Let $f:X\rightarrow C$ be a fibration of complex manifold over a smooth curve $C$. Let $D$ be an irreducible component of singular fibers. How can one prove that $X$ does not admit any fibration with general fiber class $D$? Or any counterexample?
I think this is intuitively true because $D$ is an irreducible component of singular fibers, so it cannot $move$. But I don't know how to prove it.
I would appreciate it if you can provide me a proof or reference.