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 $s\to 0$?
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.
