Principe local-global pour les z\'ero-cycles sur certaines fibrations au-dessus d'une courbe : I

Let $X$ be a smooth projective variety over a number field, fibered over a curve, with geometrically integral fibers. We prove that, supposing the finiteness of $\text{sha}(Jac(C))$, if the fibers over a generalised Hilbertian subset satisfy the Hasse principle (resp. weak approximation), then the Brauer-Manin obstruction coming from the base curve is the only obstruction to the Hasse principle (resp. to weak approximation) for zero-cycles of degree 1 on $X$.