^*\to E^*\). Then, by Oka’s coherence theorem, we obtain that \(({\rm im}(\widetilde{p})/{\rm{Tor}})^*\) is a coherent sheaf on \(X\). Hence \({\rm im}(\widetilde{p})/{\rm{Tor}}\) extends to a torsion free coherent subsheaf \(\widetilde{\mathcal{G}}\) of \(({\rm im}(\widetilde{p})/{\rm{Tor}})^{**}\) on \(X\). Then we obtain the following short exact sequence of torsion free coherent sheaves on \(X\): \[\label{weak32proj32eq2} 0\to \widetilde{\mathcal{F}}\xrightarrow{\widetilde{\iota}}E\xrightarrow{\widetilde{p}}\widetilde{\mathcal{G}}\to0\tag{26}\] where the restriction of (26 ) to \(X\setminus D\) coincides with (25 ). We can find an analytic subset \(W\subset X\) such that \({\rm im}(\widetilde{\iota})|_{X\setminus W}\subset E|_{X\setminus W}\) is a holomorphic subbundle. Let \(\widetilde{\pi}\) be an \(h_0\)-orthogonal projection from \(E|_{X\setminus W}\) to \(\widetilde{\mathcal{F}}|_{X\setminus W}\). Then we have \[\label{weak32proj32eq3} \widetilde{\pi}|_{X\setminus W}=\pialmost everywhere in X\setminus D.\tag{27}\] Hence, it suffices to show the following: \[\label{weak32proj32eq4} \int_Xc_1(\widetilde{\mathcal{F}})\wedge\alpha^{n-1}=\int_{X\setminus W}c_1(\widetilde{\mathcal{F}}, h_0|_{\widetilde{\mathcal{F}}})\wedge\langle T^{n-1}\rangle,\tag{28}\] since \[\label{weak32proj32eq4465} c_1(\widetilde{\mathcal{F}}, h_0|_{\widetilde{\mathcal{F}}})={\rm{Tr}}(\widetilde{\pi}\sqrt{-1}F_{h_0})-{\rm{Tr}}(\bar{\partial}\widetilde{\pi}\wedge(\bar{\partial}\widetilde{\pi})^{*_{h_0}}).\tag{29}\] Since (28 ) is proved by the standard way (e.g. [24]), we only describe the outline of the proof. We denote by \(q:={\rm{rk}}\widetilde{\mathcal{F}}\). Let us consider \[\widetilde{j}:=\wedge^q\widetilde{\iota}:\det\widetilde{\mathcal{F}}\to \bigwedge^qE.\] Then \(\{\widetilde{j}=0\}=W\). Since \(\widetilde{j}\) is an injection of a holomorphic line subbundle, the pull back \[u:=\widetilde{j}^*(\wedge^qh_0)\] defines a smooth hermitian metric on \(\det\widetilde{\mathcal{F}}|_{X\setminus W}\). Let \(\tau\) be a local holomorphic frame of \(\det\widetilde{\mathcal{F}}\) and \(s_1,\ldots,s_r\) be a local holomorphic frame of \(E\). Then we can denote by \[\widetilde{j}(\tau)=\Sigma_{I}\tau^Is_Iwhere s_I=s_{i_1}\wedge\ldots\wedge s_{i_p} with i_1<\cdots<i_p.\] Let \(\widetilde{u}\) be a smooth hermitian metric on \(\det\widetilde{\mathcal{F}}\) which is globally defined over \(X\). Then we define a nonnegative function \[f:=\frac{u(\tau,\tau)}{\widetilde{u}(\tau,\tau)}=\Sigma_{I,J}u_{IJ}\tau^I\overline{\tau^J}\] where \(u_{IJ}=\wedge^qh_0(s_I,s_J)/\widetilde{u}(\tau,\tau)\). Then we can see that \(f\) is independent of the choice of \(\tau\), that is, \(f\) is globally defined on \(X\), and \(\{f=0\}=W\). Let \(\mu:Y\to X\) be a composition of blow-ups such that \(\mu^{-1}(W)\) is a snc divisor in \(Y\). Since \(\mu^*f\) vanishes along \(\mu^{-1}(W)\), we have \[dd^c\log\mu^*f=[W^{\prime}]\] where \([W^{\prime}]\) is an integral current supported on \(\mu^{-1}(W)\) with nonnegative coefficients. We obtain that \[\label{weak32proj32eq5} \int_{X\setminus (D\cup W)}dd^c\log f\wedge T^{n-1}=\int_{Y\setminus \mu^{-1}(D)\cup\mu^{-1}(W)}dd^c\log\mu^*f\wedge(\mu^*T)^{n-1}=0.\tag{30}\] Since \(c_1(\det\widetilde{\mathcal{F}},\widetilde{u})=c_1(\det\widetilde{\mathcal{F}},u)+dd^c\log f\) on \(X\setminus W\), we obtain \[\begin{align} \label{weak32proj32eq6} \int_Xc_1(\det\widetilde{\mathcal{F}})\wedge\alpha^{n-1} &=\int_Xc_1(\det\widetilde{\mathcal{F}},\widetilde{u})\wedge\langle T^{n-1}\rangle\notag\\ &=\int_{X\setminus (D\cup W)}c_1(\det\widetilde{\mathcal{F}},\widetilde{u})\wedge T^{n-1}\notag\\ &=\int_{X\setminus (D\cup W)}c_1(\det\widetilde{\mathcal{F}},u)\wedge T^{n-1}\notag\\ &=\int_{X\setminus W}c_1(\det\widetilde{\mathcal{F}},u)\wedge \langle T^{n-1}\rangle. \end{align}\tag{31}\] In the third line above, we used (30 ). Since \(u=\widetilde{j}^*(\wedge^qh_0)\), we have \[\label{weak32proj32eq7} c_1(\det\widetilde{\mathcal{F}},u)=c_1(\widetilde{\mathcal{F}},h)on X\setminus W.\tag{32}\] Then by (27 ), (29 ), (31 ) and (32 ), we obtain \[\begin{align} \int_Xc_1(\widetilde{\mathcal{F}})\wedge\alpha^{n-1} &=\int_{X\setminus W}c_1(\widetilde{\mathcal{F}},h_0|_{\widetilde{\mathcal{F}}})\wedge\langle T^{n-1}\rangle\\ &=\int_{X\setminus (D\cup W)}\left({\rm{Tr}}(\pi\sqrt{-1}F_{h_0})-{\rm{Tr}}(\bar{\partial}\pi\wedge(\bar{\partial}\pi)^{*_{h_0}})\right)\wedge T^{n-1}. \end{align}\] ◻

Proof. Let us denote by \(\Psi_i:=e^{s_i}\) for some \(h_0\)-hermitian endomorphism of \((E,h_0)\) with \({\rm{Tr}}(s_i)=0\). Then, by Lemma 35, we obtain that \[\int_X\left(-\log|s_D|^2+|s_i|_{h_0}\right)\le C.\] Then, by Lemma 34, we have \[\sup_X|s_D\Psi_i|_{h_D\otimes h_0\otimes h_0^*}\le C.\] Hence, by Lemma 33, we obtain \(\Psi_{\infty}\) as in the statement. ◻

