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In the following paper , The Kahler-Ricci flow on surfaces of positive Kodaira dimension (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 directly say

"Therefore $ω_{SF}$ is a smooth family of Ricci-flat metrics over $B$". I can not see this fact from Logarithmic transformation. Can you explain more?

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

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

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    $\begingroup$ What paper? What page? $\endgroup$
    – Jason Starr
    Commented Sep 8, 2014 at 14:13
  • $\begingroup$ You may look at the Gross-Wilson paper on Large complex structure limits of $K3$ surfaces. They discussed the semi-flat metric in great detail. This is the standard Gibbons-Hawking construction. $\endgroup$
    – YHBKJ
    Commented Sep 9, 2014 at 7:10
  • $\begingroup$ Thanks user43423, But I couldn't find any connection ,you can explain more $\endgroup$
    – user21574
    Commented Sep 9, 2014 at 12:27
  • $\begingroup$ Perhaps your advisor can provide you with more details, since he wrote the paper? $\endgroup$
    – YangMills
    Commented Feb 4, 2015 at 21:12

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