In the following paper , in page  621 in [1][1] , I am trying to understand the proof of part 1.  

In the proof of part 1 of inequality the authors first use of logarithmic transformation due to Kodaira. After they say 

"Therefore $ω_{SF}$ is a smooth family of Ricci-flat metrics over $B$".
I can not see this fact. 

In the first part of proof they define $\tilde Y=\mathbb C\times \tilde B/L$, BUT I think we must define $\tilde Y=\mathbb C\times  B/L$. For instance see this [paper][2]

In final The proof is not clear for me


  [1]: http://www.maths.ed.ac.uk/cheltsov/cambridge/pdf/tian07.pdf
  [2]: http://www.math.nyu.edu/~tschinke/papers/yuri/04ellhyp/ellhyp6.pdf