Fujita decomposition and Hodge loci

TL;DR

This paper explores properties of Hodge loci in the moduli space of curves, showing conditions under which certain substructures appear and establishing that Hodge loci have codimension at least 2.

Contribution

It provides new results linking Fujita decomposition, Hodge loci, and the geometry of moduli spaces of curves, including codimension bounds.

Findings

Hodge substructure in exterior powers of fiber cohomology

Moduli image of fibers lies in a proper Hodge locus

Hodge locus in the moduli space of curves has codimension at least 2

Abstract

This paper contains two results on Hodge loci in the moduli space of curves. The first concerns fibrations over curves with a non-trivial flat part in the Fujita decomposition. If local Torelli theorem holds for the fibres and the fibration is non-trivial, an appropriate exterior power of the cohomology of the fiber admits a Hodge substructure. In the case of curves it follows that the moduli image of the fibers is contained in a proper Hodge locus. The second result deals with divisors in the moduli space of curves. It is proved that the image of a divisor in the moduli of principally polarized abelian varieties is not contained in a proper totally geodesic subvariety. It follows that a Hodge locus in the moduli space of curves has codimension at least 2.