Genera of Algebraic Varieties and Counting of Lattice Points

This paper announces results on the behavior of some important algebraic and topological invariants — Euler characteristic, arithmetic genus, and their intersection homology analogues; the signature, etc. — and their associated characteristic classes, under morphisms of projective algebraic varieties. The formulas obtained relate global invariants to singularities of general complex algebraic (or analytic) maps. These results, new even for complex manifolds, are applied to obtain a version of Grothendieck-Riemann-Roch, a calculation of Todd classes of toric varieties, and an explicit formula for the number of integral points in a polytope in Euclidean space with integral vertices. Consider first the behavior of the classical Euler-Poincare characteristic e(X) = £(-l)'nwk#«(*) i under a (surjective) projective morphism /: X —► Y of projective (possibly singular) algebraic varieties. Such a morphism can be stratified with subvarieties as strata. In particular, there is a filtration of Y by closed subvarieties, underlying a Whitney stratification, <p C Y0 C ■ • • C Ys = Y of strictly increasing dimension, such that Y, Y¡-i is a union of smooth manifolds of the same dimension and such that the restriction of / to f~xiY¡ Y,_i) is a locally trivial map of Whitney stratified spaces. (In the results it will suffice to have dim Y¡ < dim Yi+{ and dim f~x{x) constant over "strata" Y¡ y¿_,.) We recall the definition of the normal cone CzW of an irreducible subvariety Z of a variety W : CZW = Spec (0Jr"/^"+1) , Received by the editors October 16, 1992 and, in revised form, January 4, 1993. 1991 Mathematics Subject Classification. Primary 14M25, 14C40, 14E15, 14F32, 32S20, 52B20, 55R40, 19L10; Secondary 14C30, 14F45, 32S35, 32S60, 57R45, 11F20, 57R20. Both authors were partially supported by NSF grants. 62 ©1994 American Mathematical Society 0273-0979/94 $1.00+ $.25 per page GENERA OF ALGEBRAIC VARIETIES AND COUNTING OF LATTICE POINTS 63 J2" the sheaf of ideals defining Z. Let P{Cz@\) be its projective completion, and let Pz,w be the general fiber of the canonical projective morphism [Fl]