# Gauß-Manin determinants for rank 1 irregular connections on curves

Research paper by Spencer Bloch, Hélène Esnault

Indexed on: 01 Sep '01Published on: 01 Sep '01Published in: Mathematische Annalen

#### 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.