Proof. We recall that, by the construction of \(\Psi_D\) (see Theorem 40) , it satisfies \[\label{nK32para32eq1} \sup_{X\setminus D}|s_D\Psi_D|\le Cand\int_{X\setminus D}|s_D^{m+1}\partial^{\nabla_{h_{D,0}}}\Psi_D|^2\le C.\tag{33}\] Since \(s_D^l:(\mathcal{O}_X,\bar{\partial}^{\nabla_{h_{{\rm triv}}}})\to (\mathcal{O}(lD),\bar{\partial}^{\nabla_{h_{D,0}}})\) and \(\Psi_D:(\mathcal{O}(lD),\bar{\partial}^{\nabla_{h_{D,0}}}) \to (\mathcal{O}(lD)|_{X\setminus D}, \bar{\partial}^{\nabla_D})\) is holomorphic (refer to Lemma 42) and since \(\sqrt{-1}\Lambda_TF_{h_D}=0\) by Theorem 40, we obtain \[\label{nK32para32eq2} \Delta_T|\Psi_Ds_D^l|_{h_{D,0}\otimes h_{{\rm triv}}}^2=\left|(\nabla_{D}\otimes\nabla_{h_{{\rm triv}}}^*)(\Psi_Ds_D^l)\right|^2_{h_{D,0}\otimes h_{{\rm triv}}^*,T}.\tag{34}\] We show that the integral of LHS with respect to \(T^n\) equals to 0. Let \(\eta_k:X\to [0,1]\) be a smooth cut-off functions as in Lemma 29, that is, each \(\eta_k\) satisfies \[{\rm{Supp}}(\eta_k)\Subset X\setminus D, X\setminus D_{\varepsilon_{k-1}}\subset \{\eta_k=1\}, \lim_{k\to\infty}\int_X|d\eta_k|^2_{\omega_0}\omega_0^n=0.\] Then we comute as follows: \[\begin{align} \left|\nabla|\Psi_Ds_D^l|^2\right|_T &\le 2\left|\langle(\nabla_D\otimes\nabla_{h_{D,0}}^*\Psi_D)s_D^l+l\Psi_Ds_D^{l-1}\nabla_{h_{D,0}}s_D,\Psi_Ds_D^l\rangle\right|\\ &\le |s_D|^{m+1}|\partial^{\nabla_{h_{D,0}}}\Psi_D|_T|s_D\Psi_D||s_D|^{2l-m-2} +l|s_D\Psi_D|^2|s_D|^{2m}|\nabla_{h_{D,0}}s_D|_T|s_D|^{2l-2m-3}\\ &\le C\left(|s_D^{m+1}\partial^{\nabla_{h_{D,0}}}\Psi_D|_T+1 \right)|s_D|^{2l-2m-3} \end{align}\] In the last line, we used (33 ) and the assumption that \(T\ge |s_D|^{2m}\omega_0\). Then we can estimate as follows: \[\begin{align} \left|\int_{X\setminus D}\eta_k\Delta_T|\Psi_Ds_D^l|^2T^n \right| &=\left|\int_{X\setminus D}\langle\nabla\eta_k,\nabla|\Psi_Ds_D^l|^2\rangle_TT^n \right|\\ &\le \int_{X\setminus D}|\nabla\eta_k|_T|\nabla|\Psi_Ds_D^l|^2|_TT^n\\ &\le C\left(\int_X|\nabla\eta_k|_{\omega_0}^2|s_D|^{4l-2m-6}T^n\right)^{1/2}\left(\int_{X\setminus D}|s_D^{m+1}\partial^{\nabla_{h_{D,0}}}\Psi_D|_T^2T^n \right)^{1/2}\\ &+C\int_X|\nabla\eta_k|_{\omega_0}|s_D|^{2l-4m-3}T^n\\ &\le C\left(\int_X|\nabla\eta_k|_{\omega_0}^2\omega_0^n\right)^{1/2}\left(\int_{X\setminus D}|s_D^{m+1}\partial^{\nabla_{h_{D,0}}}\Psi_D|_T^2T^n \right)^{1/2}\\ &+C\int_X|\nabla\eta_k|_{\omega_0}\omega_0^n\\ \end{align}\] In the last line, we use \(|s_D|^{k}T^n\le C\omega_0^n\) (see Lemma 23). Hence we obtain \[\label{nK32para32eq3} \int_{X\setminus D}\Delta_T|\Psi_Ds_D^l|^2T^n =\lim_{k\to\infty}\int_{X\setminus D}\eta_k\Delta_T|\Psi_Ds_D^l|^2T^n=0.\tag{35}\] Then, by (34 ) and (35 ), we obtain that \[(\nabla_D\otimes\nabla_{h_{{\rm triv}}}^*)\Psi_Ds_D^l=0on X\setminus D.\] ◻

Proof. Let \(h_1=h_0\Psi_1^2\) and \(h_2=h_0\Psi_2^2\) be two \(T\)-adapted HYM metrics with \(\det\Psi_i=1\) for \(i=1,2\). Suppose that the \(T\)-HYM constants \(\lambda_i\) of \(h_i\) satisfies \(\lambda_1\ge \lambda_2\). Let us consider a connection \(\nabla_i:=\Psi_i\circ\nabla_{h_i}\circ\Psi_i^{-1}\) on \(E\). Then \(\nabla_i\) is a \(T\)-admissible HYM connection of the complex hermitian vector bundle \((E,h_0)\) and \[\Psi_i: (E,\bar{\partial}^E)|_{X\setminus D}\to (E|_{X\setminus D},\bar{\partial}^{\nabla_i})\] is a holomorphic isomorphism (see Lemma 42) Furthermore, by definition of \(T\)-adapted HYM metrics, we have that for \(m\) in Notation 41, \[\label{unique32conn32eq1} \sup_{X\setminus D}|s_D^{l}\Psi_i|_{h_0}\le Cfor some l\in\mathbb{Z}_{\ge m}\tag{36}\] and \[\label{unique32conn32eq2} \int_{X\setminus D}\left|(\nabla_i\otimes\nabla_{h_0}^*)(s_D^l\Psi_i)\right|^2_{h_0^*\otimes h_0\otimes h_{D,0}^{l},T}T^n\le C.\tag{37}\] By Theorem 40, the holomorphic line bundle \(\mathcal{O}((4l+1)D)\) admits a \(T\)-adapted HYM metric \(h_{D}=h_{D,0}\Psi_{D}^2\) with \(\det \Psi_D=1\). Let us define \(\nabla_D:=\Psi_D\circ\nabla_{h_D}\circ\Psi_D^{-1}\) a \(T\)-admissible HYM connection of \((\mathcal{O}((4l+1)D),h_{D,0})\). Then, by Theorem 40, we have that \(\Psi_D\) satisfies \[\label{unique32conn32eq3} \sup_{X\setminus D}|s_D\Psi_{D}|_{h_{D,0}^{4l+1,*}\otimes h_{D,0}^{4l+1}\otimes h_{D,0}}\tag{38}\] and \[\label{unique32conn32eq4} \int_{X\setminus D}\left|s_D^{m+1}\partial^{\nabla_{h_{D,0}}}\Psi_{D}\right|^2_{h_{D,0}^*\otimes h_{D,0}\otimes h_{D,0}^{m+1},T}T^n\le C.\tag{39}\] Let us consider the following diagram. All vector bundles are restricted to \(X\setminus D\):

