In this note we explain that the conjecture of the pinching of the bisectional curvature mentioned in [HGY] and [CHY] is proved by a combination of the arguments from the proofs of the Theorem 1.2 in [CHY], the Theorem 2 in [HGY] and the Proposition 4 in [SY]. Moreover, we prove that any compact Kähler-Einstein surface M is a quotient of the complex two dimensional unit ball or the complex two dimensional plane if (1) M has nonpositive Einstein constant and (2) at each point, the average

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... the average holomorphic sectional curvature is closer to the minimal than to the maximal. √ 1/6] < 0.476. We also observed in Theorem 2 that if a ≤ 1 2 , then there is a ball-like point P . That is, at P , K max = K min . We notice that 1/6 > 1/3. Therefore, we conjectured that M is a quotient of the complex ball if a = 1 2 . In general, we believe that we