Lê cycles and Milnor classes

Abstract

The purpose of this work is to establish a link between the theory of Chern classes for singular varieties and the geometry of the varieties in question. Namely, we show that if Z is a hypersurface in a compact complex manifold, defined by the complex analytic space of zeroes of a reduced non-zero holomorphic section of a very ample line bundle, then its Milnor classes, regarded as elements in the Chow group of Z, determine the global Lê cycles of Z; and viceversa: The Lê cycles determine the Milnor classes. Morally this implies, among other things, that the Milnor classes determine the topology of the local Milnor fibres at each point of Z, and the geometry of the local Milnor fibres determines the corresponding 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.,…”

On the Chern classes of singular complete intersections

“…Clearly, this is also an isomorphism. We know from [, Section 3] that we have a commutative diagram: The commutativity of this diagram amounts to saying: for any function which is constructible for the given Whitney stratification, where , with being the complex of sheaves such that . Substituting in equation , we get …”

“…Remark In , there is a concept of global Lê classes of a singular hypersurface in a smooth complex submanifold of , and a formula relating these with the Milnor classes of . The Lê classes extend the notion of the local Lê cycles introduced in .…”

“…Now consider the projectivized cotangent bundles P(T * M ) and P(T * (M (r) )); we denote by P((T * M ) ⊕r ), the bundle P(T * M ⊕ · · · ⊕ T * M ). Note that one has a fibre square diagram (see [11, p. 428]): (8) where π (r) is the natural proper map. Let i : P(T * M ) → P((T * M ) ⊕r ) be the morphism induced by the diagonal embedding T * M → T * M ⊕ · · · ⊕ T * M .…”