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

TL;DR

This paper provides a pointwise method to determine the flat unitary subbundle of a weight one PVHS using associated Higgs fields, with applications to families of curves and introducing a new second-order Kodaira-Spencer class.

Contribution

It introduces a novel family of morphisms derived from Higgs fields to identify the flat subbundle pointwise, and defines a new second-order Kodaira-Spencer class for geometric PVHS.

Findings

The morphisms' kernels intersect at the flat subbundle fiber.

The first morphism acts as cup-product with a second-order Kodaira-Spencer class.

In the case of plane curves, the new condition is shown to be non-trivial.

Abstract

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.