Let $(\mathcal X,\mathcal D)\to T$ be a surjective holomorphic fibre space of K\"ahler manifolds of pairs such that fibers $(X\_s,D\_s)$ admit Ricci flat metric in bounded geometric sense (conic, Poincare singularities),i.e $Ric(\omega\_s)=[D\_s]$


Then, is there any asymptotic formula for $Ric(\omega\_s)$ and $\omega\_s$ when $s\to 0$?



Asymptotic formula for Ricci flat metric
On each fiber $(X\_s,D\_s)$ , if you kindly assume $D\_s$ is snc with conic singularities such that $K\_{X\_s}+D\_s$ is trivial, then $Ric(\omega\_s)=[D\_s]$ where $[D\_s]$ is the current of integration of divisor $D\_s$. Hence you need to know the asymptotic formula of current of integration $[D\_s]$. See page 150 , 3rd line,


*D.Barlet , Dévelopment asymptotique des fonctions obtenues par intégration sur les fibres, Inventiones mathematicae (1982) Volume: 68, page 129-174.* 


there is an asymptotic formula, hence you get an asymptotic formula when $s$ approaches to zero, some people from Stanford published a paper in Annals of math in 2016 as same as this formula of Barlet about asymptotic formula of Ricci flat metric on pair $(X,D)$ (Kähler–Einstein metrics with edge singularities, Annals of Mathematics,
Pages 95-176 from Volume 183 (2016), Issue 1 by Thalia Jeffres, Rafe Mazzeo, Yanir A. Rubinstein).


The important point is that, Ricci flat metric is a good metric in the sense of Mumford and if you don't assume such a mild singularities on central fiber in the sense of MMP, you loose goodness of such metric on central fiber.


**Good metric in the sense of Mumford**


Let $\omega$ be a smooth local $p$-form defined on $V\_\alpha$ we say
$\omega$ has Poincaré growth if there is a constant
$C\_\alpha>0$ depending on $\omega$ such that


$$|\omega(t\_1,t\_2,...,t\_p)|^2\leq C\_\alpha \prod\_{i=1}^p||t\_i||^2$$
for any point
$z
\in V\_\alpha$
and $t\_1,...,t\_p\in T\_zX$ where $||.||$ is taken on Poincaré metric. We say $\omega$
is Mumford metric if both
$\omega$ and $d\omega$ have Poincaré growth.


