Fefferman showed that the Bergman kernel of strictly pseudoconvex domains admit logarithmic singularity.  For the case of the ball, the log term vanishes and it is conjectured that the log term vanishes only for the ball.  Robin Graham proved it in 2-dimensions, while for higher dimensions, counter examples were found for domains which are not Stein.  However, I still believe that the conjecture is true for domains in C^n.  In this talk, I use Kuranishi's deformation complex of CR structures to determine the moduli space of the deformation of the ball (based on the works of Bland-Ducahmp) and apply the result to prove the conjecture for domains sufficiently close to the ball.