Unitary flat part in polarized variations of the Hodge structure of families of curves

In the geometric variation of the Hodge structure associated to a semistable 1-dimensional family or curves the polarization given by the intersection form defines a unitary flat part on the canonical Deligne extension of the (1,0)-Hodge bundle. In the seminar I will summarize some results obtained on the rank and on the monodromy of this bundle and I will discuss some applications and related problems. The results are in collaboration with Víctor González-Alonso, Gian Pietro Pirola and Lidia Stoppino.