International Congress of Mathematicians

Classes and Gauss-Manin Bundle 475If U is such that m(Ua, n ,u) is isomorphic as an abstract group to 7Ti(V an ,w),where V is an Artin neighborhood on a rational variety, then Ej n \u becomes arepresentation of 7Ti(V an ,w), and since then H° (V ,H?p R ~ 1 ) = 0 for n > 2 and V C Va good compactification, one obtains c n (Ej n \c( X )) = 0,n > 2 in this case. Such anexample is provided by a product of smooth affine curves of any genus. It has thefundamental group of a product of P 1 minus finitely many points.Question 2.6. In view of the previous discussion, we may ask what complexsmooth varieties X are dominated by h :Y —¥ X proper, with Y smooth, such thatY has an Artin neighborhood, the fundamental group of which is the fundamentalgroup of an Artin neighborhood on a rational variety, or more generally of a varietyfor which H° (smooth compactification, HQH) = 0 for n > 1. We have seen thatthis would imply vanishing modulo torsion of c n (Ej n \c( X )), n > 2, or equivalentlyCS n ((E,V)) £ HOiX^-HQln))).On the other hand, if X is projective smooth, Reznikov's theorem ([19])shows vanishing modulo torsion of the Chern-Cheeger-Simons classes in Jf 2 "^1 (X an ,C/Q(n)). It is a consequence of Simpson's nonabelian Hodge theory on smooth projectivevarieties. Our classes CS n ((E, Vj) live at the generic point of X. We don'thave a nonabelian mixed Hodge theory at disposal. Yet one may ask whether it isalways true that CS n ((E, Vj) £ H°'(X,H 2n - 1 (Q(n))) for n > 2, even if many Xdon't have the topological property explained above.3. The behavior of the algebraic Chern-Simonsclasses in families in the regular singular caseThe algebraic Chern-Simons invariants CS n ((E,Vj) have been studied in afamily in [5]. Given / : X —t S a proper smooth family, and (F, V) a flat connectionon X, the Gauß-Manin bundlesF7*(Ûx/ S ®F,V x/s )carry the Gauß-Manin connection GM l (V). We give a formula for the invariantsCS„((GM*(V)-rank (V) • GAF(d))) on S, as a function of CS n ((E,Vj) and ofcharacteristic classes of /. Here (0,d) is the trivial connection.More generally, we may assume that / is smooth away from a normal crossingsdivisor T c S such that Y = / _1 (F) c X is a normal crossings divisor with theproperty that 0^., s (logF) is locally free. Then (E,V) has logarithmic poles alongY U Z where Y U Z C X is a normal crossings divisor, still with the property that0^. , s (log(F + Zj) is locally free. That is Z is the horizontal divisor of singularitiesof V. The formula involves the top Chern class Cd(ii x , s (log(Y + Zj) £ M d (X, K,d —^®iXz Ld ), rigidified by the residue maps ii x , s (log(Y + Zj) —t Oz t , as defined by T.Saito in [20]. One of its main features is that CS n ((GM*(V) — rank (V) • GM % (d)))vanishes if CS n ((E,Vj) vanishes. It is