Punctual characterization of the unitary flat bundle of weight 1 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.