Multiplicity of Functions On Singular Varieties

Let V be a local complete intersection in a complex manifold W. For a function g on W, we set f = g|V and f′ = g|V′, where V′ denotes the non-singular part of V. For each compact connected component S of the union of the singular set of V and the critical set of f′, we define the virtual multiplicity [Formula: see text] of f at S as the residue of the localization by df′ of the Chern class of the virtual cotangent bundle of V. The multiplicity m(f, S) of f at S is then defined by [Formula: see text], where μ(V, S) is the (generalized) Milnor number of [2]. If S = {p} is an isolated point and if g is holomorphic, we give an explicit expression of [Formula: see text] as a Grothendieck residue on V. In the global situation, where we have a holomorphic map of V onto a Riemann surface, we prove a singular version of a formule of B. Iversen [13].

Smooth Formal Embeddings and the Residue Complex

Let π:X → S be a finite type morphism of noetherian schemes. A smooth formal embedding of X (over S) is a bijective closed immersion X ⊂ 𝖃 , where 𝖃 is a noetherian formal scheme, formally smooth over S. An example of such an embedding is the formal completion 𝖃 = Y/X where X ⊂ Y is an algebraic embedding. Smooth formal embeddings can be used to calculate algebraic De Rham(co)homology.Our main application is an explicit construction of the Grothendieck residue complex when S is a regular scheme. By definition the residue complex is the Cousin complex of π!OS, as in [RD]. We start with I-C. Huang's theory of pseudofunctors on modules with 0-dimensional support, which provides a graded sheaf .We then use smooth formal embeddings to obtain the coboundary operator . We exhibit a canonical isomorphism between the complex (K·x/s, δ ) and the residue complex of [RD]. When π is equidimensional of dimension n and generically smooth we show that H-nK·x/s is canonically isomorphic to to the sheaf of regular differentials of Kunz-Waldi [KW].Another issue we discuss is Grothendieck Duality on a noetherian formal scheme 1d583 . Our results on duality are used in the construction of K·x/s.