The canonical structures of the limit of the Yang-Mills flows for nef and big classes

Abstract

In the previous paper \cite{Jin26}, the author introduced the notions of an adapted current $T$ and an adapted Hermitian-Einstein metric to establish the Kobayashi-Hitchin correspondence for a nef and big class $\alpha$. As a continuation of the previous work, this paper studies the solvability and the convergence of the Yang-Mills flow for a nef and big class $\alpha$ on a holomorphic vector bundle $E$ over a compact K\"{a}hler manifold $X$. In particular, we show that the limit of the Yang-Mills flow at infinity is determined by the holomorphic structure of $E$ and the nef and big class $\alpha$. More precisely, if we fix an integrable unitary connection $A_0$ on $E$, we show that the $T$-Yang-Mills flow on $E$ with initial condition $A_0$ is solvable for all time and it converges to a $T$-Yang-Mills connection $A_{\infty}$ in the sense of Uhlenbeck limit. Furthermore, we also show that, on the ample locus of $\alpha$, $A_{\infty}$ is complex-gauge equivalent to the direct sum of the Chern connections of the $T$-adapted Hermitian-Einstein metrics on the factors of the graded sheaf associated with the $\alpha^{n-1}$-Harder-Narasimhan-Seshadri filtration of $E$.