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.