Let (M, g, J) be a closed Kahler manifold of complex dimension m > 1. We denote by Spec(M,g) the spectrum of the real Laplace-Beltrami operator. DELTA. acting on functions on M. The following characterization problem on the spectral rigidity of the complex projective space (CP

, g

, J

) with the standard complex structure J

and the Fubini-Study metric g

has been attacked by many mathematicians : if (M,g,J) and (CP

,g

,J

) are isospectral then is it true that (M,g,J) is holomorphically isometric to (CP

,g

,J

)\ulcorner In [BGM], [LB], it is proved that if (M,J) is (CP

, J

) then the answer to the problem is affirmative. Tanno ([Ta]) has proved that the answer is affirmative if m .leq. 6. Recently, Wu([Wu]) has showed in a more general sense that if (M, g) and (CP

,g

) are (-4/m)-isospectral, m .geq. 4, and if the second betti number b

(M) is equal to b

(CP

).