Connections and restrictions to curves

TL;DR

The paper constructs a vector bundle on a complex surface that restricts to algebraically connected curves but does not admit an algebraic connection itself, highlighting a nuanced distinction in algebraic geometry.

Contribution

It provides a counterexample demonstrating that restrictions of a vector bundle can admit algebraic connections even when the bundle itself does not.

Findings

Constructed a specific vector bundle with the described properties

Showed the restriction to any smooth curve admits an algebraic connection

Proved the bundle itself does not admit an algebraic connection

Abstract

We construct a vector bundle $E$ on a smooth complex projective surface $X$ with the property that the restriction of $E$ to any smooth closed curve in $X$ admits an algebraic connection while $E$ does not admit any algebraic connection.