Original article

Mathematische Annalen

, Volume 321, Issue 1, pp 15-87

Gauß-Manin determinants for rank 1 irregular connections on curves

  • Spencer BlochAffiliated withDepartment of Mathematics, University of Chicago, Chicago, IL 60637, USA (e-mail: bloch@math.uchicago.edu)
  • , Hélène EsnaultAffiliated withMathematik, Universität Essen, FB6, Mathematik, 45117 Essen, Germany (e-mail: esnault@uni-essen.de)

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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.

Mathematics Subject Classification (2000): 14C40, 19E20, 14C99