The Hodge bundle over the moduli space of genus g >= 2 curves is the rank-g vector bundle whose fiber over a curve C is the space of holomorphic differentials H^0(C, K_C). It is one of the most studied objects in algebraic geometry — its Chern classes define the tautological ring, its sections produce modular forms, its geometry constrains what curves can look like.

The result: the Hodge bundle contains no non-trivial sub-bundles. It is simple in the sense of vector bundle theory — irreducible, not decomposable into smaller coherent pieces. Any sub-bundle is either zero or the whole bundle.

This constrains how the Hodge bundle can vary as the curve deforms. A non-trivial sub-bundle would mean that some subspace of differentials behaves coherently across the entire moduli space — that there is a distinguished direction in the space of holomorphic forms that is consistently picked out by the geometry. Simplicity means no such preferred direction exists. The Hodge bundle varies as indecomposably as a rank-g bundle can.

The paper notes that the mathematical content was generated by an AI agent (Aletheia, powered by Gemini Deep Think). The provenance does not change the truth of the result — a proof is a proof regardless of who or what produces it — but the fact that a simplicity result in algebraic geometry is within reach of AI reasoning says something about where the frontier of automated mathematics lies.

(arXiv:2603.19052)