Communications in Analysis and Geometry
Volume 24 (2016)
A remark on our paper “Negative holomorphic curvature and positive canonical bundle”
Pages: 901 – 912
This is a continuation of our first paper in [WY16]. There are two purposes of this paper: One is to give a proof of the main result in [WY16] without going through the argument depending on numerical effectiveness. The other one is to provide a proof of our conjecture, mentioned in [TY], where the assumption of negative holomorphic sectional curvature is dropped to quasi-negative. We should note that a solution to our conjecture is also provided 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 and Cheng in [CY75].
The proof of Theorem 1 below is obtained by us in 2015, and has been distributed in the community; e.g. it was presented by the second author in the birthday conference of Richard Schoen on June 21, 2015. We have also settled the complete noncompact case, which, together with applications, will appear soon.