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We say a smooth projective surface  X  satisfies the bounded cohomology property if there exists a positive constantsuch thatfor every prime divisor  C  on  X . Let the closed Mori conesuch thatandwithare some curves on  X . If either (i) the Kodaira dimensionor (ii), the irregularityand the Iitaka dimension, then we prove that  X  satisfies the bounded cohomology property.

Bounded cohomology property on a smooth projective surface with Picard number two