Here \(s:=({\rm{id}}_E\otimes \Psi_D)\circ ({\rm{id}}_E\otimes s_D^{4l+1})\circ \Psi_2\circ\Psi_1^{-1}\) is a holomorphic morphism from \((E,\bar{\partial}^{\nabla_1})\) to \((E\otimes\mathcal{O}((4l+1)D),\bar{\partial}^{\nabla_2\otimes\nabla_D})\). We remark that the \(T\)-HYM constant of \(\nabla_D\) equals to \(0\) by Theorem 40. Then the usual calculation shows the following point-wise equation on \(X\setminus D\): \[\label{unique32conn32eq5} \Delta_T|s|^2_{h_0\otimes h_0^*\otimes h_{D,0}} =\left|(\nabla_2\otimes\nabla_1^*\otimes\nabla_D)s\right|^2_{h_0\otimes h_0^*\otimes h_{D,0},T}-(\lambda_2-\lambda_1)|s|^2_{h_0\otimes h_0^*\otimes h_{D,0}}.\tag{40}\] By (36 ), (37 ), (38 ) and (39 ), we can see that each term in RHS in (40 ) is integrable over \(X\setminus D\) with respect to \(T^n\). Let \(\eta_k\) be smooth cut-off functions on \(X\) as in Lemma 29, in particular they satisfies \({\rm{Supp}}(\eta_k)\Subset X\setminus D\), \(\eta_k\nearrow 1\) and \(\int_X|d\eta_k|^2_{\omega_0}\omega_0^n\to0\). Then, we obtain the following equation by the same way with (35 ): \[\label{unique32conn32eq6} \int_{X\setminus D}\Delta_T|s|_{h_0\otimes h_0^*\otimes h_{D,0}}^2T^n=0.\tag{41}\] Here we recall that the assumption \(\lambda_1\ge \lambda_2\). Then, by integrating both sides of (40 ), we obtain \[\lambda_1=\lambda_2 and\left|(\nabla_2\otimes\nabla_1^*\otimes\nabla_D)s\right|^2_{h_0\otimes h_0^*\otimes h_{D,0},T}=0.\] In particular, we have \[\begin{align} \label{unique32conn32eq7} 0 &=\left(\nabla_2\otimes\nabla_1^*\otimes\nabla_D\right)s\notag\\ &=\left(\nabla_2\otimes\nabla_1^*\otimes\nabla_D\right)(s_D^{4l+1}\circ\Psi_D\circ\Psi_2\circ\Psi_1^{-1})\notag\\ &=(\nabla_D\otimes\nabla_0^*)(s_D^{4l+1}\Psi_D)\circ\Psi_2\circ\Psi_1^{-1} +s_D^{4l+1}\Psi_D\circ(\nabla_2\otimes\nabla_1^*)(\Psi_2\circ\Psi_1^{-1}). \end{align}\tag{42}\] By Lemma 43, we have \((\nabla_D\otimes\nabla_0^*)(s_D^{4l+1}\Psi_D)=0\). Then, by (42 ), we have \[\label{unique32conn32eq8} (\nabla_2\otimes\nabla_1^*)(\Psi_2\circ\Psi_1^{-1})=0.\tag{43}\] We remark that \[\label{unique32conn32eq8465} |\Psi_2\circ\Psi_1^{-1}|^2_{h_0\otimes h_0^*}=|{\rm{id}}_E|^2_{h_2\otimes h_1^*}\tag{44}\] and \[\label{unique32conn32eq9} \Delta_T|{\rm{id}}_E|^2_{h_2\otimes h_1^*} =|(\nabla_{h_2}\otimes \nabla_{h_1}^*)({\rm{id}}_E)|^2_{h_2\otimes h_1^*,T}-(\lambda_2-\lambda_1)|{\rm{id}}_E|^2_{h_2\otimes h_1^*},\tag{45}\] furthermore \[\label{unique32conn32eq10} \Delta_T|\Psi_2\circ\Psi_1^{-1}|_{h_0\otimes h_0^*} =|(\nabla_2\otimes\nabla_1^*)(\Psi_2\circ\Psi_1^{-1})|_{h_0\otimes h_0^*,T} -(\lambda_2-\lambda_1)|\Psi_2\circ\Psi_1^{-1}|_{h_0\otimes h_0^*}\tag{46}\] at each point of \(X\setminus D\). Then by (44 ), (45 ) and (46 ) together with (43 ), we obtain \[\label{unique32conn32eq11} |(\nabla_{h_2}\otimes \nabla_{h_1}^*){\rm{id}}_E|^2_{h_2\otimes h_1^*,T} =\left|(\nabla_2\otimes\nabla_1^*)(\Psi_2\circ\Psi_1^{-1})\right|^2_{h_0\otimes h_0^*,T}=0.\tag{47}\] The equation (47 ) shows \(\nabla_{h_1}=\nabla_{h_2}\). ◻

Proof. Let \(h\) be the \(T\)-adapted HYM metric on \(E\). By Theorem 40, the holomorphic line bundle \(\det E\) admits a \(T\)-adapted HYM metric \(h\rq{}\) whose \(T\)-HYM constant \(\lambda\rq{}\) satisfies \[\lambda\rq{}=\frac{1}{\alpha^n}c_1(E)\cdot\alpha^{n-1}.\] Let \(h\) be the given \(T\)-adapted HYM metric on \(E\) with \(T\)-HYM constant \(\lambda\). Then it induces a \(T\)-adapted HYM metric \(\det h\) on \(\det E\) with \(T\)-HYM constant \({\rm{rk}}E\lambda\). Then by Theorem 44, we have \(\nabla_{h\rq{}}=\nabla_{\det h}\), and thus the \(T\)-HYM constant \(\lambda\rq{}\) of \(\nabla_{h\rq{}}\) coincides with the \(T\)-HYM constant \({\rm{rk}}E\lambda\) of \(\nabla_{\det h}\). Hence we obtain \[\lambda=\frac{\lambda\rq{}}{{\rm{rk}}E}=\frac{1}{\alpha^n}\frac{c_1(E)\cdot\alpha^{n-1}}{{\rm{rk}}E}.\] Then we can prove the first equation of the statement by integrating the \(T\)-HYM equation \(\sqrt{-1}\Lambda_TF_h=\lambda{\rm{id}}\) over \(X\setminus D\) against \(T^n\). ◻

Proof. Let us consider the image sheaf \(\mathcal{G}\) of \(s_D^lg\), which is a coherent subsheaf of \(E_2\otimes\mathcal{O}(lD)|_{X\setminus D}\). Since \(s_D^lg\) is bounded and \(E_1\) is a locally free sheaf defined all over \(X\), the coherent sheaf \(\mathcal{G}\) on \(X\setminus D\) is generated by bounded sections of \(E_2\otimes\mathcal{O}(lD)|_{X\setminus D}\) around \(D\). Then, since \(E_2\otimes\mathcal{O}(lD)\) is defined all over \(X\), bounded generators of \(\mathcal{G}\) extends across \(D\) as a sections of \(E_2\otimes\mathcal{O}(lD)|_{X\setminus D}\). Hence, the image sheaf \(\mathcal{G}\) extends to a coherent subsheaf of \(E_2\otimes\mathcal{O}(lD)\), also denoted by \(\mathcal{G}\). We remark that \(s_D^lg\in H^0(X\setminus D,E_2\otimes E_1^*\otimes\mathcal{O}(lD))\) also extends to a holomorphic morphism defined all over \(X\), since \(s_D^lg\) is bounded and \(E_1\) and \(E_2\otimes\mathcal{O}(lD)\) are both defined over \(X\). Then we obtain the following short exact sequence of coherent sheaves on \(X\): \[0\rightarrow \mathcal{F}\rightarrow E_1\xrightarrow{s_D^lg} \mathcal{G}\rightarrow 0.\] Then, since \(E_1\) and \(E_2\) are \(\alpha^{n-1}\)-slope stable with same \(\alpha^{n-1}\)-slope, we can see that \(\mathcal{F}=0\). In fact, the \(\alpha^{n-1}\)-slope stability of \(E_2\) ensures that \(E_2\otimes\mathcal{O}(lD)\) is also \(\alpha^{n-1}\)-slope stable. We recall that \(\mathcal{G}\subset E_2\otimes\mathcal{O}(lD)\). Then if \(\mathcal{F}\ne 0\), we have \(\mu_{\alpha}(E_1)<\mu_{\alpha}(\mathcal{G})<\mu_{\alpha}(E_2\otimes\mathcal{O}(lD))=\mu_{\alpha}(E_1)\), that is a contradiction. Therefore \(s_D^lg:E_1\to E_2\otimes\mathcal{O}(lD)\) is injective. Hence its determinant gives the short exact sequence \[0\to\det E_1\xrightarrow{\det(s_D^lg)}\det (E_2\otimes\mathcal{O}(lD))\to \tau_V\to 0.\] Since \(s_D^lg\) is injective, the quotient sheaf \(\tau_V\) is a torsion sheaf supported on some analytic subvariety \(V\subset X\). We recall the assumptions that \({\rm{rk}}E_1={\rm{rk}}E_2\) and \(\mu_{\alpha}(E_1)=\mu_{\alpha}(E_2)\). Then we have \[0=c_1(\det (E_2\otimes\mathcal{O}(lD)))\alpha^{n-1}-c_1(\det E_1)\alpha^{n-1} =c_1(\tau_V)\alpha^{n-1}.\] Hence we obtain a decomposition \(V=V_1\cup V_2\) such that \(V_1\subset D\) and \({\rm{codim}}V_2\ge 2\). In particular we have that \(\det(s_D^lg)\) is isomorphic over \(X\setminus D\) and thus it has no nontrivial zero locus in \(X\setminus D\). As a consequence \(g: E_1|_{X\setminus D}\to E_2|_{X\setminus D}\) is isomorphic. Assume that \(E=E_1=E_2\) and let \(a\in \mathbb{C}\setminus\{0\}\) be an eigenvalue of \(g\) at \(x\in X\setminus D\). If \(g-a{\rm{id}}_E\ne 0\), then it is an isomorphism by the above proof. But it is a contradiction. Hence \(g=a{\rm{id}}_E\). ◻

