In the following paper , [The Kahler-Ricci flow on surfaces of positive Kodaira dimension][1] (Sung and Tian) in page  621, I am trying to understand the proof of part 1 of Lemma 3.4.  

In the part 1 of proof 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. Can you explain for me?

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][3]

In final the strategy of proof is not clear for me. I need to more details


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