International Congress of Mathematicians

ICM 2002 • Vol. II • 471-181Characteristic Classes of Flat Bundlesand Determinant of theGauss-Manin ConnectionHélène Esnault*2000 Mathematics Subject Classification: 14C22, 14C25, 14C40,14C35, 14C99.1. IntroductionThe purpose of this note is to give a survey on recent progress on characteristicclasses of flat bundles, and how they behave in a family.2. Characteristic classesLet X be a smooth algebraic variety over a field k. In [13] and [15], we definedthe ringAD(X)= ® n AD n (X)= ® n W(X,K^d^ü n x/k A.-.^Q^1)(2.1)of algebraic differential characters. Here the Zariski sheaf K.ff is the kernel of theresidue map from Milnor F'-theory at the generic point of X to Milnor F'-theory atcodimension 1 points. More precisely, K.^ satisfies a Gersten type resolution (see[16] and [18])/Cf 4(* fc( x),.^(feW)^e ie x(i)* I ,.^i(«(x))-^• • • ® œG xw iz,*K%_ a (K(x)) -•...-•® xexW i x ,,K^(K(x))).Here X^ means the free group on points in codimension a, while i x : x —ï X is theembedding. The map dlog({ai,... ,a n }) = dlogai A • • • Adloga„ from Kff(k(X))* Mathematik, Universität Essen, FB6, Mathematik, 45117 Essen, Germany. E-mail:esnault@uni-essen.de