Proof. Let \(h_1\) and \(h_2\) be two \(T\)-adapted HYM metrics on a holomorphic vector bundle \(E\). We denote by \(h_2=h_1g\) for some positive definite \(h_1\)-hermitian endomorphism \(g\). Let \(l\in\mathbb{Z}_{>0}\) be a sufficiently large integer. Let \(h_D\) be a \(T\)-adapted HYM metric on \(\mathcal{O}(lD)\) whose existence is proved in Theorem 40. Let us define a smooth section \(\overline{s_D^l}:=h_D(\cdot,s_D^l)\) of \(\mathcal{O}(lD)^*=\mathcal{H}om(\mathcal{O},\mathcal{O}(lD)^*)\) over \(X\setminus D\). If we denote by \(h_D=h_{D,0}\Psi_{D}^2\), then \(\overline{s_D^l}=h_{D,0}(\Psi_D\circ\cdot,\Psi_Ds_D^l)\) and we have, by applying Lemma 43 in the first equality below, \[\begin{align} 0 &=(\nabla_D\otimes \nabla_{triv}^*)(\Psi_Ds_D^l)\\ &=(\Psi_D\circ\nabla_{h_D}\circ\Psi_D^{-1})\circ \Psi_Ds_D^l-\Psi_Ds_D^l\circ\nabla_{triv}\\ &=\Psi_D\circ(\nabla_{h_D}\otimes\nabla_{triv}^*) s_D^l\\ &=\Psi_D\circ(\nabla_{h_D}s_D^l) \end{align}\] Here \(\nabla_D=\Psi_D\circ\nabla_{h_D}\circ\Psi_D^{-1}\) the \(T\)-adapted HYM connection on \((\mathcal{O}(lD),h_{D,0})\) and \(\nabla_{triv}\) is the trivial connection on the trivial bundle \(\mathcal{O}\). Since \(\Psi_D\) does not have zero locus on \(X\setminus D\), we obtain \[\label{unique32metric32eq-1} \nabla_{h_D}s_D^l=0.\tag{48}\] Then we also obtain that for any smooth section \(t\) of \(\mathcal{O}(lD)\), \[\begin{align} \label{unique32metric32eq-0465} (\nabla_{h_D}\overline{s_D^l})(t) &=d\left(h_D(t,s_D^l)\right)-h_D(\nabla_{h_D}t,s_D^l)\notag\\ &=-h_D(t,\nabla_{h_D}s_D^l)=0 \end{align}\tag{49}\] by (48 ). Then, we can see the following:

Since \(h_1\) and \(h_2\) are \(T\)-adapted and \(g\) is defined by \(h_2=h_1g\), there is \(l\in \mathbb{Z}_{\ge 0}\) such that \(\sup_{X\setminus D}|s_D^lg|_{h_0\otimes h_0^*\otimes h_{D,0}}<\infty\). Then, by Claim 48 and Lemma 46, we can conclude that \(g=a{\rm{id}}_E\) over \(X\setminus D\) for some \(a\in\mathbb{R}\setminus \{0\}\) since now we assume that \(E\) is \(\alpha^{n-1}\)-slope stable. As a consequence, we obtain \(h_2=ah_1\) and thus we obtain the uniqueness of \(T\)-adapted HYM metrics on \(E\). ◻

