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 Gaus-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.