Macpherson’s and Fulton’s Chern Classes of Hypersurfaces

In this note we compare two notions of Chern class of an algebraic scheme X (over C) specializing to the Chern class of the tangent bundle c(TX) ∩ [X] when X is nonsingular. The first of such notions is MacPherson’s Chern class, defined by means of Mather-Chern classes and local Euler obstructions [5]. MacPherson’s Chern class is functorial with respect to a push-forward defined via topological Euler characteristics of fibers; in particular, mapping to a point shows that the degree of the zero-dimensional component of MacPherson’s Chern class of a complete variety X equals the Euler characteristic χ(X) of X. We denote MacPherson’s Chern class of X by cMP (X). The second notion is Fulton’s intrinsic class of schemes X ′ that can be embedded in a nonsingular variety M : Fulton shows ([3], Example 4.2.6) that the class cF (X ′) = c(TM) ∩ s(X ′,M) is independent of the choice of embedding of X ′. This class has the advantage of being defined over arbitrary fields and in a completely algebraic fashion, but does not satisfy at first sight nice functorial properties: cf. [3], p. 377. (MacPherson’s class can also be defined algebraically over any field of characteristic 0: this is done in [4].) To state our result we need to remind the reader that if W is a scheme supported on a Cartier divisor X of a nonsingular variety M , then the Segre class of W in M can be written in terms of the Segre class of X and the Segre class of the residual scheme J to X in W : for a precise statement of this fact, see [3], Proposition 9.2, or section 2 below. By modifying this expression, we can make sense of the “Segre class” in M of an object “X \J” in which J is intuitively speaking “removed” from X. Since this object has a Segre class, we can define its Fulton–Chern class as above. Here is our result: