Let $\pi:T\rightarrow \Delta$ be a family of elliptic curves over a disc $\Delta=\{|z|=1\}$. Suppose that this family has a double fiber over the origin. What is the (higher) direct image sheaves $R^i\pi_*\mathcal{O}_T$. Is it the same as those of a smooth family? I think the (higher) direct image sheaves reflect holomorphic structure of the central fiber. 

Moreover, what if one replaces $\Delta$ with a neighbourhood of a singularity? Say a Du Bois singularity. Are they still locally constant?

Thank you for your assistance.