Holomorphic sectional curvature and Kähler-like metric

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This paper focuses on the Kähler-like manifolds with holomorphic sectional curvature condition. Firstly, we prove thatKähler-like manifold with non-zero constant holomorphic sectional curvature must be Kähler. Secondly, we prove the parabolic Schwarz lemmas on Hermite manifolds.Thirdly, we give several vanishing theorems on compact Kähler-like manifold with positive holomorphic sectional curvature.