Twisted holomorphic chains and vortex equations over non-compact Kähler manifolds

In this paper, we study twisted holomorphic chains and related gauge equations over non-compact Kähler manifolds. We use the heat flow method to solve the Dirichlet boundary problem for the related gauge equations, and prove a Hitchin–Kobayashi type correspondence for twisted holomorphic chain over some non-compact Kähler manifolds.