Proof. We denote by \(h_E\) the \(T\)-adapted HYM metric on \(E\). Let \(\iota:\mathcal{F}\hookrightarrow E\) be an \(\alpha^{n-1}\)-slope stable torsion free subsheaf of rank \(q\) with torsion free quotient \(\mathcal{G}:=E/\mathcal{F}\) such that \[\label{HYM32stable32eq1} \mu_{\alpha}(E)\le \mu_{\alpha}(\mathcal{F}).\tag{57}\] Let \(\pi:Y\to X\) be a sequence of blow-ups so that \(F:=\pi^*\mathcal{F}/{\rm{Tor}}\) is locally free. Then we have \(\mu_{\pi^*\alpha}(F)=\mu_{\alpha}(\mathcal{F})\ge\mu_{\alpha}(E)=\mu_{\pi^*\alpha}(\pi^*E)\). We denote by \(i:F\to \pi^*E\) the induced morphism from the inclusion \(\iota:\mathcal{F}\hookrightarrow E\). We remark that \(\pi^*T\) is an adapted current on \(Y\) (Lemma 24), and \(F\) is \(\pi^*\alpha^{n-1}\)-slope stable (Lemma 16), \(F\) admits a \(T\)-adapted HYM metric \(h_F\) by Theorem 40. Let \(D^{\prime}:=E_{nK}(\pi^*\alpha)\) be a snc divisor in \(Y\) given by the non-Kähler locus of \(\pi^*\alpha\). We denote by \(h_{D^{\prime}}\) the \(\pi^*T\)-adapted HYM metric on \(\mathcal{O}(lD)\) for sufficiently large \(l\in\mathbb{Z}_{>0}\). Then we consider the following composition of morphisms of \(\pi^*T\)-adapted HYM vector bundles: \[(F,h_F)\xrightarrow{i} (\pi^*E,\pi^*h_E)\xrightarrow{s_{D^{\prime}}^l} (\pi^*E\otimes\mathcal{O}(lD^{\prime}), \pi^*h_E\otimes h_{D^{\prime}}).\] Then, since \(s_{D^{\prime}}^l\circ i\) is holomorphic, we have \[\begin{align} \label{HYM32stable32eq2} &\Delta_T|s_{D^{\prime}}^li|^2_{\pi^*h_E\otimes h_F\otimes h_{D^{\prime}}}\notag\\ &=|(\nabla_{\pi^*h_E}\otimes\nabla_{h_F}\otimes\nabla_{h_{D^{\prime}}})s_{D^{\prime}}^li|^2_{\pi^*h_E\otimes h_F\otimes h_{D^{\prime}},\pi^*T} -(\lambda_E-\lambda_F)|s_{D^{\prime}}^li|^2_{\pi^*h_E\otimes h_F\otimes h_{D^{\prime}}} \end{align}\tag{58}\] where \(\lambda_E\) is the \(T\)-HYM constant of \(h_E\) and \(\lambda_F\) is the \(\pi^*T\)-HYM constant of \(h_F\). By Corollary 45 and (57 ), we have \(\lambda_E\le \lambda_F\). Since \(\pi^*h_E\), \(h_F\) and \(h_{D^{\prime}}\) are all \(\pi^*T\)-adapted, each term in (58 ) is integrable over \(Y\setminus D^{\prime}\) with respect to \(T^n\) (refer to (40 )). Furthermore, by the cut-off argument as (35 ), we obtain that \[\int_{Y\setminus D^{\prime}}\Delta_T|s_{D^{\prime}}^li|^2_{\pi^*h_E\otimes h_F\otimes h_{D^{\prime}}}(\pi^*T)^n=0\] Then we obtain that \(\lambda_E=\lambda_F\), in particular \[\label{HYM32stable32eq2465} \mu_{\alpha}(E)=\mu_{\alpha}(\mathcal{F}),\tag{59}\] and \[\label{HYM32stable32eq3} 0 =(\nabla_{\pi^*h_E}\otimes\nabla_{h_F}\otimes\nabla_{h_{D^{\prime}}})s_{D^{\prime}}^li =(\nabla_{h_{D^{\prime}}}s_{D^{\prime}}^l)i+ s_{D^{\prime}}^l(\nabla_{\pi^*h_E}\otimes\nabla_{h_F}^*)i.\tag{60}\] By (48 ), we have \(\nabla_{h_{D^{\prime}}}s_{D^{\prime}}^l=0\). Hence (60 ) implies \[\label{HYM32stable32eq4} (\nabla_{\pi^*h_E}\otimes\nabla_{h_F}^*)i=0.\tag{61}\] Let us consider a short exact sequence \[\label{HYM32stable32eq5} 0\to \mathcal{F}\xrightarrow{\iota}E\rightarrow\mathcal{G}\to 0\tag{62}\] over \(X\) and a short exact sequence \[\label{HYM32stable32eq6} 0\to F|_{Y\setminus D^{\prime}}\xrightarrow{i}\pi^*E|_{Y\setminus D^{\prime}}\rightarrow G\to 0\tag{63}\] over \(Y\setminus D^{\prime}\). By (61 ), the sequence (63 ) holomorphically splits: \(\pi^*E|_{Y\setminus D^{\prime}}\simeq F|_{Y\setminus D^{\prime}}\oplus G\). Since \(F|_{Y\setminus D^{\prime}}=\pi^*(\mathcal{F}|_{X\setminus (D\cup{\rm{Sing}}\mathcal{F})})\) and \(G=\pi^*(\mathcal{G}|_{X\setminus (D\cup{\rm{Sing}}\mathcal{F})})\), and the codimension of \({\rm{Sing}}\mathcal{F}\), the non locally free locus of \(\mathcal{F}\), is at least 3, we obtain that the sequence (62 ) holomorphically splits: \(E|_{X\setminus D}\simeq (\mathcal{F}\oplus\mathcal{G})|_{X\setminus D}\). By (59 ), we have \(\mu_{\alpha}(E)=\mu_{\alpha}(\mathcal{G})\). We remark that \(\mathcal{G}\) also admits a \(T\)-adapted HYM metric, since \(E\) and \(\mathcal{F}\) both admit. Then, by repeating this argument by replacing \(E\) to \(\mathcal{G}\), we obtain the \(\alpha^{n-1}\)-slope polystability of \(E\). ◻

Proof. Let \(h_1\) and \(h_2\) be two \(T\)-adapted HYM metrics on \(E\). Then, by Theorem 49, we can see that \(E\) is \(\alpha^{n-1}\)-slope polystable. That is, there exists \(\alpha^{n-1}\)-slope stable torsion free sheaves \(\mathcal{E}_1,\ldots,\mathcal{E}_k\) on \(X\) such that \(E\) holomorphically splits to the direct sum of \(\mathcal{E}_i\) on \(X\setminus D\), that is, there is a holomorphic isomorphism \(E|_{X\setminus D}\simeq (\mathcal{E}_1\oplus\cdots\mathcal{E}_k)|_{X\setminus D}\). By Theorem 47, the \(T\)-adapted HYM metric \(g_i\) on \(\mathcal{E}_i\) is unique. As in the proof of Theorem 49, we can assume that \(\iota:\mathcal{E}_1\hookrightarrow E\) is an \(\alpha^{n-1}\)-stable subsheaf and we have by (61 ), \[(\nabla_{h_i}\otimes\nabla_{g_1}^*)\iota=0\] on the locally free locus of \(\mathcal{E}_1\) for \(i=1,2\). Thus \(\nabla_{h_i}|_{\mathcal{E}_1}=\nabla_{g_1}\) and in particular \(h_1|_{\mathcal{E}_1}=g_1=h_2|_{\mathcal{E}_1}\) by Theorem 47. Since \(\mathcal{E}_2=E/\mathcal{E}_1\), it admits three \(T\)-adapted HYM metrics \(g_2\) and \(\widetilde{h_i}\) induced by \(h_i\) and \(h_i|_{\mathcal{E}_1}=g_1\) for \(i=1,2\). Since \(\mathcal{E}_2\) is \(\alpha^{n-1}\)-slope stable, we have that \(\widetilde{h_i}=g_2\) by Theorem 47. Hence, inductively we obtain that \[h_1=h_1|_{\mathcal{E}_1}\oplus\cdots\oplus h_1|_{\mathcal{E}_k} =g_1\oplus \cdots\oplus g_k =h_2|_{\mathcal{E}_1}\oplus\cdots\oplus h_2|_{\mathcal{E}_k}=h_2.\] ◻

Proof. The \(\alpha^{n-1}\)-slope polystability is proved in Theorem 49. The uniqueness is showed in Theorem 50. ◻

Proof. Let \(h=h_0\Psi^2\) be a \(T\)-adapted HYM metric on \(E\). Then, by Theorem 49, the holomorphic vector bundle \(E\) is \(\alpha^{n-1}\)-slope polystable. By Theorem 40, we obtain a \(T\)-adapted HYM metric \(h_{\infty}=h_0{\Psi_{\infty}}^2\) such that \(\det{\Psi_{\infty}}=1\) and \(\sup_{X\setminus D}|s_D{\Psi_{\infty}}|_{h_D\otimes h_0\otimes h_0^*}<\infty\). By the uniqueness of \(T\)-adapted HYM metrics (Theorem 47), we have \(\Psi=a{\Psi_{\infty}}\). Since \(\det\Psi=1=\det{\Psi_{\infty}}\), the constant \(a\) equals to \(1\): \(\Psi={\Psi_{\infty}}\). Hence we obtain \((1)\).

