Residues and D-modules

Coleff–Herrera Currents Revisited

“…The concept and its importance were pointed out by J. E. Björk in [11,12]. The original construction of global sections for such sheaves is due to N. Coleff and M. Herrera in [14].…”

“…The original construction of global sections for such sheaves is due to N. Coleff and M. Herrera in [14]. In this section, we will recall the definition of the sheaf CH X ,V (·, E) (following the approach of J. E. Björk, M. Andersson, H. Samuelsson [11,12,2,13]), together with the local structure of its sections (which justifies their operational properties). Since our objective all along this presentation is to stick to the methods based on analytic continuation (which seems to be a natural way to introduce the objects algebraically, for example by using Bernstein-Sato functional equations as (2)), the approach we adopt here follows that developped by M. Andersson in [2].…”

“…Note that the morphism above is surjective (at the level of germs at x ∈ V ) as soon as O X ,x /I V,x is Cohen-Macaulay [12].…”

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Abstract In the present paper, we describe the recent approach to residue currents by M. Andersson, J. E. Björk, H. Samuelsson [2,12,13], focusing primarily on the methods inspired by analytic continuation (which were initiated in a quite primitive form in [8]). Coleff-Herrera currents (with or without poles) play indeed a crucial role in Lelong-Poincaré type factorization formulas for integration currents on reduced closed analytic sets.

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“…The reason is that the singularities under the integral in (12) are only holomorphic singularities. The same remains true if ϕ = hψ, where h ∈ I V ∩S∩U , since in this case any ξ ι,j , j = 1, ..., n − m, such that β ι,j = 0 divides π * h, which implies that all antiholomorphic singularities in the term under the integral in (11) are thus canceled.…”