RIEMANN-ROCH THEOREMS FOR FLAT BUNDLES

SPENCER BLOCH AND HÉLÈNE ESNAULT

Abstract. Let f : U → Spec (K) be a smooth open curve over a field K ⊃ k, where k is an algebraically closed field of characteristic

0. Let ∇ : L → L ⊗ Ω_{U/k}^{1} be a (possibly irregular) absolutely integrable connection on a line bundle L. In the lecture, we explained how to use the polar part of the connection ∇ to determine the determinant of de Rham cohomology with its Gauß-Manin connection ( det Rf_{∗}(L ⊗ Ω_{U/K}^{1}), det ∇_{GM}) in Ω_{K}^{1}/d log K^{∗} modulo torsion. We showed that this expression, which is obtained as the trace of a certain cup-product, might be also understood as a sum of local contributions in the singularities of the connection, after chosing a rational section of the relative 1-forms. We showed how this leads naturally to consider a generalized twisted jacobian in the higher rank case, and stated a conjecture on the general shape of the determinant of the relative de Rham cohomology.

References

[1] Bloch, S.; Esnault, H.: A Riemann-Roch theorem for flat bundles, with values in the algebraic Chern-Simon theory, preprint, (to appear in the Annals of Mathematics), 1998, 47 pages.

[2] Bloch, S., Esnault, H.: Gauß-Manin determinants for rank 1 irregular connections on curves.

Dept. of Mathematics, University of Chicago, Chicago, IL 60637, USA

E-mail address: bloch@math.uchicago.edu

Mathematik, Universität Essen, FB6, Mathematik, 45117 Essen, Germany

E-mail address: esnault@uni-essen.de

Date: June 19, 1999.