Abstract
The purpose of this paper is twofold. We first solve the Dirichlet problem for $$\tau $$-Hermitian–Einstein equations on holomorphic filtrations over compact Hermitian manifolds. Secondly, by using Uhlenbeck–Yau’s continuity method, we prove the existence of approximate $$\tau $$-Hermitian–Einstein structure on holomorphic filtrations over closed Gauduchon manifolds.
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