Definable structures on flat bundles

TL;DR

This paper proves that the natural definable structures on flat vector bundles over algebraic varieties coincide under certain conditions, especially for bundles underlying variations of Hodge structures.

Contribution

It establishes the equivalence of flat and algebraic definable structures on flat bundles under a specific monodromy condition, clarifying their relationship.

Findings

The two definable structures coincide under the given condition.

The condition on local monodromy at infinity is satisfied for bundles from variations of Hodge structures.

The result applies broadly to flat bundles with underlying Hodge structures.

Abstract

A flat vector bundle on an algebraic variety supports two natural definable structures given by the flat and algebraic coordinates. In this note we show these two structures coincide, subject to a condition on the local monodromy at infinity which is satisfied for all flat bundles underlying variations of Hodge structures.