Symplectic and Isometric SL(2,R) invariant subbundles of the Hodge bundle

TL;DR

This paper proves that the Forni bundle of an affine SL(2,R)-invariant submanifold in the moduli space of curves is always flat and orthogonal to the tangent space, leading to the semisimplicity of the Hodge bundle.

Contribution

It establishes the flatness and orthogonality of the Forni bundle for affine SL(2,R)-invariant submanifolds, revealing the semisimplicity of the Hodge bundle.

Findings

The Forni bundle is always flat.

It is orthogonal to the tangent space of the submanifold.

The Hodge bundle of the submanifold is semisimple.

Abstract

Suppose N is an affine SL(2,R)-invariant submanfold of the moduli space of pairs (M,w) where M is a curve, and w is a holomorphic 1-form on M. We show that the Forni bundle of N (i.e. the maximal SL(2,R)-invariant isometric subbundle of the Hodge bundle of N) is always flat and is always orthogonal to the tangent space of N. As a corollary, it follows that the Hodge bundle of N is semisimple.