  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.
