The authors study asymptotic properties of the Hermitian-Yang-Mills flow \(H^{-1} \frac{\partial H}{\partial t}= 2(i\Lambda_{\omega}F_H - \lambda Id_E), H(0)=H_0\) over a compact non-Kähler manifold \((X, g)\) with the Kähler form \(\omega\) satisfying the Gauduchon and Astheno-Kähler condition where \((E,H_0)\) is a Hermitian vector bundle over \(M\) and \(F_H\) is the curvature of the Chern connection for \((E,H)\).
Let \(A\) be a connection on \(E\) and \(F_A\) be its curvature. The authors prove:
Theorem. Let \((X, \omega)\) be an \(n\)-dimensional compact Hermitian manifold with \(\omega\) satisfying \(\partial\bar{\partial}\omega^{n-1} = \partial\bar{\partial}\omega^{n-2} = 0\).
Suppose \(A(t)\) is the global smooth solution of the heat flow \(\frac{\partial A}{\partial t} = i(\bar{\partial}_A -\partial_A)\Lambda_{\omega}F_A\), on the Hermitian vector bundle \((E,H_0)\) with the initial data \(A_0=(\bar{\partial}_E,H_0)\) over \((X, \omega)\). Then
(1) for every sequence \(t_k \rightarrow\infty\), there is a subsequence \(t_{k_j}\) such that as \(t_{k_j} \rightarrow\infty\), \(A(t_{k_j} )\) converges modulo gauge transformations to a connection \(A_{\infty}\) satisfying \(D_{A_{\infty}}\Lambda_{\omega}F_{A_{\infty}} = 0\) on the Hermitian vector bundle \((E_{\infty},H_{\infty})\) in \(C^{\infty}_{loc}\) topology outside \(\Sigma\), where \(\Sigma\) is a closed set of Hausdorff codimension at least 4;
(2) the limiting \((E_{\infty},H_{\infty}, \bar{\partial}_{A_{\infty}})\) can be extended to the whole X as a reflexive sheaf with a holomorphic orthogonal splitting \[(E_{\infty},H_{\infty},A_{\infty}) =\bigoplus_{i=1}^l(E^i_{\infty},H^i_{\infty},A^i_{\infty}),\] where \(l\) is the number of distinct eigenvalues of \(\Lambda_{\omega}F_{A_{\infty}}\) on \(X -\Sigma\) and \(H^i_{\infty}\) is an admissible Hermitian-Einstein metric on the reflexive sheaf \(E^i_{\infty}\).