International Congress of Mathematicians

ICM 2002 • Vol. II • 57^66Algebraic CobordismM. Levine*AbstractTogether with F. Morel, we have constructed in [6, 7, 8] a theory of algebraiccobordism,., an algebro-geometric version of the topological theory ofcomplex cobordism. In this paper, we give a survey of the construction andmain results of this theory; in the final section, we propose a candidate for atheory of higher algebraic cobordism, which hopefully agrees with the cohomologytheory represented by the P 1 -spectrum MGL in the Morel-Voevodskystable homotopy category.2000 Mathematics Subject Classification: 19E15, 14C99, 14C25.Keywords and Phrases: Cobordism, Chow ring , _R"-theory.1. Oriented cohomology theoriesFix a field k and let Seh/, denote the category of separated finite-type k-schemes. We let Sm^ be the full subcategory of smooth quasi-projective fc-schemes.We have described in [7] the notion of an oriented cohomology theory on Smj.Roughly speaking, such a theory A* consists of a contravariant functor from Sm^to graded rings (commutative), which is also covariantly functorial for projectiveequi-dimensional morphisms f :Y —¥ X (with a shift in the grading):/, :A*(Y)^A*- dimxY (X).The pull-back g* and push-forward /» satisfy a projection formula and commute intransverse cartesian squares. If L —¥ X is a line bundle with zero-section s : X —t L,we have the first Chern class of L, defined byCi(L):=s*(s*(l x ))£A 1 (X),where lx £ A°(X) is the unit. A* satisfies the projective bundle formula:* Department of Mathematics, Northeastern University, Boston, MA 02115, USA. E-mail:marc@neu.edu