Gauß-Manin determinants for rank 1 irregular connections on curves

Abstract.

Let $f:U \to{\rm Spec}(K)$ be a smooth open curve over a field $K\supset k$, where k is an algebraically closed field of characteristic 0. Let $\nabla : L \to L\otimes \Omega^1_{U/k}$ be a (possibly irregular) absolutely integrable connection on a line bundle L. A formula is given for the determinant of de Rham cohomology with its Gauß-Manin connection $\Big(\det Rf_*(L\otimes\Omega^1_{U/K}), \det\nabla_{GM}\Big)$. The formula is expressed as a norm from the curve of a cocycle with values in a complex defining algebraic differential characters [7], and this cocycle is shown to exist for connections of arbitrary rank.