Lê cycles and Milnor classes

On characteristic classes of singular hypersurfaces and involutive symmetries of the Chow group

“…where d is the dimension of the singular locus of X and a kth subscript on a class denotes its component of dimension k. This fact seems to suggest the existence of some non-trivial involutive symmetry of A * X which exchanges M(X) and Λ(X), which we show in §4 is the case. In fact, both M(X) and Λ(X) may be recovered from the relative Segre class (see Definition 2.1) s(X s , M ) of the singular scheme X s of X (i.e., the subscheme of X whose ideal sheaf is locally generated by the partial derivatives of a local defining equation for X), and we show that there exists a countable infinity of such involutive symetries of A * X which exchange M(X) and classes closely related to s(X s , M ), for which the result of [8] is but one of them.…”

“…Both the Fulton class and CSM class are elements of the Chow group A * X which are generalizations of Chern classes to the realm of singular varieties in the sense that the classes both agree with the total homology Chern class in the case that X is smooth 1 . Another characteristic class suppoerted on the singular locus of a hypersurface X is the Lê-class of X, denoted Λ(X) ∈ A * X, which was first defined in [8] and named as such as they are closely related to the so-called Lê-cycles of X, which were initially defined and studied independent of Milnor classes [12]. The attractive result of [8] is that both M(X) and Λ(X) determine each other in a completely symmetric way, i.e., (1.1)…”

“…Another characteristic class suppoerted on the singular locus of a hypersurface X is the Lê-class of X, denoted Λ(X) ∈ A * X, which was first defined in [8] and named as such as they are closely related to the so-called Lê-cycles of X, which were initially defined and studied independent of Milnor classes [12]. The attractive result of [8] is that both M(X) and Λ(X) determine each other in a completely symmetric way, i.e., (1.1)…”

“…The proofs in [8] of formulas 1.1 and 1.2 are topological in nature, span many pages and involve heavy machinery such as derived categories and Whitney stratifications, obfuscating any conceptual insight as to why such formulas should hold. However, in an unpublished note [1], Aluffi shows that the proofs may be reduced to merely a few lines using his 'intersection-theoretic calculus' (which is purely algebraic).…”

“…As such, the theme of this note is that the symmetric formulas 1.1 and 1.2 are but a special case of a more general phenomenon of characteristic classes of hypersurfaces, which are induced by involutive symmetries of their Chow groups. [8], and for our discussions on Milnor classes. We also thank Paolo Aluffi for sharing with us the unpublished note [1], for which this article is little more than a mere rewriting of.…”