Fix any integer \(k\in\mathbb{Z}_{>0}\). Then \(h^{\otimes k}=h_0^{\otimes k}\Psi^{\otimes k}\) gives a \(T\)-adapted HYM metric on \(E^{\otimes k}\) with \(\det\Psi^{\otimes k}=1\). Hence, there exists a constant \(C_k>0\) such that \[\label{HYM32est32eq1} \sup_{X\setminus D}\log(|s_D|_{h_D}|\Psi|^k_{h_0\otimes h_0^*}) =\sup_{X\setminus D}\log|s_D\Psi^{\otimes k}|\le C_k.\tag{64}\] We denote by \(\Psi=e^S\) for some \(h_0\)-hermitian endomorphism \(S\). Let \(s_1,\ldots, s_r\) be the eigenvalues of \(S\) at \(x\in X\setminus D\) and diagonalize \(S\) at \(x\) so that \(S={\rm diag}(s_1,\ldots,s_r)\). Then, by (64 ) and by \(|\Psi|^{2k}\ge e^{2ks_1}+\cdots+e^{2ks_r}\), we obtain \[\begin{align} \label{HYM32est32eq2} C_k &\ge \log|s_D|^2+\log(e^{2ks_1}+\cdots+e^{2ks_r})\notag\\ &\ge \log|s_D|^2+\log(e^{2ks_i})\notag\\ &\ge \log|s_D|^2+2ks_i \end{align}\tag{65}\] for any \(i=1,\ldots,r\). By Theorem 40, the inverse morphism \(\Phi=\Psi^{-1}\) also satisfies
\(\sup_{X\setminus D}|s_D\Psi^{-1}|_{h_D\otimes h_0^*\otimes h_0}<\infty\). Then by applying the computation of (65 ) to \((\Psi^{\otimes k})^{-1}\), we obtain that for any \(i=1,\ldots,r\), \[\label{HYM32est32eq3} \log|s_D|^2-2ks_i\le C_k.\tag{66}\] By (65 ) and (66 ), we obtain that \(k|s_i|\le C_k-\log|s_D|^2\). Hence we have \[\label{HYM32est32eq4} k|S|_{h_0\otimes h_0^*}\le r(C_k-\log|s_D|^2).\tag{67}\] Let us fix a sequence of constants \(\varepsilon_k>0\) such that \(\varepsilon_k\searrow 0\) and such that \[\frac{C_k}{-\log|s_D|^2}\le 1\] on the \(\varepsilon_k\)-neighborhood \(D_{\varepsilon_k}\) of \(D\). Then by (67 ), we obtain \[\label{HYM32est32eq5} \sup_k\sup_{D_{\varepsilon_k}}\frac{k|S|_{h_0\otimes h_0^*}}{-\log|s_D|^2}\le 2r.\tag{68}\] Here we suppose that \(\inf_k\sup_{D_{\varepsilon_k}}\frac{|S|_{h_0\otimes h_0^*}}{-\log|s_D|^2}\ge c>0\) for some constant \(c>0\). Then \[\infty=\sup_k(kc)\le\sup_k\sup_{D_{\varepsilon_k}}\frac{k|S|_{h_0\otimes h_0^*}}{-\log|s_D|^2},\] contradicting to (68 ). Hence we obtain \[\inf_k\sup_{D_{\varepsilon_k}}\frac{|S|_{h_0\otimes h_0^*}}{-\log|s_D|^2}=0.\] ◻

Proof. By Lemma 16, we can assume that \(\mathcal{E}\) is locally free.
(1) The existence of a Jordan-H\(\rm\ddot{o}\)lder filtration is established in [8]. We include the proof for the completeness. Assume that \(\mathcal{E}\) is \(\alpha^{n-1}\)-slope semistable, but not stable, Then, by Claim 18 (2) (see also [22]), there exists a saturated subsheaf \(\mathcal{E}_1\subset \mathcal{E}\) such that

• \(\mu_{\alpha}(\mathcal{E}_1)=\mu_{\alpha}(\mathcal{E})\) and \(0<{\rm{rk}}\mathcal{E}_1<{\rm{rk}}\mathcal{E}\).

• For any saturated subsheaf \(\mathcal{F}\subset \mathcal{E}\) satisfying the two above conditions, then \(\mathcal{E}_1\not\subset\mathcal{F}\) or \(\mathcal{E}_1=\mathcal{F}\).

Then the rest of the proof of (1) is standard as follows. By (a) above, we have \(\mu_{\alpha}(\mathcal{E}/\mathcal{E}_1)=\mu_{\alpha}(\mathcal{E})\). We prove that \(\mathcal{E}/\mathcal{E}_1\) is \(\alpha^{n-1}\)-slope stable. Let \(\mathcal{F}\subset\mathcal{E}/\mathcal{E}_1\) be a saturated subsheaf with \(0<{\rm{rk}}\mathcal{F}<{\rm{rk}}(\mathcal{E}/\mathcal{E}_1)\). We consider the following short exact sequences

Since \(\mathcal{E}\) is \(\alpha^{n-1}\)-slope semistable, we have \(\mu_{\alpha}(\mathcal{E}_1+\mathcal{F})\le \mu_{\alpha}(\mathcal{E})\). Then by the exact sequence in the second line, we have \[\begin{align} 0 &={\rm{rk}}\mathcal{F}(\mu_{\alpha}(\mathcal{E}_1+\mathcal{F})-\mu_{\alpha}(\mathcal{F}))+{\rm{rk}}\mathcal{E}_1(\mu_{\alpha}(\mathcal{E}_1+\mathcal{F})-\mu_{\alpha}(\mathcal{E}_1))\\ &\le {\rm{rk}}\mathcal{F}(\mu_{\alpha}(\mathcal{E})-\mu_{\alpha}(\mathcal{F}))+{\rm{rk}}\mathcal{E}_1(\mu_{\alpha}(\mathcal{E})-\mu_{\alpha}(\mathcal{E}_1))\\ &={\rm{rk}}\mathcal{F}(\mu_{\alpha}(\mathcal{E}/\mathcal{E}_1)-\mu_{\alpha}(\mathcal{F})). \end{align}\] Hence we have \(\mu_{\alpha}(\mathcal{F})\le \mu_{\alpha}(\mathcal{E}/\mathcal{E}_1)\). If \(\mu_{\alpha}(\mathcal{F})= \mu_{\alpha}(\mathcal{E}/\mathcal{E}_1)\), then the first line above shows \[{\rm{rk}}\mathcal{F}(\mu_{\alpha}(\mathcal{E}_1+\mathcal{F})-\mu_{\alpha}(\mathcal{E}_1))=0.\] By (b) above, we obtain \(\mathcal{E}_1+\mathcal{F}=\mathcal{E}\). Then, the above commutative diagram shows \(\mathcal{F}=\mathcal{E}/\mathcal{E}_1\). It contradicts to \({\rm{rk}}\mathcal{F}<{\rm{rk}}(\mathcal{E}/\mathcal{E}_1)\). Therefore we have \(\mu_{\alpha}(\mathcal{F})< \mu_{\alpha}(\mathcal{E}/\mathcal{E}_1)\) and thus \(\mathcal{E}/\mathcal{E}_1\) is \(\alpha^{n-1}\)-slope stable. Then, inductively we obtain a Jordan-H\({\rm \ddot{o}}\)lder filtration of \(\mathcal{E}\).
(2) By definition of Jordan-H\({\rm \ddot{o}}\)lder filtration, the graded sheaf associated to the Jordan-H\({\rm \ddot{o}}\)lder filtration is \(\alpha^{n-1}\)-slope polystable. Hence it admits a unique \(T\)-adapted HYM metric by Theorem 1. ◻

