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This is a continuation of our first paper in [WY16]. There are two purposesof this paper: One is to give a proof of the main result in [WY16] withoutgoing through the argument depending on numerical effectiveness. The other oneis to provide a proof of our conjecture, mentioned in [TY], where theassumption of negative holomorphic sectional curvature is dropped toquasi-negative. We should note that a solution to our conjecture is alsoprovided by Diverio-Trapani [DT]. Both proofs depend on our argument in [WY16].But our argument here makes use of the argument given by the second author andCheng in [CY75].

A remark on our paper "Negative Holomorphic curvature and positive canonical bundle"