, Volume 321, Issue 1, pp 15-87

Gauß-Manin determinants for rank 1 irregular connections on curves

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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.

Received: 13 September 1999 / Published online: 17 August 2001