Note on the Riemann-hurwitz Type Formula for Multiplicative Sequences

We give a formula of the Riemann-Hurwitz type for classes defined by multiplicative sequences as corollaries of the Chern number formula for ramified coverings. 1. 1.1. In the classical results on Riemann surfaces, we have the Riemann-Hurwitz formula which relates the topological Euler numbers of the covering space and base space. The difference of the Euler numbers can be expressed by local residues of ramification points. In the higher-dimensional case, we can see that the difference of the Euler numbers can be expressed by Euler numbers of degeneration loci ([Y]). In the simplest case that a covering map f : X −→ Y between n-dimensional compact complex manifolds with covering multiplicity μ has the non-singular branch locus B, the above formula is χ(Y )− μ · χ(X) = −χ(B). In such cases, we can also consider the genera of B determined by the multiplicative sequence Kj(c1, · · · , cj) of Chern classes. Thus we expect that the difference Kn(c1(X), · · · , cn(X))− μKn(c1(Y ), · · · , cn(Y )) can be explessed by datas of {Kj}1≤j≤n−1 of B. But there are no results for this type of formulas. In this note, we show that this type of formula follows immediately from the definition of the multiplicative sequence and the Chern number formula for ramified coverings ([Iz]). 1.2. First we recall the definition of multiplicative sequence. (For more detail, see [Hz1].) Let A be a commutative algebra with identity and consider A[p1, p2, · · · ], the ring of polynomials in the pi. Let {Kj} be a sequence of polynomials in the Received by the editors April 6, 2001. 2000 Mathematics Subject Classification. Primary 57R20; Secondary 32J25.