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### On the kernel bundle of the Higgs field of families of curves and the Jacobian locus

##### Sara Torelli (Pavia)

Aula Beltrami - Thursday, March 21, 2019 h.16:00

**Abstract. ** We consider two vector subbundles of the Hodge bundle naturally associated to any polarized variation of Hodge structures of weight 1: the flat bundle U with the Gauss-Manin connection and the kernel bundle K of the associated Higgs-field (fibrewise describing its infinitesimal variation). By definition they always fit into an inclusion of U in K, so one can ask in which cases this must be strict and why. In the seminar we answer the question for variations coming from families of curves. More precisely, for any smooth projective curve C of genus greater than 2 and for any integer k from 0 to g-1, we construct a family of deformations of C parametrized by a quasi-projective smooth curve such that the rank of K is k, the rank of U is less or equal than (g+1)/2 and the monodromy of U is finite.

As an application we use the modular interpretation of such families to study their properties in the Jacobian locus, following some recent improvements in collaboration with Alessandro Ghigi and Gian Pietro Pirola.

The result is a joint work with Victor Gonzàlez-Alonso.

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