Proof. (1) By the argument in [8], it suffices to show in the case that \(\mathcal{E}\) is \(\alpha^{n-1}\)-slope stable (see also Remark 19). The proof is similar to [4]. Let \(T\) be an adapted current in \(\alpha\). Let \(\pi:Y\to X\) be a composition of blow-ups along \(E_{nK}(\alpha)\) and \({\rm{Sing}}(\mathcal{E})\) such that \(E:=\pi^*\mathcal{E}/{\rm{Tor}}\) is locally free, \(D=E_{nK}(\pi^*\alpha)\) is a snc divisor in \(Y\) and \(\pi^*T\) admits an adapted approximation \(\omega_i\in\pi^*\alpha+(1/i)\{\omega_Y\}\) where \(\omega_Y\) is a Kähler metric on \(Y\). By Lemma 16, the vector bundle \(E\) is \((\pi^*\alpha)^{n-1}\)-slope stable. Then, by Lemma 17, we can find an \(\omega_i\)-HYM metric \(h_i=h_0\Psi_i^2\) on \(E\) with \(\det \Psi_i=1\). And by Theorem 40, we can also find a \(T\)-adapted HYM metric \(h_{\infty}=h_0\Psi_{\infty}^2\) on \(E\) with \(\det \Psi_{\infty}=1\). Then, by Theorem 39, we have that \(h_i\) subsequencially converges to \(h_{\infty}\) in \(C^{1,\alpha}\)-topology on any relatively compact open subset in \(X\setminus D\). Since \[\int_X{\rm{Tr}}(\sqrt{-1}F_{h_i}\wedge \sqrt{-1}F_{h_i})\wedge\omega_i^{n-2} \le \int_X|F_{h_i}|_{h_i,\omega_i}\omega_i^n\le C,\] and \[\int_X\left({\rm{Tr}}(\sqrt{-1}F_{h_i})\wedge {\rm{Tr}}(\sqrt{-1}F_{h_i})\right)\wedge\omega_i^{n-2} \le \int_X|F_{h_i}|_{h_i,\omega_i}\omega_i^n\le C,\] we can choose a further subsequence of \(\{h_i\}_i\) so that \[\label{BG32eq32eq1} {\rm{Tr}}(\sqrt{-1}F_{h_i}\wedge \sqrt{-1}F_{h_i})\wedge\omega_i^{n-2}\to {\rm{Tr}}(\sqrt{-1}F_{h_{\infty}}\wedge \sqrt{-1}F_{h_{\infty}})\wedge T^{n-2}\tag{69}\] and \[\label{BG32eq32eq2} \left({\rm{Tr}}(\sqrt{-1}F_{h_i})\wedge {\rm{Tr}}(\sqrt{-1}F_{h_i})\right)\wedge\omega_i^{n-2}\to \left({\rm{Tr}}(\sqrt{-1}F_{h_{\infty}})\wedge {\rm{Tr}}(\sqrt{-1}F_{h_{\infty}})\right)\wedge T^{n-2}\tag{70}\] on \(Y\setminus D\) in the sense of current. We denote by \(c_i(h_i)\) the \(i\)-th Chern form defined by \(h_i\). Since a sequence of measures \[d\mu_i:=(2rc_2(h_i)-(r-1)c_1(h_i)^2)\wedge\omega_i^{n-2}\] has uniformly bounded mass, we can find a subsequence such that \(d\mu_i\) weakly converges to a measure \(d\mu\) and, by (69 ) and (70 ), \[d\mu=(2rc_2(h_{\infty})-(r-1)c_1(h_{\infty})^2)\wedge T^{n-2}\] on \(Y\setminus D\). Since \(d\mu\) has uniformly bounded mass on \(Y\), the following computation works by the Fatou’s lemma: \[\begin{align} \label{BG32eq32eq3} (2rc_2(E)-(r-1)c_1(E)^2)\cdot(\pi^*\alpha)^{n-2} &=\lim_{i\to\infty}(2rc_2(E)-(r-1)c_1(E)^2)\cdot\{\omega_i\}^{n-2}\notag\\ &=\lim_{i\to\infty}\int_{Y\setminus D}(2rc_2(h_i)-(r-1)c_1(h_i)^2)\wedge\omega_i^{n-2}\notag\\ &\ge\int_{Y\setminus D}(2rc_2(h_{\infty})-(r-1)c_1(h_{\infty})^2)\wedge T^{n-2}\notag\\ &\ge 0. \end{align}\tag{71}\] Since \(\mathcal{E}\) is locally free in codimension 2, we have \[(2rc_2(\mathcal{E}-(r-1)c_1(\mathcal{E})^2)\cdot\alpha^{n-2} =(2rc_2(E)-(r-1)c_1(E)^2)\cdot(\pi^*\alpha)^{n-2}.\] Hence \(E=\pi^*\mathcal{E}/{\rm{Tor}}\) is projectively flat on \(Y\setminus D\). Since \(\pi:Y\setminus D\to X\setminus (E_{nK}(\alpha)\cup{\rm{Sing}}(\mathcal{E}))\) is isomorphic, we have that \(\mathcal{E}|_{X\setminus (E_{nK}(\alpha)\cup{\rm{Sing}}(\mathcal{E}))}\) is projectively flat. That is, \(\mathcal{E}\) is given by the representation \(\rho:\pi_1({\rm{Amp}}(\alpha)\setminus {\rm{Sing}}(\mathcal{E}))\to PU(n)\). Since \({\rm{codim}}{\rm{Sing}}(\mathcal{E})\ge 3\), we have \(\pi_1({\rm{Amp}}(\alpha)\setminus {\rm{Sing}}(\mathcal{E}))=\pi_1({\rm{Amp}}(\alpha))\) and thus \(\rho\) extends to the representation \(\rho\rq{}:\pi_1({\rm{Amp}}(\alpha))\to PU(n)\). Let \(E\rq{}\) be the projectively flat vector bundle over \({\rm{Amp}}(\alpha)\) given by \(\rho\rq{}\). Then \(\mathcal{E}|_{{\rm{Amp}}(\alpha)\setminus {\rm{Sing}}(\mathcal{E})}\simeq E\rq{}|_{{\rm{Amp}}(\alpha)\setminus {\rm{Sing}}(\mathcal{E})}\). Since \({\rm{codim}}{\rm{Sing}}(\mathcal{E})\ge 3\) and \(\mathcal{E}\) is reflexive, we have \(\mathcal{E}|_{{\rm{Amp}}(\alpha)}\simeq E\rq{}\). In particular, we have that \(\mathcal{E}\) is locally free on \({\rm{Amp}}(\alpha)\) and is projectively flat on \({\rm{Amp}}(\alpha)\).

(2) By Theorem 55, we know that \(\mathcal{E}\) admits a Jordan-H\(\ddot{o}\)lder filtration \[0\subset \mathcal{E}_{k}\subset\cdots\subset\mathcal{E}_1\subset \mathcal{E}.\] Then the argument in [8] shows that each \(\mathcal{E}_{i}/\mathcal{E}_{i+1}\) satisfies the Bogomolov-Gieseker equality. By definition, each quotient \(\mathcal{E}_{i}/\mathcal{E}_{i+1}\) is \(\alpha^{n-1}\)-slope stable. Hence, by (1) of this theorem, we see that \(\mathcal{E}_{i}/\mathcal{E}_{i+1}\) is projectively flat on \(X_{{\rm{reg}}}\). ◻

Proof. As Theorem 56, it suffices to show when \(\mathcal{E}\) is \(\langle\alpha^{n-1}\rangle\)-slope stable. Let \(\pi:Y\to X\) be a resolution such that \(Y\) is a smooth projective manifold and \(\beta:=\langle\pi^*\alpha\rangle\) is a nef and big class. Then, we have \[\langle(\pi^*\alpha)^k\rangle=\beta^k\] for any \(k\) by [8]. By [35], we have \[E_{nK}(\pi^*\alpha)=\pi^{-1}(E_{nK}(\alpha))\cup {\rm{Exc}}(\pi).\] Hence, using Lemma 59, we have \[\beta^{n-1}\cdot[D\rq{}]=\langle(\pi^*\alpha)^{n-1}\rangle\cdot[D\rq{}]=0\] for any exceptional divisor \(D\rq{}\). Thus, we can see that \(\pi^{[*]}\mathcal{E}\) is \(\beta^{n-1}\)-slope stable as Lemma 16. Let \(h_{\infty}\) be a \(T\)-adapted HYM metric on \(\pi^{[*]}\mathcal{E}\) where \(T\in\beta\) is an adapted current. Then, by the definition of the discriminant (72 ) and the arugument in Theorem 56 (refer to 71 ), we have \[0=\Delta(\mathcal{E})\langle\alpha^{n-2}\rangle \ge\int_{{\rm{Amp}}(\pi^*\alpha)}(2rc_2(h_{\infty})-(r-1)c_1(h_{\infty})^2)\wedge T^{n-2}\ge 0.\] Hence, we have that \(\pi^{[*]}\mathcal{E}\) is projectively flat on \({\rm{Amp}}(\pi^*\alpha)\). Since \(\pi:{\rm{Amp}}(\pi^*\alpha)\to{\rm{Amp}}(\alpha)\) is isomorphic and \(E|_{{\rm{Amp}}(\pi^*\alpha)}=\pi^*(\mathcal{E}|_{{\rm{Amp}}(\alpha)})\), we obtain that \(\mathcal{E}\) is projectively flat on \({\rm{Amp}}(\alpha)\).

