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In this paper we consider the problem of pointwise determining the fibres ofthe flat unitary subbundle of a PVHS of weight one. Starting from theassociated Higgs field, and assuming the base has dimension $1$, we construct afamily of (smooth but possibly non-holomorphic) morphisms of vector bundleswith the property that the intersection of their kernels at a general point isthe fibre of the flat subbundle. We explore the first one of these morphisms inthe 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-orderKodaira-Spencer class" which we introduce, and check in the case of a family ofsmooth plane curves that this additional condition is non-trivial.

Punctual characterization of the unitary flat bundle of weight 1 PVHS and application to families of curves