Abstract : The Jacobian locus is defined as the closure in the moduli space of principally polarized abelian varieties (p.p.a.v.) of the image of the Torelli map, mapping any smooth projective curve to its Jacobian variety. In the talk, we are interested in studying its geometry by using locally constant periods. More precisely, any curve B in the moduli space of p.p.a.v. is associated to a polarized variation of hodge structure (p.v.h.s.) of weight 1 and unitary flat local periods with respect to its hodge bundle of type (1,0) define a unitary flat bundle U. In particular, for curves in the Jacobian locus one can study such a bundle by looking at the associated family of curves. I will present some results on its rank and its relation with another interesting bundle involving periods fixed at the first order, and after that I will discuss their application to the study of the Colemann-Oort conjecture.