(2) Since now \(X\) is Moishezon, \(\mathcal{E}\) is \(\langle\alpha^{n-1}\rangle\)-slope semistable if and only if \(\pi^*\mathcal{E}/{\rm{Tor}}\) is \(\langle(\pi^*\alpha)^{n-1}\rangle\)-slope semistable for any resolution \(\pi:Y\to X\) so that \(Y\) is smooth projective, \(\pi^*\mathcal{E}/{\rm{Tor}}\) is locally free and \(\langle\pi^*\alpha\rangle\) is nef and big by Lemma 59. Then the push-forward of the Jordan-H\(\rm \ddot{o}\)lder filtration of \(\pi^*\mathcal{E}/{\rm{Tor}}\) gives the Jordan-H\(\rm \ddot{o}\)lder filtration of \(\mathcal{E}\), \[0\subset \mathcal{E}_{k}\subset\cdots\subset\mathcal{E}_1\subset \mathcal{E}.\] We recall that \(\mathcal{E}_i/\mathcal{E}_{i+1}\) is \(\langle\alpha^{n-1}\rangle\)-slope stable. Then, by the argument in [8] and (?? ), we have \[0=\Delta(\mathcal{E})\langle\alpha^{n-2}\rangle\ge \Delta(\mathcal{E}_i/\mathcal{E}_{i+1})\langle\alpha^{n-2}\rangle\ge0.\] Then, by (1) in this theorem, we see that \(\mathcal{E}_i/\mathcal{E}_{i+1}\) is projectively flat on \({\rm{Amp}}(\alpha)\). ◻

Proof. As in the proof of Theorem 56 (1), we can assume that \(\mathcal{E}\) is \(\alpha^{n-1}\)-slope stable. Then, by Lemma 17, \(\mathcal{E}\) is \((\alpha+\varepsilon\omega_0)^{n-1}\)-slope stable. Since \(c_1(\mathcal{E})=0\) and \(c_2(\mathcal{E})=0\), we have \(\Delta(\mathcal{E})\cdot(\alpha+\varepsilon\omega_0)^{n-2}=0\). Thus \(\mathcal{E}\) is locally free on \(X\) and flat globally on \(X\). ◻

Proof. This proof is motivated by [15]. We use the same notation as in [8]. Since \(X\) is K-stable, it follows from [10] that the anticanonical model \(Z\) is a K-stable klt Fano variety. Moreover, by [37], the canonical extension sheaf \(\mathcal{E}_Z\) is \(c_1(-K_Z)^{n-1}\)-polystable.

Let \(f \colon X \dashrightarrow Z\) be the natural birational map, and take a resolution \(W\) with morphisms \(p \colon W \to X\) and \(q \colon W \to Z\) resolving \(f\): \[\xymatrix@C=40pt@R=30pt{ & W \ar[ld]_p \ar[rd]^q & \\ X \ar@{-->}[rr]^f && Z }\] By the argument of [8], there exists an effective \(q\)-exceptional divisor \(B\) on \(W\) such that \[\label{eq-contraction} p^*(-K_X) = q^*(-K_Z) + B\tag{74}\] and \(p({\rm{Supp}}(B))\) coincides with the \(f\)-exceptional locus. Therefore, we have an isomorphism \[\label{eq-can32sheaf} p^*\mathcal{E}_X \cong q^*\mathcal{E}_Zoutside the support of B\tag{75}\] and \(-K_X\) admits a birational Zariski decomposition. Since \(c_i(X)=c_i(\mathcal{E}_X)\) and now \(X\) is smooth in codimension 2, the assumption (?? ) coincides with \[\label{eq-MY32is32BG} \Delta(\mathcal{E}_X)\langle(-K_X)^{n-2}\rangle=0.\tag{76}\] Thus, by Theorem 60, we obtain that \(\mathcal{E}_X\) is projectively flat on \(X_{{\rm{reg}}}\). By (75 ) and the fact that \(B\) is \(q\)-exceptional, we obtain the following claim.

Since \(Z\) has at most klt singularities, by [38] (see also [15]), there exists a finite quasi-étale Galois cover \(\nu \colon \widehat{Z} \to Z\) such that \(\widehat{Z}\) is maximal quasi-étale. In particular, \(-K_{\widehat{Z}} = \nu^{*}(-K_{Z})\) (refer to [15]), so \(-K_{\widehat{Z}}\) is ample.

We will show that \(\widehat{Z} \cong \mathbb{C}\mathbb{P}^{n}\). Let \(\widehat{Z}^{\circ} := \nu^{-1}(Z_{{\rm{reg}}})\). Then \(\mathcal{E}_{\widehat{Z}}|_{\widehat{Z}^{\circ}}\) is also projectively flat since \(\nu \colon \widehat{Z}^{\circ} \to Z_{{\rm{reg}}}\) is étale by purity of the branch locus (refer to [38]), and we have \(\mathcal{E}_{\widehat{Z}}|_{\widehat{Z}^{\circ}} \cong \pi^{*}(\mathcal{E}_{Z}|_{Z_{{\rm{reg}}}})\) by [15]. Hence by Definition 53, the reflexive sheaf \({\rm{End}}(\mathcal{E}_{\widehat{Z}})\) is flat and locally free on \(\widehat{Z}^{\circ}\). As \({\rm{codim}}(\widehat{Z} \setminus \widehat{Z}^{\circ}) \ge 2\), we have \(\pi_1(\widehat{Z}_{{\rm{reg}}}) \cong \pi_1(\widehat{Z}^{\circ})\), so \({\rm{End}}(\mathcal{E}_{\widehat{Z}})\) is also flat locally free on \(\widehat{Z}_{{\rm{reg}}}\). Since \(\widehat{Z}\) is maximally quasi-étale, [38] implies that \({\rm{End}}(\mathcal{E}_{\widehat{Z}})\) is flat locally free on \(\widehat{Z}\).

Consider the following locally split exact sequence: \[\label{can32ext32of32Z} 0 \to \mathcal{O}_{\widehat{Z}} \to \mathcal{E}_{\widehat{Z}} \to \mathcal{T}_{\widehat{Z}} \to 0.\tag{77}\] Since (77 ) is locally splittable by the construction [15], we have that \(\mathcal{T}_{\widehat{Z}}\) is a direct summand of \({\rm{End}}(\mathcal{E}_{\widehat{Z}})\). Since \({\rm{End}}(\mathcal{E}_{\widehat{Z}})\) is locally free, so is \(\mathcal{T}_{\widehat{Z}}\). Thus, by [39], \(\widehat{Z}\) is smooth. Since \(\mathcal{E}_{\widehat{Z}}\) is projectively flat and \(\widehat{Z}\) is Fano, there exists a line bundle \(L\) with \(\mathcal{E}_{\widehat{Z}} \cong L^{\oplus (n+1)}\). (Note that Fano manifolds are simply connected and thus the representation \(\pi_1(\widehat{Z})\to PGL(n+1,\mathbb{C})\) corresponding to \(\mathcal{E}_{\widehat{Z}}\) is the trivial representations.) Moreover, since \(\det \mathcal{E}_{\widehat{Z}} \cong \mathcal{O}_{\widehat{Z}}(-K_{\widehat{Z}})\), we obtain \[\mathcal{O}_{\widehat{Z}}(-K_{\widehat{Z}}) \cong L^{\otimes (n+1)}.\] Since \(-K_{\widehat{Z}}\) is ample, \(L\) is also ample. Therefore, by [40], we conclude that \(\widehat{Z} \cong \mathbb{C}\mathbb{P}^{n}\). ◻