Punctual characterization of the unitary flat bundle of weight one PVHS and application to families of curves

In this paper we consider the problem of pointwise determining the fibres of the flat unitary subbundle of a PVHS of weight one. Starting from the associated Higgs field, and assuming the base has dimension $1$, we construct a family of (smooth but possibly non-holomorphic) morphisms of vector bundles with the property that the intersection of their kernels at a general point is the fibre of the flat subbundle. We explore the first one of these morphisms in the case of a geometric PVHS arising from a family of smooth projective curves, showing that it acts as the cup-product with some sort of"second-order Kodaira-Spencer class"which we introduce, and check in the case of a family of smooth plane curves that this additional condition is non-trivial.