Derived equivalence and fibrations over curves and surfaces

We prove that the bounded derived category of coherent sheaves on a smooth projective complex variety reconstructs the isomorphism classes of fibrations onto smooth projective curves of genus g >= 2. Moreover, in dimension at most four, we prove that the same category reconstructs the isomorphism classes of fibrations onto nor-mal projective surfaces with positive holomorphic Euler characteristic and admitting a finite morphism to an abelian variety. Finally, we study the derived invariance of a class of fibrations with minimal base dimension under the condition that all the Hodge numbers of type h0,p(X) are derived invariant.